CBSE Class 10 Science Notes Chapter 7 Control and Coordination Pdf free download is part of Class 10 Science Notes for Quick Revision. Here we have given NCERT Class 10 Science Notes Chapter 7 Control and Coordination. According to new CBSE Exam Pattern, MCQ Questions for Class 10 Science pdf Carries 20 Marks.

CBSE Class 10 Science Notes Chapter 7 Control and Coordination

Control and Co-ordination in Animals: Nervous system and endocrine system.
In animals, the nervous system and hormonal system are responsible for control and co¬ordination.

Receptors: Receptors are the specialized tips of the nerve fibres that collect the information to be conducted by the nerves.
Receptors are in the sense organs of the animals.
These are classified as follows :

• Phono-receptors: These are present in inner ear.
  Functions: The main functions are hearing and balance of the body.
• Photo-receptors: These are present in the eye.
  Function: These are responsible for visual stimulus.
• Thermo-receptors: These are present in skin.
  Functions: These receptors are responsible for pain, touch and heat stimuli.
  These receptors are also known as thermoreceptors.
• Olfactory-receptors: These are present in nose.
  Functions: These receptors receive smell.
• Gustatory-receptors: These are present in the tongue.
  Functions: These helps in taste detection.

Nervous System: The nervous system is composed of specialized tissues, called nervous tissue. The nerve cell or neuron is the functional unit of the nervous system. It is the nervous system which is mainly responsible for control and coordination in complex animals.

Functions of the nervous system

• Nervous system receives information from the environment.
• To receive the information from the various body.
• To act according to through muscles and glands.

A neuron is the structural and functional unit of the nervous system.

Neuron: Neuron is a highly specialized cell which is responsible for the transmission of nerve impulses. The neuron consists of the following parts
(i) Cyton or cell body: The cell body or cyton is somewhat star-shaped, with many hair like structures protruding out of the margin. These hair-like structures are called dendrites. Dendrites receive the nerve impulses.
(ii) Axon: This is the tail of the neuron. It ends in several hair-like structures, called axon terminals. The axon terminals relay nerve impulses.
(iii) Myelin sheath: There is an insulator cover around the axon. This is called myelin sheath. The myelin sheath insulates the axon against nerve impulse from the surroundings.

Types of neuron

• Sensory neuron: These neurons receive signals from a sense organ.
• Motor neuron: These neurons send signals to a muscle or a gland.
• Association or relay neuron: These neurons relay the signals between sensory neuron and motor neuron.

Neuron Diagram Class 10

Synapse: The point contact between the terminal branches of axon of one neuron with the dendrite of another neuron is called synapse.

Neuromuscular Junction (NMJ): NMJ is the point where a muscle fibre comes in contact with a motor neuron carrying nerve impulse from the control nervous system.

Transmission of nerve impulse: Nerve impulses travel in the following manner from one neutron to the next :
Dendrites → cell body → axon → nerve endings at the tip of axon → synapse → dendrite of next neuron.
Chemical released from axon tip of one neuron, cross the synapse or neuromuscular junction to reach the next cell.

Human Nervous System: The nervous system in humans can be divided into three main parts
1. Central Nervous System: The central nervous system is composed of the brain and the spinal cord. The brain controls all the functions in the human body. The spinal cord works as the relay channel for signals between the brain and the peripheral nervous system.

2. Peripheral Nervous System: The peripheral nervous system is composed of the cranial nerves and spinal nerves. There are 12 pairs of cranial nerves. The cranial nerves come our of the brain and go to the organs in the head region. There are 31 pairs of spinal nerves. The spinal nerves come out of the spinal cord and go to the organs which are below the head region.

3. Autonomous Nervous System: The autonomous nervous system is composed of a chain of nerve ganglion which runs along the spinal cord. It controls all the involuntary actions in the human body. The autonomous nervous system can be divided into two parts :

• Sympathetic nervous system.
• Parasympathetic nervous system.

Sympathetic Nervous System: This part of the autonomous nervous system heightens the activity of an organ as per the need. For example, during running, there is an increased demand for oxygen by the body. This is fulfilled by an increased breathing rate and increased heart rate. The sympathetic nervous system works to increase the breathing rate the heart rate, in this case.

Parasympathetic Nervous System: This part of the autonomous nervous system slows the down the activity of an organ and thus has a calming effect. During sleep, the breathing rate slows down and so does the heart rate. This is facilitated by the parasympathetic nervous system. It can be said that the parasympathetic nervous system helps in the conservation of energy.

Human Brain: Human brain is a highly complex organ, which is mainly composed of nervous tissue. The tissues are highly folded to accommodate a large surface area in less space. The brain is covered by a three-layered system of membranes, called meninges. Cerebrospinal fluid is filled between the meninges. The CSF providers cushion the brain against mechanical shocks. Furthermore, protection. The human brain can be divided into three regions, viz. forebrain, midbrain and hindbrain.

Parts of Human Brain :

• Fore-brain: It is composed of the cerebrum.
• Mid-brain: It is composed of the hypothalamus.
• Hind-brain: It is composed of the cerebellum, pons, medulla, oblongata.

Some main structures of the human brain are explained below :
Cerebrum: The cerebrum is the largest part in the human brains. It is divided into two hemispheres called cerebral hemispheres.

Functions of cerebrum

• The cerebrum controls voluntary motor actions.
• It is the site of sensory perceptions, like tactile and auditory perceptions.
• It is the seat of learning and memory.

Hypothalamus: The hypothalamus lies at the base of the cerebrum. It controls sleep and wake cycle (circadian rhythm) of the body. It also controls the urges for eating and drinking.

Cerebellum: Cerebellum lies below the cerebrum and at the back of the whole structure. It coordinates the motor functions. When you are riding your bicycle, the perfect coordination between your pedalling and steering control is achieved by the cerebellum.

• It controls posture and balance.
• It controls the precision of voluntary action.

Medulla: Medulla forms the brain stem, along with the pons. It lies at the base of the brain and continues into the spinal cord. The medulla controls various involuntary functions, like hear beat respiration, etc.
It controls involuntary actions.
Example: Blood pressure, salivation, vomiting.

Pons: It relays impulses between the lower cerebellum and spinal cord, and higher parts of the brain like the cerebrum and midbrain, also regulates respiration.

Spinal cord: Spinal cord controls the reflex actions and conducts massages between different parts of the body and brain.

Reflex Action: Reflex action is a special case of involuntary movement involuntary organs. When a voluntary organ is in the vicinity of sudden danger, it is immediately pulled away from the danger to save itself. For example, when your hand touches a very hot electric iron, you move away your hand in a jerk. All of this happens in flash and your hand is saved from the imminent injury. This is an example of reflex action.

Reflex Arc: The path through which nerves signals, involved in a reflex action, travel is called the reflex arc. The following flow chart shows the flow of signal in a reflex arc.
Receptor → Sensory neuron → Relay neuron → Motor neuron → Effector (muscle)
The receptor is the organ which comes in the danger zone. The sensory neurons pick signals from the receptor and send them to the relay neuron. The relay neuron is present in the spinal cord. The spinal cord sends signals to the effector via the motor neuron. The effector comes in action, moves the receptor away from the danger.

The reflex arc passes at the level of the spinal cord and the signals involved in reflex action do not travel up to the brain. This is important because sending signals to the brain would involve more time.
Although every action is ultimately controlled by the brain, the reflex action is mainly controlled at the level of spinal cord.

Reflex Arc Diagram Class 10

Protection of brain and spinal cord
Brain is protected by a fluid filled balloon which acts as shocks absorber and enclosed in cranium (Brain box)
Spinal chord is enclosed in vertebral column.

Muscular Movements and Nervous Control: Muscle tissues have special filaments, called actin and myosin. When a muscle receives a nerve signal, a series of events is triggered in the muscle. Calcium ions enter the muscle cells. It result in actin and myosin filaments sliding towards each other and that is how a muscle contracts. Contraction in a muscle brings movement in the related organ.

Endocrine System: The endocrine system is composed of several endocrine glands. A ductless gland is called endocrine gland. Endocrine gland secretes its product directly into the bloodstream. Hormones are produced in the endocrine glands. Hormone is mainly composed of protein. Hormones assist the nervous system in control and co-ordination. Nervous do not react to every nook and corner of the body and hence hormones are needed to affect control and coordination in those parts. Moreover, unlike nervous control, hormonal control is somewhat slower.

Hormones: These are the chemical messengers secreted in very small amounts by specialised tissues called ductless glands. They act on target tissues/organs usually away from their source. Endocrine System helps in control and coordination through chemical compounds called hormones.

Endocrine Gland: A ductless gland that secretes hormones directly into the bloodstream.

Iodised salt is necessary because: Iodine mineral is essential part of thyronine hormone so it is important that we must consume iodised salt as in turn it is essential for thyroid gland as it controls carbohydrate, proteins and fat metabolism for best balance of growth deficiency of iodine might cause disease called goitre.

Diabetes: Cause : It is due to deficiency of insulin hormone secreted by pancreas that is responsible to lower/control the blood sugar levels.

Treatment : Patients have to internally administer injections of insulin hormone which helps in regulating blood-sugar level.

In case of flight or fight reaction to an emergency situation, Adrenal glands → release adrenaline into blood → which acts on heart and other tissues → causes faster heart beat → more oxygen to muscles → reduced blood supply to digestive system and skin → diversion of blood to skeletal muscles → increase in breathing rate.

Feedback mechanism: A type of self-regulating mechanism in which the level of one substance in body influences the level of another.

Control and Co-ordination in Plants: Movements in plants and plant harmones.
Co-ordination in Plants: Unlike animals, plants do not have a nervous system. Plants use chemical means for control and co-ordination. Many plant hormones are responsible for various kinds of movements in plants. Movements in plants can be divided into two main types :

1. Tropic movement
2. Nastic movement

1. Tropic Movement: The movements which are in a particular direction in relation to the stimulus are called tropic movements. Tropic movements happen as a result of growth of a plant part in a particular direction. There are four types of tropic movements.
(i) Geotropic movement: The growth in a plant part in response to the gravity is called geotropic movement. Roots usually show positive geotropic movement, i.e. they grow in the direction of the gravity. Stems usually show negative geotropic movement.

(ii) Phototropic Movement: The growth in a plant part in response to light is called phototropic movement. Stems usually show positive phototropic movement, while roots usually show negative phototropic movement. If a plant is kept in a container in which no sunlight reaches and a hole in the container allows some sunlight; the stem finally grows in the direction of the sunlight. This happens because of a higher rate of cell division in the part of stem which is away from the sunlight. As a result, the stem bends towards the light. The heightened rate of cell division is attained by increased secretion of the plant hormone auxin in the which is away from sunlight.

(iii) Hydrotropic Movement: When roots grow in the soil, they usually grow towards the nearest source of water. This shows a positive hydrotropic movement.

(iv) Thigmotropism Movement: The growth in a plant part in response to touch is called thigmotropism movement. Such movements are seen in tendrils of climbers. The tendril grows in a way so as it can coil around a support. The differential rate of cell division in different parts of the tendril happens due to action of auxin.

2. Nastic Movement: The movement which do not depend on the direction from the stimulus acts are called nastic movement. For example, when someone touches the leaves of mimosa, the leaves droop. The drooping is independent of the direction from which the leaves are touched. Such movements usually happen because of changing water balance in the cells. When leaves of mimosa are touched, the cells in the leaves lose- water and become flaccid, resulting in drooping of leaves.

Plant hormones: Plant hormones are chemical which help to co-ordinate growth, development and responses to the environment.
Type of plant hormones: Main plant hormones are

• Auxin: (Synthesized at shoot tip).
  Function: Helps in growth.
  Phototropism: more growth of cells towards the light.
• Gibberellin: Helps in the growth of the stem.
• Cytokinins: Promotes cell division.
• Abscisic acid: Inhibits growth, cause wilting of leaves. (Stress hormone)

Control and Coordination in Plants

• Stimuli: The change in the environment to which an organism responds.
• Co-ordination: Working together of various organs of an organism in a systematic manner to produce a proper response.
• Phyto-hormones: These are plant hormones.
• Auxin: It is a plant hormone which promotes cell enlargement and growth in plants.
• Gibberellins: A plant hormone which promotes cell differentiation and breaking dormancy of seeds and buds.
• Cytokinin: A plant hormone which promotes cell division and the opening of stomata.
• Abscisic Acid: It helps in inhibiting the growth of the plant and promotes wilting and falling of leaves and food.
• Tropism: A growth movement of a plant which determines direction with the stimulus.
• Nastism: A growth movement of a plant which does not determine direction with a stimulus.
• Phototropism: Movement of plants towards a light.
• Geotropism: Movement of plants towards the gravity of earth.
• Chemotropism: Movement of plants towards chemicals.
• Hydrotropism: Movement of plants towards the water.
• Thigmotropism: Movement of plants towards a response to the touch of an object.

Control and Coordination in Animals

• Stimuli: The change in the environment to which the organism responds.
• Co-ordination: Working together of various organs of an organism in a systematic manner to produce a proper response.
• Neuron: Functional unit of the nervous system.
• Synapse: A microscopic gap between a pair of adjacent neurons.
• Receptor: A cell in a sense organ which is sensitive to stimuli.
• Motor nerves: It carries the message from the brain to body parts for action.
• Sensory nerves: It carries the message from body to brain.
• Olfactory receptor: It detects smell by the nose.
• Gustatory receptor: It detects taste by a tongue.
• Thermoreceptor: It detects heat and cold by a skin.
• Photoreceptor: It detects light by eye.
• Reflex action: Sudden movement or response to the stimulus which occurs in a very short duration of time and does not involve any will or thinking of the brain.
• Brain: An organ present in the skull which controls and regulates the activity of the whole body and is known as president of the body.
• Cerebrum: Main thinking part of brain present in the forebrain area which controls all voluntary actions.
• Cerebellum: It is present in the hindbrain area and helps in maintaining posture and balance of the body.
• Medulla: It is present in the hindbrain area and helps in controlling voluntary actions of the brain.
• Spinal cord: It is a cylindrical structure of nerve fibres enclosed in the vertebral column which helps in the conduction of nerve impulses to and from the brain.

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NCERT Solutions for Class 8 Science Chapter 13 Sound

Topics and Sub Topics in Class 8 Science Chapter 13 Sound:

Sound Class 8 Science NCERT Textbook Questions

Question 1.
Choose the correct answer.
Sound can travel through
(a) gases only
(b) solids only
(c) liquids only
(d) solids, liquids, and gases
Answer:
(d) solids, liquids, and gases.

Question 2.
Voice of which of the following is likely to have a minimum frequency?
(a) Baby girl
(b) Baby boy
(c) A man
(d) A woman
Answer:
(c) A man

Question 3.
In the following statements, tick ‘T’ against those which are true and ‘F’ against those which are false.

1. Sound cannot travel in a vacuum.
2. The number of oscillations per second of a vibrating object is called its time period.
3. If the amplitude of vibration is large, the sound is feeble.
4. For human ears, the audible range is 20 Hz to 20,000 Hz.
5. The lower the frequency of vibration, the higher is the pitch.
6. Unwanted or unpleasant sound is termed as music.
7. Noise pollution may cause partial hearing impairment.

Answer:

1. True
2. False
3. False
4. True
5. False
6. False
7. True

Human Ear Diagram Class 8

Question 4.
Fill in the blanks with suitable words.

1. Time taken by an object to complete one oscillation is called _______
2. Loudness is determined by the ________ of vibration.
3. The unit of frequency is ________
4. Unwanted sound is called _______
5. The shrillness of a sound is determined by the ______ of vibration.

Answer:

1. Time period
2. Amplitude
3. Hertz (Hz)
4. Noise
5. Frequency

Question 5.
A pendulum oscillates 40 times in 4 seconds. Find its time period and frequency.
Answer:
No. of oscillation = 40
Total time is taken = 4 seconds

Question 6.
The sound from a mosquito is produced when it vibrates its wings at an average rate of 500 vibrations per second. What is the time period of the vibration?
Answer:
Number of vibrations per second = 500

Question 7.
Identify the part which vibrates to produce sound in the following instruments.

1. Dholak
2. Sitar
3. Flute

Answer:

1. Stretched membrane
2. String of sitar
3. Air column

Question 8.
What is the difference between noise and music? Can music become noise sometimes?
Answer:
The type of sound which are unpleasant to listen is known as noise whereas music is a pleasant sound, which produces a sensation.
Yes, music can become noise when it’s too loud.

Question 9.
List the sources of noise pollution in your surroundings.
Answer:
Following are the major sources of noise pollution:

• Sound of vehicles
• Sound of kitchen appliances
• Sound of bursting crackers
• Sound of loudspeakers, TV, transistors

Question 10.
Explain in what way noise pollution is harmful to humans.
Answer:
Noise pollution causes:
(a) Lack of sleep
(b) Anxiety
(c) Hypertension
and these are harmful to health.

Question 11.
Your parents are going to buy a house. They have been offered one on the roadside and another three lanes away from the roadside. Which house would you suggest your parents should buy? Explain your answer.
Answer:
I would suggest my parents buy a house three lanes away from the roadside because house on the roadside would be much noisy in both days and night due to running vehicles. Whereas, a house three lanes away would be comparatively quieter as the intensity of noise decreases with the distance between the source and the listener.

Question 12.
Sketch larynx and explain its function in your own words.
Answer:
Larynx is also known as voice box. It is at the upper end of the windpipe. Two vocal cords are stretched across the voice box or larynx in such a way that it leaves a narrow slit between them for passage of air (Fig. 13.12). When lung force air through the slit, the vocal cords vibrate, producing sound. Muscles attached to the vocal cords can make the cords tight or loose.

When the vocal cords are tight and thin, the type or quality of voice is different from that when they are loose and thick.

Question 13.
Lightning and thunder take place in the sky at the same time and at the same distance from us. Lightning is seen earlier and thunder is heard later. Can you explain why?
Answer:
The speed of light is more than that of the speed of sound. Thus, due to more speed of light it reaches us before sound. So, lightning is seen earlier and thunder is heard later.

Sound Class 8 Science NCERT Intext Activities Solved

Activity 1 (NCERT Textbook, Page 158)
Take a metal plate (or a shallow pan). Hang it at a convenient place in such a way that it does not touch any wall. Now strike it with a stick (Fig. 13.1). Touch the plate or pan gently with your finger. Do you feel the vibrations? Again strike the plate with the stick and hold it tightly with your hands immediately after striking. Do you still hear the sound? Touch the plate after it stops producing sound. Can you feel vibrations now?

Solution:
When we touch the pan gently with our finger after striking we feel the vibration. When we hold the pan tightly after striking it, we do not hear the sound. When the pan stops producing sound it also stops vibrating. Thus, we can conclude that vibrating body produces sound.

Activity 2 (NCERT Textbook, Page 758)
Jake a rubber band. Put it around the, longer side of a pencil box (Fig. 13.2). Insert two pencils between the box and the stretched rubber. Now, pluck the rubber band somewhere in the middle. Do you hear any sound? Does the band vibrate?

Solution:
Yes, we hear the sound on plucking the rubber band. Also, we find that the band is vibrating. Thus, all vibrating bodies produce sound.

Activity 3 (NCERT Textbook, Page 758-759)
Take a metal dish. Pour water in it. Strike it at its edge with a spoon (Fig. 13.3). Do you hear a sound? Again strike the dish and then touch it. Can you feel the dish vibrating? Strike the dish again. Look at the surface of water. Do you see any waves there? Now hold the dish. What change do you observe on the surface of water? Can you explain the change? Is there a hint to connect sound with the vibrations of a body?

Solution:
On striking the metal dish we hear sound and on touching it we feel the dish vibrating. Striking the dish with water we see circular wave are produced. Thus vibrating object produces sound.

Activity 4 (NCERT Textbook, Page 159)
Take a hollow coconut shell and make a musical instrument ektara. You can also make it with the help of an earthen pot (Fig. 13.4). Play this instrument and identify its vibrating part.

Solution:
We observed that the vibrating part of the musical instrument ektara is stretched string.

Activity 5 (NCERT Textbook, Page 160)
Take 6-8 bowls or tumblers. Fill them with water upto different levels, increasing gradually from one end to the other. Now take a pencil and strike the bowls gently. Strike all of them in succession. You will hear pleasant sounds. This is your Jaltarang (Fig. 13.5).

Solution:
We can hear a pleasant sound. This is due to different levels of water in the bowls.
Thus, we find that shorter the length of the vibrating air column, higher is the pitch of the sound produced.

Activity 6 (NCERT Textbook, Page 161)
Take two rubber strips of the same size. Place these two pieces one above the other and stretch them tight. Now blow air through the gap between them [Fig. 13.6(a)]. As the air blows through the stretched rubber strips, a sound is produced. You can also take a piece of paper with a narrow slit and hold it between your fingers as shown in [Fig. 13.6(b)]. Now blow through the slit and listen to the sound.

Solution:
This activity shows that vocal cords also produce sound in a similar manner when they vibrate.

Activity 7 (NCERT Textbook, Page 161)
Take a metal or glass tumbler. Make sure that it is dry. Place a cell phone in it. Ask your friend to give a ring on this cell phone from another cell phone. Listen to the ring carefully.
Now, surround the rim of the tumbler with your hands (Fig. 13.7). Put your mouth on the opening between your hands. Indicate to your friend to give a ring again. Listen to the ring while sucking air from the tumbler. Does the sound become fainter as you suck air?
Remove the tumbler from your mouth. Does the sound become loud again?

Solution:
We observed that sound becomes fainter than earlier when we try to suck air. But when we remove tumbler from our mouth the sound again becomes loud. Thus, sound needs a medium to travel.

Activity 8 (NCERT Textbook, Page 162)
Take a bucket or a bathtub. Fill it with clean water.
Take a small bell in one hand. Shake this bell inside the water to produce sound. Make sure that the bell does not touch the body of the bucket or the tub. Place your ear gently on the water surface (Fig. 13.8). Can you hear the sound of the bell? Does it indicate that sound can travel through liquids?

Solution:
We can hear the sound of the bell which indicates that sound can travel through liquids.

Activity 9 (NCERT Textbook, Page 162)
Take a metre scale ora long metal rod and hold its one end to your ear. Ask your friend to gently scratch or tap at the other end of the scale (Fig. 13.9).
Can you hear the sound of the scratching? Ask your friends around you if they were able to hear the same sound?

Solution:
Yes, we find that we can hear the sound of the scratch. But, the people standing around us cannot hear the same sound or we can say that it is limping not audible to them.

Activity 10 (NCERT Textbook, Page 163)
Take a plastic or tin can. Cut its ends. Stretch a piece of rubber balloon across one end of the can and fasten it with a rubber band. Put four or five grains of dry cereal on the stretched rubber. Now ask your friend to speak”Hurrey, Hurrey”from the open end (Fig. 13.10). Observe what happens to the grain. Why do the grain jump up and down?

Solution:
The grain jump up and down because of the vibration caused underneath the stretched rubber. Thus when sound waves fall on the eardrum, it starts vibrating back and forth rapidly.

Activity 11 (NCERT Textbook, Page 164-165)
Take a metallic tumbler and a tablespoon. Strike the tablespoon gently at the brim of the tumbler. Hear the sound produced. Now bang the spoon on the tumbler and hear the sound produced again. Is the sound louder when the tumbler is struck hard?
Now suspend a small thermocol ball touching the rim of the tumbler (Fig. 13.11). Vibrate the tumbler by striking it. See how far the ball is displaced. The displacement of the ball is a measure of the amplitude of vibration of the tumbler.
Now, strike the tumbler gently and then with some force. Compare the amplitudes of vibrations of the tumbler in the two cases. In which case is the amplitude larger?

Solution:
The sound produced is louder when the tumbler is struck hard. The amplitude of vibration of the tumbler is larger when the glass is struck hard.Thus the loudness of sound depends upon the amplitude of vibration.

NCERT Solutions for Class 8 Science Chapter 13 – 1 Mark Questions and Answers

Question 1.
Choose the correct answer. Sound can travel through

• Gases only
• Solids only
• Liquids only
• Solids, liquids and gases.

Answer:
Solids, liquids and gases.

Question 2.
Voice of which of the following is likely to have minimum frequency ? [NCERT]

• Baby girl
• Baby boy
• A man
• A woman

Answer:
Aman

Question 3.
Identify the part which vibrates to produce sound in the following instruments. [NCERT]

• Dholak
• Sitar
• Flute

Answer:

• Dholak – stretched membrane
• Sitar – stretched string
• Flute – air column

Question 4.
In the following statements, tick ‘T’ against those which are true and ‘F’ against those which are false. [NCERT]

1. Sound cannot travel in vacuum. (T/F)
2. The number of oscillations per second of a vibrating object is called its time period. (T/F)
3. If the amplitude of vibration is large, sound is feeble. (T/F)
4. For human ears, the audible range is 20 Hz to 20,000 Hz. (T/F)
5. The lower the frequency of vibration, the higher is the pitch. (T/F)
6. Unwanted or unpleasant sound is termed as music. (T/F)
7. Noise pollution may cause partial hearing impairment. (T/F)

Answer:

1. T
2. F
3. F
4. T
5. F
6. F
7. T

Question 5.
Fill in the blanks with suitable words. [NCERT]

1. Time taken by an object to complete one oscillation is called …………
2. Loudness is determined by the ………….. of vibration.
3. The unit of frequency is ……………
4. Unwanted sound is called …………….
5. Shrillness of a sound is determined by the …………….. of vibration.

Answer:

1. time period
2. amplitude
3. Hertz
4. noise
5. frequency

Question 6.
Define vibration.
Answer:
Vibration is the to and fro or back and forth motion of an object.

Question 7.
Which part of the human body is responsible for producing sound ? [NCT 2011]
Answer:
In humans, the sound is produced by the voice box or larynx

Question 8.
What is the length of vocal cords in men ?
Answer:
The vocal cords in men are about 20 mm long.

Question 9.
Can sound travel in vacuum ?
Answer:
No, sound cannot travel in vacuum.

Question 10.
What is meant by oscillatory motion ?
Answer:
The to and fro motion of an object is known as oscillatory motion.

Question 11.
Define frequency.
Answer:
The number of oscillations per second is called the frequency of oscillation.

Question 12.
Define 1 hertz.
Answer:
A frequency of 1 hertz means one oscillation per second.

Question 13.
How are frequency of a sound and pitch related ?
Answer:
If the frequency of vibration is higher then the sound has a higher pitch.

Question 14.
Whose voice has a higher frequency – man or woman ?
Answer:
The voice of woman has higher frequency.

Question 15.
What is range of audible sound ?
Answer:
Sound of frequency 20 Hz to 20,000 Hz is the audible range.

Question 16.
Which animal can hear sounds of frequencies higher than 20,000 Hz ?
Answer:
Dogs can hear frequencies higher than 20,000 Hz.

Question 17.
What is meant by base loudness level ?
Answer:
The base loudness level is defined as that loudness of sound that the human ear can just perceive.

Question 18.
What is meant by noise pollution ?
Answer:
Presence of excessive or unwanted sound in the atmosphere is called noise pollutipn.

Question 19.
If the frequency of a sound is below 20 Hz, will it be audible to human beings ?
Answer:
No, it will not be audible.

Question 20.
In which state of matter does sound travel the

• slowest
• fastest ?

Answer:

• Air.
• Solids.

Question 21.
What happens to sound when it strikes a surface ?
Answer:
Sound gets reflected on striking a surface.

Question 22.
Why do we hear the sound of the hom of an approaching car before the car reaches us ?
Answer:
This happens because the speed of sound is much greater than the speed of the car.

NCERT Solutions for Class 8 Science Chapter 13 – 2 Mark Questions and Answers

Question 1.
The sound from a mosquito is produced when it vibrates its wings at an average rate of 500 vibrations per second. What is the time period of the vibration ? [NCERT]
Answer:
Time taken for 500 vibrations = 1 second
Time taken for 1 vibration = 1/500 second.
∴ Time period = 1/500 second.

Question 2.
How do plants help in reducing noise pollution ?
Answer:
Plants absorb sound and so help us in minimizing noise pollution.

Question 3.
How can we control the sources of noise pollution ?
Answer:
We can control noise pollution by designing and installing silencing devices in machines.

Question 4.
How can a hearing impaired child communicate ?
Answer:
A hearing impaired child can communicate effectively by using sign language.

Question 5.
If the amplitude increases 3 times, by how much will the loudness increase ?
Answer:
If the amplitude increases three times, the loudness will increase by a factor of 9.

Question 6.
The frequency of a given sound is 1.5 kHz. How many vibrations is it completing in one second ?
Answer:
Frequency = No.of vibrations/time
∴ No. of vibrations = Frequency x time = 1.5 x 1000 x 1 = 1500 vibrations

Question 7.
Which characteristic of a vibrating body determines

1. loudness
2. pitch of the sound produced by it ?

Answer:

1. Amplitude.
2. Frequency.

Question 8.
Why do we not hear echoes in our ordinary surroundings ?
Answer:
We do not hear echoes in our ordinary surroundings because the distance to hear echo should be more than 17 m.

Question 9.
We cannot hear the sound of the exploding meteors in the sky, though we can see them. Why ?
Answer:
Sound cannot travel through vacuum. In space there is vacuum. Light can travel through vacuum, so we can see the exploding meteor but cannot hear the explosion.

Question 10.
We can hear the supersonic jet planes flying. How ?
Answer:
The supersonic jet planes fly in the air. Since sound can travel through air, we can hear then flying.

Question 11.
What are vocal cords ? What is their function ? [NCT 2011]
Answer:
The larynx has a pair of membranes known as vocal cords stretched across their length. The vocal cords vibrate and produce sound.

Question 12.
When does a thud become music ?
Answer:
When thuds are repeated at’regular intervals, it becomes music, e.g., beating of drums or wood.

Question 13.
How do birds and insects produce sound ?
Answer:
Birds chirp with the help syrinx in their wind pipe. Insects produce sound by flapping their wings.

Question 14.
What is the function of eusfachian tube in human ear ?
Answer:
The vibrations of the spoken words reach our ears through eustachian tubes.

Question 15.

1. In our body which part of the ear receives sound waves ?
2. What may happen if the eardrum is absent from our ear ?

Answer:

1. Pinna helps in receiving sound waves.
2. If the eardrum is absent we would not be able to hear.

Question 16.
Can a hearing impaired child speak ? If not why ?
Answer:
A child having hearing impairment can not speak because if he is able to hear, he will leam to speak.

Question 17.
Give an example each of:

1. stringed instrument
2. percussion instrument
3. wind instrument
4. striking instrument

Answer:

1. Violin
2. Drums
3. Flute
4. Jal Tarang

Question 18.
Can sound travel through water ? How do whales communicate under water ?
Answer:
Yes, sound can travel through water. Since sound can travel through water, the whales can communicate with each other.

Question 19.
How is the pressure variation in a sound wave amplified in human ear ?
Answer:
The pressure variation in a sound wave causes vibrations in the eardrum. These vibrations are amplified several times by the three bones. (The hammer, anvil and stirrup).

Question 20.
How is that you can hear a friend talking in another room without seeing him ?
Answer:
Sound can travel in all directions and around comers. Light cannot travel around comers. Therefore, we can hear a friend talking in another room but cannot see him.

NCERT Solutions for Class 8 Science Chapter 13 – 3 Mark Questions and Answers

Question 1.
List sources of noise pollution in your surroundings. [NCERT]
Answer:
The major sources of noise pollution are sounds of vehicles, explosions, machines, loudspeakers.

Question 2.
What are the effects of noise pollution ?
Answer:
Due to noise pollution a person may suffer from lack of sleep, hypertension and anxiety. If a person is exposed to noise continuously he may get temporary or permanent deafness.

Question 3.
How can the noise pollution be controlled in residential area ?
Answer:

1. The noisy operations must be conducted away from residential areas.
2. Noise producing industries should be set away from such areas.
3. Use of automobile horns be minimized.
4. TV and music systems should be run at lower volumes.

Question 4.
A pendulum oscillates 40 times in 4 seconds. Find its time period and frequency. [NCT 2011, NCERT]
Answer:
40 vibrations in 4 seconds.
10 vibrations in 1 second
Frequency =10 vibrations/sec. or 10 Hz.
∴ Time period = 1/10 sec.

Question 5.
Your parents are going to buy a house. They have been offered one on the roadside and another three lanes away from the roadside. Which house would you suggest your parents should buy ? Explain your answer. [NCERT]
Answer:
I would advise my parents to buy the house three lanes away from the roadside because there the noise from automobiles would be much less.

Question 6.
What happens when we pluck the strings of a sitar ?
Answer:
When we pluck the strings of a sitar, the whole instrument vibrates and the sound is heard.

Question 7.
Why is the voice of men, women and children different ?
Answer:
The voice of men, women and children are different because the length of vocal cords are different. The length of vocal cords is longest in men and shortest in children.

Question 8.
How are we able to hear sound ?
Answer:
The eardrum is like a stretched rubber sheet. Sound vibrations make the eardrum vibrate. The eardrum sends vibrations to the inner ear. From there, the signal goes to the brain and we are able to hear.

Question 9.
What sources in the home may lead to noise ?
Answer:
Television and transistor at high volumes, some kitchen appliances, desert coolers, air conditioners all contribute to noise pollution.

Question 10.
What is the-difference between noise and music ? Can music become noise sometimes,?
Answer:
Unpleasant sounds are called noise.
Music is a sound which produces a pleasing sensation.
If the music is too loud, it becomes noise.

Question 11.
Draw a labelled diagram showing the structure of the human ear.
Answer:

Question 12.
What is the function of:

1. External ear.
2. Internal ear.

Answer:

1. The external ear helps us in receiving the sound waves and directing them to the eardrum.
2. The internal ear has cochlea which is filled with a fluid and having tiny hair cells inside. The hairy cells change the sound vibrations into nerve impulse which travels to the brain.
   The internal ear also helps us in balancing the body.

Question 13.
Give some suggestions by which we can keep our ears healthy.
Answer:

1. Never insert any pointed object into the ear. Tt can damage the eardrum and make a person deaf.
2. Never shout loudly in someone’s ear.
3. Never hit anyone hard on their ear.

Question 14.
Can you hear the sound on the moon ? Explain.
Answer:
We cannot hear the sound on the moon because sound requires a material medium to travel. On the moon there is no atmosphere and sound cannot travel in vacuum.

Question 15.
What are ultrasounds ? How are they useful to us?
Answer:
Sound having frequency higher than 20kHz is known as ultrasound, is used for

• detecting finer faults in metal sheets.
• scanning and imaging the body for stones, tumour and foetus.

NCERT Solutions for Class 8 Science Chapter 13 – 5 Mark Questions and Answers

Question 1.
Sketch larynx and explain its function in your own words. [NCERT]
Answer:
We produce sound in the larynx of our throats. The larynx has two vocal cords, which are folds of tissue with a slit like opening between them. When we speak, air passes through the opening and the vocal cords vibrate to produce sound.

Question 2.
Lightning and thunder take place in the sky at the same time and at the same distance from us. Lightning is seen earlier and thunder is heard later. Can you explain why ? [NCERT]
Answer:
The speed of light is more than the speed of sound. Therefore, even though thunder and lightning take place simultaneously, we see the lightning earlier.

Question 3.

1. What is SONAR?
2. What is the basic principle of its working ?
3. Explain its use.

Answer:

1. SONAR refers to Sound Navigation and Ranging.
2. The principle of reflection of sound is used in SONAR.
3. SONAR is used to measure the depth of the ocean. Ultrasonic waves are sent from the ship down into the sea. They are received back after reflection from the sea bed. The depth is calculated by noting the time period.

Question 4.
What is the use of ultrasound in medicine and industry ?
Answer:
Use of ultrasound in medicine :

• For scanning and imaging the body for stones, tumour and foetus.
• For relieving pain in muscles and joints.

Use of ultrasound in industry :

• For detecting finer faults in metal sheets.
• In dish washing machines where water and detergent are vibrate with ultrasonic vibrators.
• For homogenising milk in milk plants.

Question 5.
What is a sonogram ? Why is it preferred to X-rays ?
Answer:
Sonogram is image of the internal organs. Ultrasound can pass through the human body and are reflected back. The reflections are recorded by computer and images are generated on the screen.
Sonogram is not harmful and is therefore used for studying the foetus or stone or tumor in the organs. On the other hand, X-rays can be harmful if humans are exposed for longer time.

Question 6.

1. Name a property of sound which is
   (i) similar to the property of light.
   (ii) different from that of light.
2. Why do some people have hearing impairment ? How do they communicate with others ?

Answer:

1. (i) The property of sound similar to light is that in both reflection takes place.
   (ii) Sound can travel around comers but light cannot.
2. Some people suffer from hearing impairment because their ear drum is damaged or absent. This can be from birth or may occur later on. Such people communicate with “sign language”. They can also use “hearing aids”.

NCERT Solutions for Class 8 Science Chapter 13 MCQs

Question 1.
The maximum displacement of a vibrating body on either side of its mean position, is known as its
(a) Frequency
(b) Loudness
(c) Amplitude
(d) Pitch
Answer:
(c)

Question 2.
The frequency of a given sound is 1.5 kHz. The vibrating body is
(a) completing 1,500 vibrations in one second.
(b) taking 1,500 seconds to complete one vibration.
(c) taking 1.5 seconds to complete one vibration.
(d) completing 1.5 vibrations in one second
Answer:
(a)

Question 3.
A given sound is inaudible to the human ear, if
(a) its amplitude is too small.
(b) its frequency is below 20 Hz.
(c) its frequency is above 20 kHz.
(d) it has any of the three characteristics listed above.
Answer:
(d)

Question 4.
Sound can propagate
(a) through vacuum as well as gases
(b) only through gases and liquids
(c) only through gases and solids
(d) any of the three states of the matter.
Answer:
(d)

Question 5.
When lightning and thunder take place, they
(a) occur together and are also observed together.
(b) occur one after the other but are observed together.
(c) occur together but the thunder is observed a little after the lightning.
(d) occur together but the thunder is observed a little before the lightning
Answer:
(c)

Question 6.
Soundshaving frequency more than 20 Hz are called
(a) Infrasonic
(b) Supersonic
(c) Ultrasonic
(d) None of these
Answer:
(c)

Question 7.
Hertz is the unit of
(a) Amplitude
(b) Frequency
(c) Pitch
(d) Wavelength
Answer:
(b)

Question 8.
Loudness of sound is expressed in
(a) Hertz
(b) Decibel
(c) Seconds
(d) None of these
Answer:
(b)

More CBSE Class 8 Study Material

• NCERT Solutions for Class 8 Maths
• NCERT Solutions for Class 8 Science
• NCERT Solutions for Class 8 Social Science
• NCERT Solutions for Class 8 English
• NCERT Solutions for Class 8 English Honeydew
• NCERT Solutions for Class 8 English It So Happened
• NCERT Solutions for Class 8 Hindi
• NCERT Solutions for Class 8 Sanskrit
• NCERT Solutions

The post NCERT Solutions for Class 8 Science Chapter 13 Sound appeared first on Learn CBSE.

Click here to access the best NCERT Solutions Class 3 English Santoor Unit 4 The Sky Chapter 12 Chandrayaan textbook exercise questions and answers.

Chandrayaan NCERT Class 3rd English Santoor Chapter 12 Questions and Answers

Chandrayaan Class 3 Question Answer

Let us think (Page 122)

A. Answer the following.

Question 1.
Rani was very curious. How do you know?
Answer:
Rani was frequently asking many questions because she was curious.

Question 2.
What did Pratik say in excitement?
Answer:
Pratik said that India has landed on the Moon.

Question 3.
Who was Nandini Aunty?
Answer:
Nandini Aunty was a scientist and mother of Pratik’s friend Vivaan.

Question 4.
What do the words Chandra and Yaan mean?
Answer:
Chandra means the moon and yaan means vehicle.

Question 5.
Complete the sentence:
Chandrayaan-3 landed on the Moon on ____________
Answer:
Chandrayaan-3 landed on the Moon on 14 July 2023.

B. Think and say.
Imagine you are invited by the scientists to spend 10 days on the moon. Make a list of items that you would like to take with you.

Answer:

1. Binoculars
2. Camera
3. Water
4. Food
5. Medicine
6. Spacesuit
7. Joystick
8. Drawing book
9. Ball
10. Umbrella

Let us speak (Page 123)

Question 1.
Chandrayaan travelled the distance between the Earth and the Moon. You may ask elders at home about their experiences of travelling to places that are far from home.
Answer:
Discuss with the teacher and give answers.

Question 2.
Have you ever travelled to far-off places yourselves? If yes, share your experience with others.
Answer:
Discuss with the teacher and give answers.

Let us learn (Page 123)

Read the following lines.
There was a girl named Rani who lived in a village.
A hanky for an elephant.
Notice that we used ‘a’ before girl, village, and hanky, and ‘an’ before an elephant.
We use ‘a’ before singular nouns that begin with consonants.
‘An’ is used before singular nouns that begin with vowel sounds (a, e, i, o, u).

A. Fill in the blanks using ‘a’.

Question 1.
_________ dog is barking at the postman.

Answer:
A dog is barking at the postman.

Question 2.
My mother gives chapattis to _________ cow every day.

Answer:
My mother gives chapattis to a cow every day.

Question 3.
In summer, _________ sparrow builds its nest on the mango tree near our house.

Answer:
In summer, a sparrow builds its nest on the mango tree near our house.

B. Fill in the blanks using ‘an’.

Question 1.
I saw _________ eagle yesterday.

Answer:
I saw an eagle yesterday.

Question 2.
Maya bought _________ umbrella for the monsoon.

Answer:
Maya bought an umbrella for the monsoon.

Question 3.
_________ ice-cream man brings his cart in the evenings.

Answer:
An ice cream man brings his cart in the evenings.

C. Fill in the blanks using ‘a’ or ‘an’.

Question 1.
_________ brown hen laid _________ egg.

Answer:
A brown hen laid an egg.

Question 2.
_________ eagle sat on _________ building.

Answer:
An eagle sat on a building.

Question 3.
Mary ate _________ apricot, _________ chikoo, and _________ orange.

Answer:
Mary ate an apricot, a chikoo, and an orange.

Let us write (Page 125)

A. Fill in the blanks by forming words using ‘sk’, ‘sw’, ‘sp’, and ‘st’.

Question 1.
There are swings in the park.
Answer:
There are swings in the park.

Question 2.
The children throw _______ones in the river.
Answer:
The children throw stones in the river.

Question 3.
I need a _______oon and a bowl for soup.
Answer:
I need a spoon and a bowl of soup.

Question 4.
Have you heard the _______ory of the boy who cried wolf?
Answer:
Have you heard the story of the boy who cried wolf?

Question 5.
The blue _______irt matched with the pink shirt.
Answer:
The blue skirt matched with the pink shirt.

B. Describing the Moon.
Draw a picture of the Moon. With the help of your teacher, write a few lines about the Moon.

Answer:

The Moon is Earth’s closest buddy in space, and it goes around our planet. It looks like a big, round rock and has lots of craters. At night, it can be bright and change shape.

Fun with Words (Page 126)

A. Read these sentences aloud with your teacher.

There are seven days in a week.
Sunday is the first day of the week.
Monday is the second day of the week.
Tuesday is the third day of the week.
Wednesday is the fourth day of the week.
Thursday is the fifth day of the week.
Friday is the sixth day of the week.
Saturday is the seventh day of the week.
Answer:
Do yourself.

B. Look at the words given below. Read them aloud. Find their position in a week using the text given above. One has been done for you.

Answer:

• Sunday – first
• Monday – second
• Tuesday – third
• Wednesday – fourth
• Thursday – fifth
• Friday – sixth
• Saturday – seventh

Let us do (Page 127)

A. Divide yourselves into small groups. Each group shall collect newspaper or magazine clippings related to the Chandrayaan-3 mission. Paste it on a chart paper and display it in your class.

Answer:
Do it in the class with the help of a teacher.

Let us explore (Page 127)

A. Do you know a song or a story about the moon in your mother tongue? If not, ask your family members and share it in the class.

Answer:
Moon up high, so round and bright,
You’re my lantern in the night.
We play peek-a-boo with the clouds, softly whispering, never loud.

Self Assessment

Question 1.
I can recite the poems
in a group.
in pairs.
by repeating after my teacher.
by myself.
Answer:
I can recite the poems
in a group.
in pairs.
by repeating after my teacher.
by myself.

Question 2.
I can read the stories
by myself.
with my classmates.
with the help of my teacher.
Answer:
I can read the stories
by myself.
with my classmates.
with the help of my teacher.

Question 3.
I can write
by copying from the book or the board.
as my teacher speaks.
with the help of my classmates.
by myself.
Answer:
I can write
by copying from the book or the board.
as my teacher speaks.
with the help of my classmates.
by myself.

Question 4.
I can tell the stories that I have read or listened
to by myself (without any help).
with the help of the teacher.
by using the book.
Answer:
I can tell the stories that I have read or listened
to by myself (without any help).
with the help of the teacher.
by using the book.

Question 5.
I was able to do
all the exercise questions.
most of the exercise questions.
a few exercise questions.
Answer:
I was able to do
all the exercise questions.
most of the exercise questions.
a few exercise questions.

Chandrayaan Class 3 Summary in English and Hindi

Chandrayaan Class 3 Summary in English

Rani was a curious girl from a village, who constantly asked questions about the world. around her. One evening, she asked her mother how far the Moon was after hearing a lullaby. The next morning, her brother Pratik excitedly told her that India had landed on the Moon, which he had seen on TV. Curious about the Moon. Landing Rani and Pratik visited their friend Vivaan’s house, where his mother, Aunt Nandini, a scientist explained the Chandrayaan mission. She told them that a rocket was used to reach the Moon and that “Chandrayaan” means “Moon Vehicle”. Rani expressed her dream of going to the Moon one day, to which Aunt Nandini replied that it might be possible.

Chandrayaan Class 3 Summary in Hindi

एक गाँव की लड़की रानी जो बहुत जिज्ञासु थी यह जानने के लिए कि उसके आस-पास जो हो रहा है कैसे और क्यों हो रहा है। एक शाम उसकी माँ एक लोरी सुना रही थी तब रानी ने पूछा “माँ ये चंदा हमसे कितनी दूर है।” अगली सुबह उसके भाई प्रतीक ने उसको बताया जो तुम टेलीविज़न पर चंद्रयान की तस्वीरें देख रही थी। हमारा यान चंद्रमा पर पहुँच गया है। जिज्ञासा में रानी और उसका भाई प्रतीक अपने दोस्त विवान के घर पहुँचे। विवान की माताजी एक वैज्ञानिक हैं वो उनको बताती हैं कि चंद्रयान मिशन क्या होता है। वह उन बच्चों को बताती हैं कि एक रॉकेट जिसको चाँद पर जाने वाला यान कहा जाता है उसको चंद्रयान कहते हैं। रानी अपने विचार व्यक्त करती है। कि मुझे भी चंद्रमा पर जाना है। तभी नंदिनी कहती है एक दिन तुम जरूर जाओगी।

Chandrayaan Class 3 Word Meanings

Pages 109 – 111

• Curious – wanting to know – जिज्ञासु
• Excitement – a feeling full of joy – उत्तेजना
• Machines – a piece of equipment with moving parts – यंत्र
• Confused – not able to think clearly – परेशान / व्याकुल
• Scientist – a person who studies science – वैज्ञानिक
• Rocket – space vehicle – प्रक्षेपास्त्र

Chandrayaan Class 3 Hindi Translation

1. There was a girl named Rani who lived in a village. She was very curious and used to ask many questions to her family, teachers, and friends.
“Why is the sky blue?”
“Where does the Sun go at night?”

अनुवाद – एक गाँव में रानी नाम की एक लड़की रहती थी। वह बहुत जिज्ञासु थी और अपने परिवार, शिक्षकों और दोस्तों से कई प्रश्न पूछती थी।
“आकाश नीला क्यों है?”
“सूरज रात में कहाँ जाता है?”

2. One evening, Rani heard her mother singing, “Chanda mama door ke …..” Hearing this, Rani asked her mother, “ Amma, how far is the Moon?” Amma said, “ Rani, the Moon is very far in the sky”.
Rani kept on thinking about the moon and fell asleep.

अनुवाद – एक शाम, रानी ने अपनी माँ को गाते हुए सुना, “चंदा मामा दूर के…” यह सुनकर रानी ने अपनी माँ से पूछा, “अम्मा, चाँद कितनी दूर है?” अम्मा ने कहा, “रानी, चाँद बहुत दूर आसमान में है।”
रानी चाँद के बारे में सोचती है और सो जाती है।

3. The next morning Rani woke up hearing the loud voice of her elder brother, Pratik. He was pointing to the TV and shouting in excitement, “Look Rani, India is on the Moon!”

अनुवाद – अगली सुबह रानी अपने बड़े भाई प्रतीक की तेज़ आवाज़ सुनकर उठी। वह टी.वी. की ओर इशारा करके उत्साह से चिल्ला रहा था, “देखो रानी, भारत चंद्रमा पर है”।

4. Rani looked at the TV. She could see photos of some machines on the Moon. She heard a few people talking about the landing on the Moon. She was confused and asked Pratik, “How did Chandrayaan reach the Moon?”

अनुवाद – रानी ने टी.वी. की तरफ देखा। वह चंद्रमा पर कुछ मशीनों की तस्वीरें देख रही थी। उसने कुछ लोगों को चंद्रमा पर उतरने के बारे में बात करते हुए सुना। वह भ्रमित हो गई और उसने प्रतीक से पूछा, “चंद्रयान चंद्रमा तक कैसे पहुँचा?”

5. Pratik replied that his friend Vivaan’s mother, Aunt Nandini, is a scientist. “Shall we go and talk to her?” asked Pratik. Rani happily agreed. Both Rani and Pratik decided to visit Vivaan’s house.

अनुवाद – प्रतीक ने उत्तर दिया कि उसके दोस्त विवान की माँ, आंटी नंदिनी, एक वैज्ञानिक हैं। “क्या हम जाकर उससे बात करें?” प्रतीक ने पूछा। रानी खुशी से सहमत हो गई। रानी और प्रतीक दोनों ने विवान के घर जाने का फैसला किया।

6. Vivaan and his mother welcomed them. Rani could see many models and photographs of rockets in their house.
Nandini Aunty offered sharbat to the children and said, “Rani, what do you want to know?”
Rani replied, “People say India is on the Moon. Can you please tell us more about this?”

अनुवाद – विवान और उसकी माँ ने उनका स्वागत किया। रानी को उनके घर में रॉकेट के कई मॉडल और तस्वीरें देखने को मिली।
नंदिनी आंटी ने बच्चों को शरबत पिलाया और बोली, “रानी, तुम क्या जानना चाहती हो?”
रानी ने जवाब दिया, “लोग कहते हैं कि भारत चंद्रमा पर है। क्या आप कृपया हमें इसके बारे में कुछ और बता सकते हैं?”

7. Aunty said, “Yes, of course. Do you know which vehicle is used to go to the Moon?”
Pratik immediately said, “I know Aunty, it is a rocket. I had seen it on the TV”.
Rani pointed to the models and photos of rockets in the room.
Nandini said, “Very good. Yes, a rocket is used to reach the Moon.” Rani asked, “What does Chandrayaan mean?”

अनुवाद – आंटी ने कहा, “हाँ बिलकुल। क्या तुम जानते हो चंद्रमा पर जाने के लिए कौन-से वाहन का इस्तेमाल किया गया?”
प्रतीक ने तुरंत कहा, “मुझे पता है आंटी, यह एक रॉकेट है। मैंने टी.वी. पर देखा था।”
रानी ने कमरे में रॉकेट के मॉडलों और तस्वीरों की ओर इशारा किया।
नंदिनी ने कहा, “बहुत अच्छा। हाँ, चंद्रमा तक पहुँचने के लिए रॉकेट का उपयोग किया जाता है।” रानी ने पूछा, “चंद्रयान का क्या मतलब है?”

8. Nandini replied, “Chandra means Moon and Yaan means vehicle. On 14 July, 2023 India became the first country to land on the far side of the Moon.”
Rani jumped with excitement and said, “Can I go to the Moon, too?”
Nandini smiled, “Maybe, one day.”

अनुवाद – नंदिनी ने उत्तर दिया, “चंद्र का अर्थ है चंद्रमा और यान का अर्थ है वाहन। 14 जुलाई 2023 को भारत के सुदूर हिस्से पर उतरने वाला पहला देश बन गया।”
रानी उत्साह से उछल पड़ी और बोली, “क्या मैं भी चाँद पर जा सकती हूँ?”
नंदिनी मुस्कुराई, “शायद, एक दिन।”

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Class 6 Maths Chapter 1 Notes Patterns in Mathematics

Class 6 Maths Notes Chapter 1 – Class 6 Patterns in Mathematics Notes

→ Mathematics may be viewed as the search for patterns and for the explanations as to why those patterns exist.

→ Among the most basic patterns that occur in mathematics are number sequences.

→ Some important examples of number sequences include counting numbers, odd numbers, even numbers, square numbers, triangular numbers, cube numbers, Virahānka numbers, and powers of 2.

→ Sometimes number sequences can be related to each other in beautiful and remarkable ways. For example, adding up the sequence of odd
numbers starting with 1 gives square numbers.

→ Visualizing number sequences using pictures can help to understand sequences and the relationships between them.

→ Shape sequences are another basic type of pattern in mathematics.

→ Some important examples of shape sequences include regular polygons, complete graphs, stacked triangles and squares, and Koch snowflake iterations. Shape sequences also exhibit many interesting relationships with number sequences.

What is Mathematics? Class 6 Notes

Mathematics is, in large part, the search for patterns, and for the explanations as to why those patterns exist. Such patterns indeed exist all around us in nature, in our homes and schools, and in the motion of the sun, moon, and stars. They occur in everything that we do and see, from shopping and cooking to throwing a ball and playing games, to understanding weather patterns and using technology.

The search for patterns and their explanations can be a fun and creative endeavor. It is for this reason that mathematicians think of mathematics both as an art and as a science. This year, we hope that you will get a chance to see the creativity and artistry involved in discovering and understanding mathematical patterns. It is important to keep in mind that mathematics aims to not just find out what patterns exist, but also the explanations for why they exist. Such explanations can often then be used in applications well beyond the context in which they were discovered, which can then help to propel humanity forward.

For example, the understanding of patterns in the motion of stars, planets, and their satellites led humankind to develop the theory of gravitation, allowing us to launch our satellites and send rockets to the Moon and Mars; similarly, understanding patterns in genomes has helped in diagnosing and curing diseases—among thousands of other such examples.

Patterns in Numbers Class 6 Notes

Among the most basic patterns that occur in mathematics are patterns of numbers, particularly patterns of whole numbers: 0, 1, 2, 3, 4, …
The branch of Mathematics that studies patterns in whole numbers is called number theory.
Number sequences are the most basic and among the most fascinating types of patterns that mathematicians study.
The table shows some key number sequences that are studied in Mathematics.

Visualizing Number Sequences Class 6 Notes

Many number sequences can be visualized using pictures. Visualizing mathematical objects through pictures or diagrams can be a very fruitful way to understand mathematical patterns and concepts. Let us represent the first seven sequences in the Table using the following pictures.

Relations among Number Sequences Class 6 Notes

Sometimes, number sequences can be related to each other in surprising ways.
Example: What happens when we start adding up odd numbers?
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36
This is a really beautiful pattern!

Why does this happen? Do you think it will happen forever?
The answer is that the pattern does happen forever. But why? As mentioned earlier, the reason why the pattern happens is just as important and exciting as the pattern itself.

A picture can explain it
Visualizing with a picture can help explain the phenomenon. Recall that square numbers are made by counting the number of dots in a square grid.
How can we partition the dots in a square grid into odd numbers of dots: 1, 3, 5, 7,…?
Think about it for a moment before reading further!
Here is how it can be done:

This picture now makes it evident that 1 + 3 + 5 + 7 + 9 + 11 = 36.
Because such a picture can be made for a square of any size, this explains why adding up odd numbers gives square numbers.
By drawing a similar picture, can you say what is the sum of the first 10 odd numbers?
Now by imagining a similar picture, or by drawing it partially, as needed, can you say what is the sum of the first 100 odd numbers?

Another example of such a relation between sequences:
Adding up and down
Let us look at the following pattern:
1 = 1
1 + 2 + 1 = 4
1 + 2 + 3 + 2 + 1 = 9
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25
1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36
This seems to be giving yet another way of getting the square numbers—by adding the counting numbers up and then down!

Patterns in Shapes Class 6 Notes

Other important and basic patterns that occur in Mathematics are patterns of shapes. These shapes may be in one, two, or three dimensions (1D, 2D, or 3D) or even more dimensions. The branch of Mathematics that studies patterns in shapes is called geometry. Shape sequences are one important type of shape pattern that mathematicians study. The table shows a few key shape sequences that are studied in Mathematics.

Relation to Number Sequences Class 6 Notes

Often, shape sequences are related to number sequences in surprising ways. Such relationships can help study and understand both the shape sequence and the related number sequence.

Example: The number of sides in the shape sequence of Regular Polygons is given by the counting numbers starting at 3, i.e., 3, 4, 5, 6, 7, 8, 9, 10,… That is why these shapes are called, respectively, regular triangles, quadrilaterals (i.e., square), pentagons, hexagons, heptagons, octagons, nonagons, decagons, etc., respectively.

The word ‘regular’ refers to the fact that these shapes have equal-length sides and also equal ‘angles’ (i.e., the sides look the same and the corners also look the same). We will discuss angles in more depth in the next chapter. The other shape sequences in the Table also have beautiful relationships with number sequences.

Class 6 Maths Notes

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Students can use Maths Mela Class 3 Solutions Chapter 13 Time Goes On Question Answer to explore alternative problem-solving methods.

Maths Mela Class 3 NCERT Solutions Chapter 13 Time Goes On

Appa bought a new calendar,
Amma points a big blunder.
Missing was the month of July,
I wonder, where did it fly?
Don’t worry, don’t worry,
I have a solution to your query.
July and January looks the same,
Let us play the calendar game.

Let us Do
Make the calendar for the month of July 2024.

Answer:

• Observe the July month and complete the following.

Question 1.
Number of Sundays _______
Answer:
Number of Sundays 4.

Question 2.
Write the dates in this month that are Thursdays _______
Answer:
The dates in this month that are Thursdays – 4, 11, 18 and 25.

Question 3.
Three days after July 22 is July ______.
The day on this date is ______
Answer:
Three days after July 22 is July 25.
The day on this date is Thursday.

Question 4.
A school closes on July 7 for 15 days. The date on which the school will open is ________
Answer:
A school closes on July 7 for 15 days. The date on which the school will open is 22.

Let us Explore

Collect calendars for the last two years. Observe the following in both the calendars:

Question 1.
What is the same? What is changing in the years?
Tick (✓) the correct answer.
(a) Names of the months Same/ Changes
(b) Days in a month Same/ Changes
(c) Days in a week Same/ Changes
(d) Number of Sundays Same/ Changes
(e) Number of weeks in a year Same/ Changes
Answer:
(a) Same (✓)
(b) Same (✓)
(c) Same (✓)
(d) Changes (✓)
(e) Same (✓)

•

Circle the festivals that fall on the same date.
Answer:

Let us Do

Question 1.
Write the names of the 12 months in a year.
Answer:

1. January
2. February
3. March
4. April
5. May
6. June
7. July
8. August
9. September
10. October
11. November
12. December

Question 2.
Months that have less than 30 days ______
Answer:
Months that have less than 30 days is February.

Question 3.
Number of days in a year ______
Answer:
Number of days in a year 365.

Question 4.
Hetal says there are 53 weeks in a year. Is she right? Yes/No. If not, how many weeks did you find in a year?
Answer:
No. We find 52 weeks in a year.

Age Fun

Talk to your mother and find the following. 
Hetal is twice as old as her brother. She is also 10 years older than her brother. Guess the age of Hetal and her brother.
Answer:
Hetal is 20 years old and her brother is 10 years old.

Let us Do

Look at the birth certificate of Bincy and answer the following question.

Question 1.
2/5/2015 shows that Bincy was born on 2 (April/May/ June/July) in the year 2015.
Answer:
2/5/2015 shows that Bincy was born on 2 (April/May/ June/July) in the year 2015.

Question 2.
How old will Bincy be on 2 May 2025? ______
Answer:
Bincy will be 10 years old on 2 May 2025.

Question 3.
How old will she be in 2030? ______
Answer:
She will be 15 years old in 2030.

Question 4.
Eighth Birthday of Bincy was on ____.
Answer:
Eighth Birthday of Bincy was on 02/05/2023.

Question 5.
Bincy was ____ months old on 2 August 2015.
Answer:
Bincy was 3 months old on 2 August 2015.

Question 6.
After how many days of her birth was the certificate issued? ____
Answer:
After 16 days of her birth the certificate was issued.

Let us Do

Question 1.
Make your own birth certificate.

Answer:
Do yourself.

Question 2.
Complete the following by writing the dates in the boxes given below:

Answer:
Do yourself.

Let us Play

Get a working analog clock or watch. Observe the face of the clock and movements of the hands. Discuss what you observe.

Let us Do

Question 1.
Hetal started her breakfast at 7 o’clock in the morning. She finished her breakfast at 07:15 in the morning.
(a) 07:00

(b) 07:15

She took _____ minutes to eat her breakfast. How do you know?
Answer:

She took 15 minutes to eat her breakfast. We know that by seeing the minute and hour hand of the clock.

Question 2.
Draw the hour hand and minute hand on the clock to show the following:
(a) 8 : 15 or quarter past 8

Answer:
(a) 8 : 15 or quarter past 8

(b) 8:30 or half past 8

Answer:
8:30 or half past 8

Let us Do

A Day in the life of Hetal.

• Match the activity with the statement shown in the picture.
Write the time and draw the minute hand and the hour hand wherever it is required.

Answer:

Visit to Nani’s House (Grandma’s House)

Let us Think

Fill the table by writing events or activities from your daily life that can take the following durations of time. One is written as an example in each column.

Answer:

Question 3.
Write the number of minutes taken for the following activities.

Answer:

Question 4.
Write down what you can do in the time frame given below.

Answer:

Question 5.
Write the number of minutes passed by looking at the movement of the minute and hour hands.

Answer:

Let us Explore

You may have seen digital watches or clocks at various places. Where have you seen them?

Question 1.
What is the difference between the above two clocks?
Answer:
Clock 1 is digital clock and the second clock 2 is analog clock.

Question 2.
Which clock do you have on your school wall?
Answer:
I have Analog clock on my school wall.

Duration of time is also measured using sand clocks. Make your own sand clock (timer), in the presence of an adult.
Material required:
(i) Two waste transparent or glass bottles of small size with caps.
(ii) Strong glue to join bottle caps.

Process:
Join the tops of the bottle caps with each other using a strong glue.
Make a small hole with a fine needle at the centre of the joined caps.
Fill one of the bottles half way with fine sand and close the bottle with the joined caps.
Attach the second bottle on the other side of the joined caps.

Find out how much time it takes to shift the sand from one bottle to the second one by looking at the clock. Your sand clock is ready for use. You may use it to time while playing different games.

NCERT Solutions for Class 3 Mathematics Chapter 13 Smart Charts!

Flowers of Different Colours
1.Have you ever been to a park?
Ans. Yes, I have been to a park.

2.What coloured flowers did you see?
Ans. I saw flowers of different colours like yellow, blue, red, orange and purple etc.

3.Were most of the flowers yellow in colour?
Ans. Yes.

What do We See on the Road?
1.Look at the traffic scene in the picture and fill in the table.
Ans.

Answer the following questions:
(а)In the picture which way of travel do you see the most?
Ans. Bus.
(b)Which way of travel (vehicle) do you see the least?
Ans. Bullock-cart.
(c)The number of people walking is more than the number of ———–
Ans. Scooters.
(d)The number of buses is less than the number of————
Ans. People walking.

How Many Times do You Get 6?
1.Have you played any games with dice?
Ans. Yes, I have played many games with dice?

2.How many dots are there on the different faces of a dice?
Ans. There are 1, 2, 3, 4, 5 and 6 dots on the different faces of a dice.
•Throw a die.
•Look at the number of dots you get on the face of your die.
•For each throw draw a mark / in front of that number in the table.
•Throw the die 30 times and mark in the table each time. For example,

(a)Which face of the die did you get the most number of times?
Ans. 2

(b)

Ans. Six .times.

(c)

Ans. No.

Getting Smart with Charts

This board shows the number of students in each class. It also shows the number of students present and absent.
1.How many children in gill are there in the school?
Ans. 121.

2.How many children in all are absent on that day?
Ans. 12.

In the chart show the absent students of class V.
Now look at the chart and fill in the blanks:
(а)The class with the highest number of absent students is IV.
(b)The class with the least number of absent students is II.
(c)The class with 3 students absent is III.
(d)The number of students absent in class IV and class V are 4 and 2.

Practice Time
Make your own smart charts about things around you.
1.Which bird has the most colours?

Ans. The table clearly shows that the peacock has the most colours

2.Which is the animal liked most as a pet?

Ans. This table clearly shows that dog is the animal which is liked most as a pet.

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Class 6 Maths Chapter 2 Notes Lines and Angles

Class 6 Maths Notes Chapter 2 – Class 6 Lines and Angles Notes

→ A point determines a location. It is denoted by a capital letter.

→ A line segment corresponds to the shortest distance between two points. The line segment joining points S and T is denoted by \(\overline{\mathrm{ST}}\).

→ A line is obtained when a line segment like \(\overline{\mathrm{ST}}\) is extended on both sides indefinitely; it is denoted by \(\stackrel{\leftrightarrow}{\mathrm{ST}}\) or sometimes by a single small letter like m.

→ A ray is a portion of a line starting at point D and going in one direction indefinitely. It is denoted by \(\overrightarrow{\mathrm{DP}}\) where P is another point on the ray.

→ An angle can be visualized as two rays starting from a common starting point. Two rays \(\overrightarrow{\mathrm{OP}}\) and \(\overrightarrow{\mathrm{OM}}\) form the angle ∠POM (also called ∠MOP); here, O is called the vertex of the angle, and the rays \(\overrightarrow{\mathrm{OP}}\) and \(\overrightarrow{\mathrm{OM}}\) are called the arms of the angle.

→ The size of an angle is the amount of rotation or turn needed about the vertex to rotate one ray of the angle onto the other ray of the angle.

→ The sizes of angles can be measured in degrees. One full rotation or turn is considered as 360 degrees and denoted as 360°.

→ Degree measures of angles can be measured using a protractor.

→ Angles can be straight (180°), right (90°), acute (more than 0° and less than 90°), obtuse (more than 90° and less than 180°), and reflex (more than 180° and less than 360°).

Point Class 6 Notes

Mark a dot on the paper with a sharp tip of a pencil. The sharper the tip, the thinner will be the dot. This tiny dot will give you an idea of a point. A point determines a precise location, but it has no length, breadth, or height. Some models for a point are given below.

If you mark three points on a piece of paper, you may be required to distinguish these three points. For this purpose, each of the three points may be denoted by a single capital letter such as Z, P, and T. These points are read as ‘Point Z’, ‘Point P’, and ‘Point T’. Of course, the dots represent precise locations and must be imagined to be invisibly thin.

Line Segment Class 6 Notes

Fold a piece of paper and unfold it. Do you see a crease? This gives the idea of a line segment. It has two end points, A and B. Mark any two points A and B on a sheet of paper. Try to connect A to B by various routes (Figure).

What is the shortest route from A to B?
This shortest path from Point A to Point B (including A and B) as shown here is called the line segment from A to B. It is denoted by either \(\overline{\mathrm{AB}}\) or \(\overline{\mathrm{BA}}\). The points A and B are called the endpoints of the line segment \(\overline{\mathrm{AB}}\).

Line Class 6 Notes

Imagine that the line segment from A to B (i.e., \(\overline{\mathrm{AB}}\)) is extended beyond A in one direction and beyond B in the other direction without any end (see Figure). This is a model for a line. Do you think you can draw a complete picture of a line? No. Why?

A line through two points A and B is written as \(\overleftrightarrow{\mathrm{AB}}\). It extends forever in both directions. Sometimes a line is denoted by a letter like l or m. Observe that any two points determine a unique line that passes through both of them.

Ray Class 6 Notes

A ray is a portion of a line that starts at one point (called the starting point or initial point of the ray) and goes on endlessly in a direction. The following are some models for a ray:

Look at the diagram (Figure) of a ray. Two points are marked on it. One is the starting point A and the other is a point P on the path of the ray. We then denote the ray by \(\overrightarrow{\mathrm{AP}}\).

Angle Class 6 Notes

An angle is formed by two rays having a common starting point. Here is an angle formed by rays \(\overrightarrow{\mathrm{BD}}\) and \(\overrightarrow{\mathrm{BE}}\) where B is the common starting point (Figure).

The point B is called the vertex of the angle, and the rays \(\overrightarrow{\mathrm{BD}}\) and \(\overrightarrow{\mathrm{BE}}\) are called the arms of the angle. How can we name this angle? We can simply use the vertex and say that it is the Angle B. To be clearer, we use a point on each of the arms together with the vertex to name the angle. In this case, we name the angle as Angle DBE or Angle EBD. The word angle can be replaced by the symbol ‘ ∠’, i.e., ∠DBE or ∠EBD. Note that in specifying the angle, the vertex is always written as the middle letter. To indicate an angle, we use a small curve at the vertex (refer to Figure). Vidya has just opened her book. Let us observe her opening the cover of the book in different scenarios.

Do you see angles being made in each of these cases? Can you mark their arms and vertex? Which angle is greater – the angle in Case 1 or the angle in Case 2?
Just as we talk about the size of a line based on its length, we also talk about the size of an angle based on its amount of rotation. So, the angle in Case 2 is greater as in this case, she needs to rotate the cover more. Similarly, the angle in Case 3 is even larger than that of Case 2, because there is even more rotation, and Cases 4, 5, and 6 are successively larger angles with more and more rotation.

Let’s look at some other examples where angles arise in real life by rotation or turn:

• In a compass or divider, we turn the arms to form an angle. The vertex is the point where the two arms are joined. Identify the arms and vertex of the angle.
• A pair of scissors has two blades. When we open them (or ‘turn them’) to cut something, the blades form an angle. Identify the arms and the vertex of the angle.
  
• Look at the pictures of spectacles, wallets, and other common objects. Identify the angles in them by marking out their arms and vertices.
  
• Do you see how these angles are formed by turning one arm over the other?

Comparing Angles Class 6 Notes

Look at these animals opening their mouths. Do you see any angles here? If yes, mark the arms and vertex of each one. Some mouths are open wider than others; the more the turning of the jaws, the larger the angle! Can you arrange the angles in this picture from smallest to largest?

Is it always easy to compare two angles?

Here are some angles. Label each of the angles. How will you compare them?
Draw a few more angles; label them and compare.

Comparing angles by superimposition
Any two angles can be compared by placing them one over the other, i.e., by superimposition. While superimposing, the vertices of the angles must overlap. After superimposition, it becomes clear which angle is smaller and which is larger.

The picture shows the two angles superimposed. It is now clear that ∠PQR is larger than ∠ABC.

Equal Angles:
Now consider ∠AOB and ∠XOY in the figure. Which is greater?

The corners of both of these angles match and the arms overlap with each other, i.e., OA OX and OB OY. So, the angles are equal in size. The reason these angles are considered to be equal in size is that when we visualize each of these angles as being formed out of rotation, we can see that there is an equal amount of rotation needed to move \(\overrightarrow{\mathrm{OB}}\) to \(\overrightarrow{\mathrm{OA}}\) and \(\overrightarrow{\mathrm{OY}}\) to \(\overrightarrow{\mathrm{OX}}\). From the point of view of superimposition, when two angles are superimposed, and the common vertex and the two rays of both angles lie on top of each other, then the sizes of the angles are equal.

Comparing angles without superimposition
Two cranes are arguing about who can open their mouth wider, i.e., who is making a bigger angle. Let us first draw their angles. How do we know which one is bigger? As seen before, one could trace these angles, superimpose them, and then check. But can we do it without superimposition? Suppose we have a transparent circle that can be moved and placed on fiures. Can we use this for comparison?

Let us place the circular paper on the angle made by the first crane. The circle is placed in such a way that its center is on the vertex of the angle. Let us mark points A and B on the edge circle at the points where the arms of the angle pass through the circle.

Can we use this to find out if this angle is greater than, equal to, or smaller than the angle made by the second crane?
Let us place it on the angle made by the second crane so that the vertex coincides with the center of the circle and one of the arms passes through OA.

Can you now tell which angle is bigger?
Which crane was making the bigger angle?
If you can make a circular piece of transparent paper, try this method to compare the angles in the Figure with each other.

Making Rotating Arms Class 6 Notes

Let us make ‘rotating arms’ using two paper straws and a paper clip by following these steps:
1. Take two paper straws and a paper clip.

2. Insert the straws into the arms of the paper clip.

3. Your rotating arm is ready!

Make several ‘rotating arms’ with different angles between the arms. Arrange the angles you have made from smallest to largest by comparing and using superimposition.

Passing through a Slit:
Collect several rotating arms with different angles; do not rotate any of the rotating arms during this activity.
Take a cardboard and make an angle-shaped slit as shown below by tracing and cutting out the shape of one of the rotating arms.

Now, shuff and mix up all the rotating arms. Can you identify which of the rotating arms will pass through the slit? The correct one can be found by placing each of the rotating arms over the slit. Let us do this for some of the rotating arms:

Only the pair of rotating arms where the angle is equal to that of the slit passes through the slit. Note that the possibility of passing through the slit depends only on the angle between the rotating arms and not on their lengths (as long as they are shorter than the length of the slit).

Special Types of Angles Class 6 Notes

Let us go back to Vidya’s notebook and observe her opening the cover of the book in different scenarios. She makes a full turn of the cover when she has to write while holding the book in her hand.

She makes a half turn off the cover when she has to open it on her table. In this case, observe the arms of the angle formed. They lie in a straight line. Such an angle is called a straight angle.

Let us consider a straight-angle ∠AOB. Observe that any ray \(\overrightarrow{\mathrm{OC}}\) divides it into two angles, ∠AOC and ∠COB.

Is it possible to draw OC such that the two angles are equal to each other in size?

Let’s Explore
We can try to solve this problem using a piece of paper. Recall that when a fold is made, it creates a straight crease. Take a rectangular piece of paper and on one of its sides, mark the straight angle AOB. By folding, try to get a line (crease) passing through O that divides ∠AOB into two equal angles. How can it be done?

Fold the paper such that OB overlaps with OA. Observe the crease and the two angles formed. Justify why the two angles are equal. Is there a way to superimpose and check? Can this superimposition be done by folding? Each of these equal angles formed is called the right angle. So, a straight angle contains two right angles.

If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?
Observe that a right angle resembles the shape of an ‘L’. An angle is a right angle only if it is exactly half of a straight angle. Two lines that meet at right angles are called perpendicular lines.

Classifying Angles
Angles are classified into three groups as shown below. Right angles are shown in the second group. What could be the common feature of the other two groups?

In the first group, all angles are less than a right angle or in other words, less than a quarter turn. Such angles are called acute angles. In the third group, all angles are greater than a right angle but less than a straight angle. The turning is more than a quarter turn and less than a half turn. Such angles are called obtuse angles.

Measuring Angles Class 6 Notes

We have seen how to compare two angles. But can we quantify how big an angle is using a number without having to compare it to another angle? We saw how various angles can be compared using a circle. Perhaps a circle could be used to assign measures for angles?

To assign precise measures to angles, mathematicians came up with an idea. They divided the angle in the center of the circle into 360 equal angles or parts. The angle measure of each of these unit parts is 1 degree, which is written as 1°. This unit part is used to assign a measure to any angle: the measure of an angle is the number of 1° unit parts it contains inside it. For example, see this figure:

It contains 30 units of 1° angle so we say that its angle measure is 30°. Measures of different angles: What is the measure of a full turn in degrees? As we have taken it to contain 360 degrees, its measure is 360°. What is the measure of a straight angle in degrees? A straight angle is half of a full turn. As a full turn is 360°, a half-turn is 180°. What is the measure of a right angle in degrees? Two right angles together form a straight angle. As a straight angle measures 180°, a right angle measures 90°.

A pinch of history
A full turn has been divided into 360°. Why 360? The reason why we use 360° today is not fully known. The division of a circle into 360 parts goes back to ancient times. The Rigveda, one of the very oldest texts of humanity going back thousands of years, speaks of a wheel with 360 spokes. Many ancient calendars, also going back over 3000 years—such as calendars of India, Persia, Babylonia, and Egypt—were based on having 360 days in a year. In addition, Babylonian mathematicians frequently used divisions of 60 and 360 due to their use of sexagesimal numbers and counting by 60s.

Perhaps the most important and practical answer for why mathematicians over the years have liked and continued to use 360 degrees is that 360 is the smallest number that can be evenly divided by all numbers up to 10, aside from 7. Thus, one can break up the circle into 1, 2, 3, 4, 5, 6, 8, 9, or 10 equal parts, and still have a whole number of degrees in each part! Note that 360 is also evenly divisible by 12, the number of months in a year, and by 24, the number of hours in a day. These facts all make the number 360 very useful. The circle has been divided into 1, 2, 3, 4, 5, 6, 8, 9 10, and 12 parts below. What are the degree measures of the resulting angles? Write the degree measures down near the indicated angles.

Degree Measures of Different Angles
How can we measure other angles in degrees? It is for this purpose that we have a tool called a protractor that is either a circle divided into 360 equal parts as shown in Figure, or a half circle divided into 180 equal parts.

Unlabelled Protractor
Here is a protractor. Do you see the straight angle at the center divided into 180 units of 1 degree? Only part of the lines dividing the straight angle are visible, though! Starting from the marking on the rightmost point of the base, there is a long mark for every 10°. From every such long mark, there is a medium-sized mark after 5°.

Labelled Protractor
This is a protractor that you find in your geometry box. It would appear similar to the protractor above except that there are numbers written on it. Will these make it easier to read the angles?

There are two sets of numbers on the protractor: one increasing from right to left and the other increasing from left to right. Why does it include two sets of numbers?

Name the different angles in the figure and write their measures.

Did you include angles such as ∠TOQ?
Which set of markings did you use – inner or outer?
What is the measure of ∠TOS?
Can you use the numbers marked to find the angle without counting the number of markings?
Here, OT and OS pass through the numbers 20 and 55 on the outer scale. How many units of 1 degree are contained between these two arms?
Can subtraction be used here?

How can we measure angles directly without having to subtract?
Place the protractor so the center is on the vertex of the angle. Align the protractor so that one of the arms passes through the 0º mark as in the picture below.

What is the degree measure of ∠AOB?

Make your Protractor!
You may have wondered how the different equally spaced markings are made on a protractor. We will now see how we can make some of them!
1. Draw a circle of a convenient radius on a sheet of paper. Cut out the circle (Figure). A circle or one full turn is 360°.
2. Fold the circle to get two halves and cut it through the crease to get a semicircle. Write ‘0°’ in the bottom right corner of the semi-circle.

3. Fold the semi-circular sheet in half as shown in Figure to form a quarter circle.

4. Fold the sheet again as shown in Figures:

When folded, this is \(\frac {1}{8}\) of the circle, or \(\frac {1}{8}\) of a turn, or \(\frac {1}{8}\) of 360°, or \(\frac {1}{4}\) of 180° or \(\frac {1}{2}\) of 90° = _________.
The new creases formed give us measures of 45° and 180° – 45° = 135° as shown. Write 45° and 135° at the correct places on the new creases along the edge of the semicircle.

5. Continuing with another half fold as shown in Figure, we get an angle of measure ______.

6. Unfold and mark the creases as OB, OC, …, etc., as shown in Figures.

Angle Bisector
At each step, we folded in halves. This process of getting half of a given angle is called bisecting the angle. The line that bisects a given angle is called the angle bisector of the angle.

Mind the Mistake, Mend the Mistake!
A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor discuss how the reading could have been made, and think about how it can be corrected.

Drawing Angles Class 6 Notes

Vidya wants to draw a 30° angle and name it ∠TIN using a protractor. In ∠TIN will be the vertex, and IT and IN will be the arms of the angle. Keeping one arm, say IN, as the reference (base), the other arm IT should take a turn of 30°.

Step 1: We begin with the base and draw \(\overrightarrow{\mathrm{IN}}\):

Step 2: We will place the center point of the protractor on I and align IN to the 0 lines.

Step 3: Now, starting from 0, count your degrees (0, 10, 30) up to 30 on the protractor. Mark point T at the label 30°.

Step 4: Using a ruler join the point I and T.
∠TIN = 30° is the required angle.

Let’s Play a Game #1
This is an angle-guessing game! Play this game with your classmates by making two teams, Team 1 and Team 2. Here are the instructions and rules for the game:

• Team 1 secretly chooses an angle measure, for example, 49°, and makes an angle with that measure using a protractor without Team 2 being able to see it.
• Team 2 now gets to look at the angle. They have to quickly discuss and guess the number of degrees in the angle (without using a protractor!).
• Team 1 now demonstrates the true measure of the angle with a protractor.
• Team 2 scores the number of points, which is the absolute difference in degrees between their guess and the correct measure.
• For example, if Team 2 guesses 39°, they score 10 points (49° – 39°).
• Each team gets five turns. The winner is the team with the lowest score!

Let’s Play a Game #2
We now change the rules of the game a bit. Play this game with your classmates by again making two teams, Team 1 and Team 2. Here are the instructions and rules:

• Team 1 announces to all, an angle measure, e.g., 34°.
• A player from Team 2 must draw that angle on the board without using a protractor. Other members of Team 2 can help the player by speaking words like ‘Make it bigger!’ or ‘Make it smaller!’.
• A player from Team 1 measures the angle with a protractor for all to see.
• Team 2 scores the number of points which is the absolute difference in degrees between Team 2’s angle size and the intended angle size.
• For example, if the player’s angle from Team 2 is measured to be 25°, then Team 2 scores 9 points (34° – 25°).
• Each team gets five turns. The winner is again the team with the lowest score.

Types of Angles and their Measures

We have read about different types of angles in this chapter. We have seen that a straight angle is 180° and a right angle is 90°. How can other types of angles — acute and obtuse — be described in terms of their degree measures?

Acute Angle:
Angles that are smaller than the right angle, i.e., less than 90° and greater than 0°, are called acute angles.

Obtuse Angle:
Angles that are greater than the right angle and less than the straight angle, i.e., greater than 90° and less than 180°, are called obtuse angles.

Have we covered all the possible measures that an angle can take? Here is another type of angle.

Reflex Angle:
Angles that are greater than the straight angle and less than the whole angle, i.e., greater than 180° and less than 360°, are called reflex angles.

Class 6 Maths Notes

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Students can use Maths Mela Class 3 Solutions Chapter 14 The Surajkund Fair Question Answer to explore alternative problem-solving methods.

The Surajkund Fair Class 3 Maths Chapter 14 Question Answer Solutions

The Surajkund Fair Class 3 Maths Question Answer

Soni and Avi are going to see a fair with their grandparents. They are going to Surajkund in Faridabad district of Haryana. Let us join them them and have fun.

Let us Discuss

• What do you see in the picture?
Answer:
We see decorated shops, a big gate, swings, and many tiles patterns on the floor.

• Spot things in the picture that look the same from the left and right side.
Answer:
Do Yourself.

Make Malas

Soni and Avi reach a stall where a man and a woman are making malas with beads.

Let us Do

Question 1.
Colour the beads in the strings using two colours to show the malas that you have made.

Answer:

Question 2.
On the previous page, tick (✓) the malas that are symmetrical.
Answer:

Question 3.
How many such malas can be made? Discuss.
Answer:
Four

a. Tick (✓) the malas that are symmetrical and cross (✗) the one(s) that are not symmetrical.

Answer:

b. Now, use 6 beads of one colour and 2 beads of another colour to make symmetrical malas.

Answer:

Vanakkam! Rangolis all around!

Soni arid Avi arrive at the stall of Tamil Nadu. Amma was making kolam in front of the hut.
Follow the steps:

1.

2.

3.

4.

5.

Let us Think

1. Observe the rangolis given below. Are all rangolis symmetrical?

2. Trace these rangolis on a paper. Fold the tracing paper in such a way that one half of the rangoli lies exactly on the other half.

3. Draw lines in the given rangolis that divide them into two identical halves.

4. Look for other symmetrical things around you. Discuss.

Let us Do

Enjoy making rangolis

Question 1.
Draw and complete the symmetrical rangolis given below.

a.

Answer:

b.

Answer:

c.

Answer:

Question 2.
Draw some more rangolis in your notebook that are symmetrical.
Answer:

Make Masks!

Tit for Tat

Soni gots her picture made by a painter.

Let us Think

Question 1.
What is the trick the painter is playing? Find things for the painter to draw so that he can no longer play the trick. Draw three such things here.
Answer:
Painter is playing the trick of making halves of symmetrical figures.
Three such things are

The Mirror Game

Soni and Avi started playing this game. Let us play with them.

Has Avi placed the counters at the right places? Check it by placing the mirror on the line drawn.

Let us Explore

Question 1.
Pick the odd one out and give reasons.

Answer:
It is odd one out because the figures of pair are not mirror images of each other.

Question 2.
Fill 4 boxes with red colour and 3 with blue colour in such a way that one side is the mirror image of the other.

In how many ways can you fill it?
Think, think!
Answer:
We can fill it with so many ways.

Question 3.
Make Micy’s side the same as that of Catty’s side. You can rearrange only three balls in Micy’s side.

Answer:

Question 4.
Which shape cutouts would fit in the given shape without overlapping and without gaps.

Answer:
Rangometry shapes like hexagon, square and rectangle could be fit in the given shape without overlapping and without gaps.

Tiling the paths

Let us Do

Question 1.
Use rangometry shapes to fill the shapes with no gaps and overlaps.

Answer:

Making Tiles, Creating Paths

Soni and Avi have started making their own tiles by joining different shapes.

Let us Do

Question 1.
Use two or more rangometry shapes to create your tiles. Now trace the tiles to create different paths.

Answer:

Question 2.
Try making these paths.

Answer:
Try yourself with the help of Teachers/Parents

Giant Wheel

Read the conversation between Soni and Avi and mark the place they are talking about.

Let us Play

Imagine yourself sitting with Soni and Avi. You think of a place or a stall and challenge your friend to find out which stall you have in your mind. You can help them guess by answering yes or no.

Search for Dada and Dadi
Soni and Avi’s Dada and Dadi were missing. They hear their announcement.

Let us Do

Question 1.
Help Soni and Avi read the map and find the following:

a. Which place does the sign show?
Answer:
The sign shows the place chaupal.

b. Circle the picture in the map that shows the play area?
Answer:
Do yourself.

c. Which place does the sign show?
Answer:
The sign shows for parking.

d. How many exit routes are there in the fair?
Answer:

Question 2.
Follow the path that Avi and Soni are following.

a. Walk on the blue lane.
b. Turn right on the green lane.
c. You will see a restaurant on your right. Don’t sit there.
d. Take a left towards the red lane.
e. Take the first left turn towards the golden lane. Stalls will be seen on the way.
f. Pass the stalls to find the Chaupal and meet Dada Dadi.
Answer:
Do yourself.

Question 3.
An uncle asks Dada ji the way to the ATM. Tell him the way to the ATM from the cha upal.
Answer:

1. Walk on the golden line.
2. Turn right on the red lane.
3. Stalls will be seen on the way. Pass the stalls and see ATM.

Let us Do

Question 1.
There are two ways to go out of the Surajkund fair.
One seems to be a maze and the other goes straight there.
Follow the maze with Soni and Avi to exit the fair.

Answer:

Question 2.
Share the way you went through the maze. Write the things you found on the way.
Answer:
I found diamond, star and cat on the way.

Dot Grids

Number Cards

Number Cards

Fraction Cards

Fraction Cards

Net of a Cube

Note: Cut along the dotted lines and fold along the dark lines.

Net of a Cuboid

Note: Cut along the dotted lines and fold along the dark lines.

Net of a Cylinder

Net of a Cone

Note: Cut along the dotted lines and fold along the dark lines.

Diene’s Blocks

Note: Cut along the dotted lines. You can make more such blocks as you need for the activities.

Diene’s Blocks

Note: Cut along the dotted lines. You can make more such blocks as you need for the activities.

NCERT Solutions for Class 3 Mathematics Chapter 14 Rupees and Paise (Old Syllabus)

Shopping
You can visit this self-service store.
A. Without using a pencil or paper, find out the cost of:
•One ball and one toy car Rs 22
•One notebook and two pencils Rs 10
•Two bananas and a glass of milk Rs 6.50
•One doll and a ball Rs 15
•One glass of is lemon juice and a packet of biscuits Rs 7.50
B.Find the total cost of:
•One toy giraffe, one copy and a glass of lemon juice Rs 14.50
•One glass of milk, one packet of biscuits and a banana Rs 9.50
•One notebook, two pencils and two erasers Rs 12
•Two tops, three toffees and two bananas Rs 8.50
C. What can you buy, if you have a twenty rupee note?
•1 toy car, 1 lemon juice, 1 banana
•1 ball, 1 doll, 1 glass milk
•1 toy car, 1 packet of biscuits, 1 toffee
•1 toy car, 1 toy giraffe, 1 glass milk
D.You need to make a cash memo for the things you bought. Before adding, first
guess how much money will be needed. Then find the total and check your guess.
Monu prepared the following cash memos:
Check the cash memos and correct them if you find a mistake.
Ans. Correct memos are given as follows:

E.If you have 30 rupees with you. Find out how much money will be left after buying the following items:

1.One hall, one doll and one toy giraffe.
Ans.Price of one ball = Rs 7.00
Price of one doll = Rs 8.00
Price of one toy giraffe = Rs 6.50
So, total cost = Rs7.00 + Rs 8.00 + Rs 6.50 = Rs 21.50
Money left = Total money – money spent
= Rs 30.0021. 50 = Rs 8.50

2.Two bananas, one pack of biscuits and two glasses of lemon juice.
Ans. Price of 1 banana = Rs 1.50
So, price of 2 bananas = 1.50 x2=Rs 3.00
Price of one pack of biscuit = Rs 4.50
Price of one glass lemon juice = Rs 3.00
So, price of two glasses of lemon juice = 3.00 x2 = Rs 6.00
Total cost = Rs 3.00 + Rs 4.50 + Rs 6.00 = 13.50
Money left = Total money – Money spent = Rs 30.00 – Rs 13.50 = Rs 16.50

3.Three notebooks, two pencils and two erasers.
Ans. Price of 1 notebook =Rs 5.00
So, price of 3 notebooks =Rs 5.00 x 3 = Rs 15.00
Price of 1 pencil =Rs 2.50
So, price of 2 pencils = Rs 2.50 x 2 = Rs 5.00
Price of 1 eraser = Rs 1.00
So, price of 2 erasers = Rs 1.00 x 2 = Rs 2.00
Total cost= Rs 15.00 + Rs 5.00 + Rs 2.00 = Rs 22.00

Practice Time
A.Three friends wanted to buy a cricket bat and ball. Bina had Rs 48.50, Raman had Rs 55.50 and Venu had Rs 38.00. How much money did they have in all?
Ans. They had X 142.00 in all.

B. Hari booked a railway ticket for Rs 62.50. He gave a 100 rupee note. How much money will he get back with the ticket?
Ans.

C. Gita and her friend went shopping. She bought things for Rs 58, Rs 37 and Rs 22. Gita had a hundred rupee note. How much money should she borrow from her friend to pay the bill?
Ans. Gita needs to pay the following amount:

Now she needs to borrow the following amount:

Train Journey
This train goes from New Jalpaiguri to Guwahati. On its way, it stops at New Mai, Alipurduar and Goalpara stations.
The cost of a rail ticket to different stations is given in the table.

Find The Distance
(а)From New Mai to Guwahati 495 km – 57 km = 438 km
(b)Between New Mai and Goalpara 666 km – 57 km = 309 km
(c)From Alipurduar to Guwahati 495 km – 175 km = 320 km
(d)Between New Mai and Alipurduar 175 km – 57 km = 118 km
(e)From Goalpara to Guwahati 495 km – 366 km = 129 km

Find The Cost of Tickets
(a) Bhupen is going from New Jalpaiguri to Alipurduar. What is the cost of his ticket?
Ans.Rs 28.00
(b) Indra has to go from New Jalpaigurei to Goalpara. How much does she pay for the ticket?
Ans. Rs 49.50
(c) Debu, Shoma and Gobind are goind from New Jalpaiguri to New Mai. What amount will they pay for three tickets? They gjve aRs 50 note for the tickets. How much money will they get back?
Ans. Cost of 1 ticket from New Jalpaiguri to New Mall =Rs 12.50 So, cost of three tickets = Rs 12.50 x 3 = 37.50 Money given – Money spent = Money got back or, Rs 50 – Rs37,50 = Rs 12. 50 They will get back Rs 12.50.

The post NCERT Class 3 Maths Mela Chapter 14 Question Answer The Surajkund Fair appeared first on Learn CBSE.

Class 6 Maths Chapter 3 Notes Number Play

Class 6 Maths Notes Chapter 3 – Class 6 Number Play Notes

→ Numbers can be used for many different purposes, including to convey information, make and discover patterns, estimate magnitudes, pose and solve puzzles, and play and win games.

→ Thinking about and formulating set procedures to use numbers for these purposes is a useful skill and capacity (called “computational thinking”).

→ Many problems about numbers can be very easy to pose, but very difficult to solve. Indeed, numerous such problems are still unsolved (e.g., Collatz’s Conjecture).

Numbers are used in different contexts and in many different ways to organize our lives. We have used numbers to count, and have applied the basic operations of addition, subtraction, multiplication, and division on them, to solve problems related to our daily lives. In this chapter, we will continue this journey, by playing with numbers, seeing numbers around us, noticing patterns, and learning to use numbers and operations in new ways.

Numbers can Tell us Things Class 6 Notes

What are these numbers telling us?
Some children in a park are standing in a line. Each one says a number.

What do you think these numbers mean?
The children now rearrange themselves, and again each one says a number based on the arrangement.

A child says ‘1’ if there is only one taller child standing next to them.
A child says ‘2’ if both the children standing next to them are taller. A
child says ‘0’, if neither of the children standing next to them are taller.
That is each person says the number of taller neighbours they have.

Supercells Class 6 Notes

Observe the numbers written in the table below. Why are some numbers coloured? Discuss.

A cell is coloured if the number in it is larger than its adjacent cells. 626 is coloured as it is larger than 577 and 345 whereas 200 is not coloured as it is smaller than 577. The number 198 is coloured as it has only one adjacent cell with 109 in it, and 198 is larger than 109.

Let’s do the supercells activity with more rows. Here the neighbouring cells are those that are immediately to the left, right, top, and bottom.

The rule remains the same: a cell becomes a supercell if the number in it is greater than all the numbers in its neighbouring cells.
In Table 1, 8632 is greater than all its neighbours 4580, 8280, 4795, and 1944.

Complete Table 2 with 5-digit numbers whose digits are ‘1’, ‘0’, ‘6’, ‘3’, and ‘9’ in some order. Only a coloured cell should have a number greater than all its neighbours.
The biggest number in the table is 96,301.
The smallest even number in the table is 19,306.
The smallest number greater than 50,000 in the table is 60,193.
Once you have filled the table above, put commas appropriately after the thousands digit.

Patterns of Numbers on the Number Line Class 6 Notes

We are quite familiar with number lines now. Let’s see if we can place some numbers in their appropriate positions on the number line. Here are the numbers: 2180, 2754, 1500, 3600, 9950, 9590, 1050, 3050, 5030, 5300 and 8400.

Playing with Digits Class 6 Notes

We start writing numbers from 1, 2, 3 … and so on. There are nine 1-digit numbers. Find out how many numbers have two digits, three digits, four digits, and five digits:

Digit Sums of Numbers
Komal observes that when she adds up digits of certain numbers the sum is the same. For example, adding the digits of the number 68 will be the same as adding the digits of 176 or 545.

Digit Detectives
After writing numbers from 1 to 100, Dinesh wondered how many times he would have written the digit ‘7’!

Pretty Palindromic Patterns Class 6 Notes

What pattern do you see in these numbers: 66, 848, 575, 797, 1111?
These numbers read the same from left to right and from right to left. Try and see. Such numbers are called palindromes or palindromic numbers. All palindromes using 1, 2, 3.
The numbers 121, 313, and 222 are some examples of palindromes using the digits ‘1’, ‘2’, ‘3’.

Reverse-and-add palindromes
Now look at these additions. Try to figure out what is happening.
Steps to follow: Start with a 2-digit number. Add this number to its reverse. Stop if you get a palindrome; otherwise, repeat the steps of reversing the digits and adding. Try the same procedure for some other numbers, and perform the same steps. Stop if you get a palindrome. There are numbers for which you have to repeat this a large number of times.

The Magic Number of Kaprekar Class 6 Notes

D.R. Kaprekar was a mathematics teacher in a government school in Devlali, Maharashtra. He liked playing with numbers very much and found many beautiful patterns in numbers that were previously unknown. In 1949, he discovered a fascinating and magical phenomenon when playing with 4-digit numbers.

Follow these steps and experience the magic for yourselves! Pick any 4-digit number, say 6382.

Take different 4-digit numbers and try carrying out these steps. Find out what happens. Check with your friends what they got.
You will always reach the magic number ‘6174’! The number ‘6174’ is now called the ‘Kaprekar constant’.
Carry out these same steps with a few 3-digit numbers. What number will start repeating?

Clock and Calendar Numbers Class 6 Notes

On the usual 12-hour clock, there are timings with different patterns.
For example, 4:44, 10:10, 12:21.

Try and find out all possible times on a 12-hour clock for each of these types.
Manish has his birthday on 20/12/2012 where the digits ‘2’, ‘0’, ‘1’, and ‘2’ repeat in that order.

Find some other dates of this form from the past.
His sister Meghana had her birthday on 11/02/2011 where the digits read the same from left to right and from right to left.

Find all possible dates of this form from the past.
Jeevan was looking at this year’s calendar. He started wondering, “Why should we change the calendar every year? Can we not reuse a calendar?” What do you think?
You might have noticed that last year’s calendar was different from this year’s. Also, next year’s calendar is different from the previous years. But, will any year’s calendar repeat again after some years? Will all dates and days in a year match exactly with that of another year?

Mental Math Class 6 Notes

Observe the figure below. What can you say about the numbers and the lines drawn?

Numbers in the middle column are added in different ways to get the numbers on the sides (1500 + 1500 + 400 = 3400). The numbers in the middle can be used as many times as needed to get the desired sum. Draw arrows from the middle to the numbers on the sides to obtain the desired sums. Two examples are given. It is simpler to do it mentally!
38,800 = 25,000 + 400 × 2 + 13,000
3400 = 1500 + 1500 + 400

Adding and Subtracting
Here, using the numbers in the boxes, we are allowed to use both addition and subtraction to get the required number. An example is shown.

An example of adding two 5-digit numbers to get another 5-digit number is 12,350 + 24,545 = 36,895.
An example of subtracting two 5-digit numbers to get another 5-digit number is 48,952 – 24,547 = 24,405.

Playing with Number Patterns Class 6 Notes

Here are some numbers arranged in some patterns. Find out the sum of the numbers in each of the below figures. Should we add them one by one or can we use a quicker way? Share and discuss in class the different methods each of you used to solve these questions.

An Unsolved Mystery – the Collatz Conjecture! Class 6 Notes

Look at the sequences below—the same rule is applied in all the sequences:
a. 12, 6, 3, 10, 5, 16, 8, 4, 2, 1
b. 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
c. 21, 64, 32, 16, 8, 4, 2, 1
d. 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
Do you see how these sequences were formed?
The rule is: one starts with any number; if the number is even, take half of it; if the number is odd, multiply it by 3 and add 1; repeat.

Notice that all four sequences above eventually reached the number 1. In 1937, the German mathematician Lothar Collatz conjectured that the sequence will always reach 1, regardless of the whole number you start with. Even today—despite many mathematicians working on it — it remains an unsolved problem as to whether Collatz’s conjecture is true! Collatz’s conjecture is one of the most famous unsolved problems in mathematics.

Make some more Collatz sequences like those above, starting with your favourite whole numbers. Do you always reach 1?
Do you believe the conjecture of Collatz that all such sequences will eventually reach 1? Why or why not?

Simple Estimation Class 6 Notes

At times, we may not know or need an exact count of things and an estimate is sufficient for the purpose at hand. For example, your school headmaster might know the exact number of students enrolled in your school, but you may only know an estimated count. How many students are in your school? About 150? 400? A thousand?

Paromita’s class section has 32 children. The other 2 sections of her class have 29 and 35 children. So, she estimated the number of children in her class to be about 100. Along with Class 6, her school also has Classes 7–10 and each class has 3 sections each. She assumed a similar number in each class and estimated the number of students in her school to be around 500.

Games and Winning Strategies Class 6 Notes

Numbers can also be used to play games and develop winning strategies. Here is a famous game called 21. Play it with a classmate. Then try it at home with your family!

Rules for Game #1:
The first player says 1, 2, or 3. Then the two players take turns adding 1, 2, or 3 to the previous number said. The first player to reach 21 wins!
Play this game several times with your classmates. Are you starting to see the winning strategy?
Which player can always win if they play correctly? What is the pattern of numbers that the winning player should say?
There are many variations of this game. Here is another common variation:

Rules for Game #2:
The first player says a number between 1 and 10. Then the two players take turns adding a number between 1 and 10 to the previous number said. The first player to reach 99 wins!
Play this game several times with your classmates. See if you can figure out the corresponding winning strategy in this case! Which player can always win? What is the pattern of numbers that the winning player should say this time?
Make your variations of this game decide how much one can add at each turn, and what number is the winning number. Then play your game several times, and figure out the winning strategy and which player can always win!

Class 6 Maths Notes

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Click here to access the best NCERT Solutions for Class 6 Sanskrit Deepakam Chapter 1 Question Answer वयं वर्णमालां पठामः textbook exercise questions and answers.

वयं वर्णमालां पठामः Class 6th Sanskrit Deepakam Chapter 1 Question Answer

NCERT Solutions for Class 6 Sanskrit Deepakam Chapter 1 वयं वर्णमालां पठामः

NCERT Solutions

The post NCERT Class 6 Sanskrit Deepakam Chapter 1 Question Answer वयं वर्णमालां पठामः appeared first on Learn CBSE.

Class 6 Maths Chapter 4 Notes Data Handling and Presentation

Class 6 Maths Notes Chapter 4 – Class 6 Data Handling and Presentation Notes

→ Facts, numbers, measures, observations, and other descriptions of things that convey information about those things are called data.

→ Data can be organized in a tabular form using tally marks for easy analysis and interpretation.

→ Frequencies are the counts of the occurrences of values, measures, or observations.

→ Pictographs represent data in the form of pictures/objects or parts of objects. Each picture represents a frequency that can be 1 or more
than 1 – this is called the scale, and it must be specified.

→ Bar graphs have bars of uniform width; the length or height then indicates the total frequency of occurrence. The scale that is used to convert length/height to frequency must again be specified.

→ Choosing the appropriate scale for a pictograph or bar graph is important to accurately and effectively convey the desired information/data and to also make it visually appealing.

→ Other aspects of a graph also contribute to its effectiveness and visual appeal, such as how colours are used, what accompanying pictures are drawn, and whether the bars are horizontal or vertical. These aspects correspond to the artistic and aesthetic side of data handling and presentation.

→ However, making visual representations of data too “fancy” can also sometimes be misleading.

→ By reading pictographs and bar graphs accurately, we can quickly understand and make inferences about the data presented.

If you ask your classmates about their favourite colours, you will get a list of colours. This list is an example of data. Similarly, if you measure the weight of each student in your class, you would get a collection of measures of weight—again data. Any collection of facts, numbers, measures, observations, or other descriptions of things that convey information about those things is called data.

We live in an age of information. We constantly see large amounts of data presented to us in new and interesting ways. In this chapter, we will explore some of the ways that data is presented, and how we can use some of those ways to correctly display, interpret, and make inferences from such data!

Collecting and Organising Data Class 6 Notes

Navya and Naresh are discussing their favourite games.


Naresh and Navya decided to go to each student in the class and ask what their favourite game was. Then they prepared a list. Navya is showing the list:

She says (happily), “I have collected the data. I can figure out the most popular game now!” A few other children are looking at the list and wondering, “We can’t yet see the most popular game. How can we get it from this list?”

Shri Nilesh is a teacher. He decided to bring sweets to the class to celebrate the new year. The sweets shop nearby has jalebi, gulab jamun, gujiya, barfi, and rasgulla. He wanted to know the choices of the children. He wrote the names of the sweets on the board and asked each child to tell him their preference. He put a tally mark ‘|’ for each student and when the count reached 5, he put a line through the previous four and marked it as .

Shri Nilesh requested one of the staff members to bring the sweets as given on the table. The above table helped him to purchase the correct number of sweets. To organize the data, we can write the name of each sweet in one column and using tally signs, note the number of students who prefer that sweet. The numbers 6, 9, and… are the frequencies of the sweet preferences for jalebi, and gulab jamun … respectively. Sushri Sandhya asked her students about the sizes of the shoes they wear. She noted the data on the board.

She then arranged the shoe sizes of the students in ascending order:
3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7

Pictographs Class 6 Notes

Pictographs are one visual and suggestive way to represent data without writing any numbers. Look at this picture — you may be familiar with it from previous classes.

This picture helps you understand at a glance the different modes of travel used by students. Based on this picture, answer the following questions:

• Which mode of travel is used by the most number of students?
• Which mode of travel is used by the least number of students?

A pictograph represents data through pictures of objects. It helps answer questions about data with just a glance. In the above pictograph, one unit or symbol ( ) is used to represent one student. There are also other pictographs where one unit or symbol stands for many people or objects.

Example: Nand Kishor collected responses from the children of his middle school in Berasia regarding how often they slept at least 9 hours during the night. He prepared a pictograph from the data:

Answer the following questions using the pictograph:

1. What is the number of children who always slept at least 9 hours at night?
2. How many children sometimes sleep at least 9 hours at night?
3. How many children always slept less than 9 hours each night? Explain how you got your answer.

Solutions

1. In the table, there are 5 pictures for ‘Always’. Each picture represents 10 children. Therefore, 5 pictures indicate 5 × 10 = 50 children.
2. There are 2 complete pictures (2 × 10 = 20) and a half picture (half of 10 = 5). Therefore, the number of children who sleep at least 9 hours only sometimes is 20 + 5 = 25.
3. There are 4 complete pictures for ‘Never’. Hence 4 × 10 = 40 children never sleep at least 9 hours in a night, i.e., they always sleep less than 9 hours.

Drawing a Pictograph
One day, Lakhanpal collected data on how many students were absent in each class.

He created a pictograph to present this data and decided to show 1 student as in the pictograph.

Meanwhile, his friends Jarina and Sangita collected data on how many students were present in each class.

If they want to show their data through a pictograph, where they also use one symbol for each student, as Lakhanpal did, what are the challenges they might face?
Jarina made a plan to address this since there were many students, and she decided to use to represent 5 students. She figured that would save time and space too.

Sangita decided to use one to represent 10 students instead. Since she used one to show 10 students, she had a problem showing 25 students and 35 students in the pictograph. Then, she realized she could use to show 5 students.

• Pictographs are a nice visual and suggestive way to represent data. They represent data through pictures of objects.
• Pictographs can help answer questions and make inferences about data with just a glance (in the examples above— about favorite games, favorite colours, most common modes of conveyance, number of students absent, etc.).
• By reading a pictograph, we can quickly understand the frequencies of the different categories (for example, cricket, hockey, etc.), and the comparisons of these frequencies.
• In a pictograph, the categories can be arranged horizontally or vertically. For each category, simple pictures and symbols are then drawn in the designated columns or rows according to the frequency of that category.
• A scale or key (for example, : 1 student or : 5 students) is added to show what each symbol or picture represents. Each symbol or picture can represent one unit or multiple units.
• It can be more challenging to prepare a pictograph when the amount of data is large or when the frequencies are not exact multiples of the scale or key.

Bar Graphs Class 6 Notes

Have you seen graphs like this on TV or in a newspaper?

Like pictographs, bar graphs can help us to quickly understand and interpret information, such as the highest value, the comparison of values of different categories, etc. However, when the amount of data is large, presenting it by a pictograph is not only time-consuming but at times difficult too. Let us see how data can be presented instead of using a bar graph.

Let’s take the data collected by Lakhanpal earlier, regarding the number of students absent on one day in each class.

He presented the same data using a bar graph.

When making bar graphs, bars of uniform width can be drawn horizontally or vertically with equal spacing between them; then the length or height of each bar represents the given number. As we saw in pictographs, we can use a scale or key when the frequencies are larger. Let us look at an example of vehicular traffic at a busy road crossing in Delhi, which was studied by the traffic police on a particular day. The number of vehicles passing through the crossing each hour from 6 am to 12:00 noon is shown in the bar graph. One unit of length stands for 100 vehicles.

We can see that the maximum traffic at the crossing is shown by the longest bar, i.e., for the time interval 7-8 a.m. The bar graph shows that 1200 vehicles passed through the crossing at that time. The second longest bar is for 8-9 a.m. During that time, 1000 vehicles passed through the crossing. Similarly, the minimum traffic is shown by the smallest bar, i.e., the bar for the time interval 6-7 a.m. During that time, only about 150 vehicles passed through the crossing. The second smallest bar is that for the time interval 11 a.m.-12 noon when about 600 vehicles passed through the crossing. The total number of cars passing through the crossing during the two-hour interval 8.00-10.00 am as shown by the bar graph is about 1000 + 800 = 1800 vehicles.

Example:

This bar graph shows the population of India in each decade over 50 years. The numbers are expressed in crores. If you were to take 1 unit length to represent one person, drawing the bars would be difficult! Therefore, we chose the scale so that 1 unit represents 10 crores. The bar graph for this choice is shown in the figure. So a bar of length 5 units represents 50 crores and of 8 units represents 80 crores.

• Based on this bar graph, what may be a few questions you may ask your friends?
• How much did the population of India increase over 50 years?
• How much did the population increase in each decade?

Drawing a Bar Graph Class 6 Notes

In a previous example, Shri Nilesh prepared a frequency table representing the sweet preferences of the students in his class. Let’s try to prepare a bar graph to present his data.

1. First, we draw a horizontal line and a vertical line. On the horizontal line, we will write the name of each of the sweets, equally spaced, from which the bars will rise by their frequencies; and on the vertical line, we will write the frequencies representing the number of students.

2. We must choose a scale. That means we must decide how many students will be represented by a unit length of a bar so that it fits nicely on our paper. Here, we will take 1 unit length to represent 1 student.

3. For Jalebi, we therefore need to draw a bar having a height of 6 units (which is the frequency of the sweet Jalebi), and similarly for the other sweets we have to draw bars as high as their frequencies.

4. We therefore get a bar graph as shown below.

When the frequencies are larger and we cannot use the scale of 1 unit length = 1 number (frequency), we need to choose a different scale as we did in the case of pictographs.

Example: The number of runs scored by Smriti in each of the 8 matches is given in the table below:

In this example, the minimum score is 0 and the maximum score is 100. Using a scale of 1 unit length = 1 run would mean that we have to go all the way from 0 to 100 runs in steps of 1. This would be unnecessarily tedious. Instead, let us use a scale where 1 unit length = 10 runs. We mark this scale on the vertical line and draw the bars according to the scores in each match. We get the following bar graph representing the above data.

Example: The following table shows the monthly expenditure of Imran’s family on various items:

To represent this data in the form of a bar graph, here are the steps:

• Draw two perpendicular lines, one horizontal and one vertical.
• Along the horizontal line, mark the ‘Items’ with equal spacing between them, and along the vertical line, mark the corresponding expenditures.
• Take bars of the same width, keeping a uniform gap between them.
• Choose a suitable scale along the vertical line. Let 1 unit length = ₹ 200, and then mark and write the corresponding values (₹ 200, ₹ 400, etc.) representing each unit length.

Finally, calculate the heights of the bars for various items as shown below.

Here is the bar graph that we obtain based on the above steps:

• Like pictographs, bar graphs give a nice visual way to represent data. They represent data through equally spaced bars, each of equal width, where the lengths or heights give frequencies of the different categories.
• Each category is represented by a bar where the length or height depicts the corresponding frequency (for example, cost) or quantity (for example, runs).
• The bars have uniform spaces between them to indicate that they are free-standing and represent equal categories.
• The bars help in interpreting data much faster than a frequency table. By reading a bar graph, we can compare frequencies of different categories at a glance.
• We must decide the scale (for example, 1 unit length = 1 student or 1 unit length = ₹ 200) for a bar graph based on the data, including the minimum and maximum frequencies so that the resulting bar graph fits nicely and looks visually appealing on the paper or poster we are preparing. The markings of the unit lengths as per the scale must start from zero.

Artistic and Aesthetic Considerations Class 6 Notes

In addition to the steps described in previous sections, there are also some other more artistic and aesthetic aspects one can consider when preparing visual presentations of data to make them more interesting and effective. First, when making a visual presentation of data, such as a pictograph or bar graph, it is important to make it fit in the intended space; this can be controlled, e.g., by choosing the scale appropriately, as we have seen earlier. It is also desirable to make the data presentation visually appealing and easy to understand so that the intended audience appreciates the information being conveyed. Let us consider an example. Here is a table naming the tallest mountain on each continent, along with the height of each mountain in meters.

How much taller is Mount Everest than Mount Kosciuszko? Are Mount Denali and Mount Kilimanjaro very different in height? This is not so easy to quickly discern from a large table of numbers. As we have seen earlier, we can convert the table of numbers into a bar graph, as shown on the right. Here, each value is drawn as a horizontal box. These are longer or shorter depending on the number they represent. This makes it easier to compare the heights of all these mountains at a glance.

However, since the boxes represent heights, it is better and more visually appealing to rotate the picture, so that the boxes grow upward vertically from the ground like mountains. A bar graph with vertical bars is also called a column graph. Columns are the pillars you find in a building that holds up the roof. Below is a column graph for our table of tallest mountains. From this column graph, it becomes easier to compare and visualize the heights of the mountains.

In general, it is more intuitive, suggestive, and visually appealing to represent heights, that are measured upwards from the ground, using bar graphs that have vertical bars or columns. Similarly, lengths that are parallel to the ground (e.g., distances between locations on Earth) are usually best represented using bar graphs with horizontal ars.

Infographics
When data visualizations such as bar graphs are further beautified with more extensive artistic and visual imagery, they are called information graphics, or infographics for short. Infographics aim to make use of attention-attracting and engaging visuals to communicate information even more clearly and quickly, in a visually pleasing way. As an example of how infographics can be used to communicate data even more suggestively, let us go back to the table above listing the tallest mountain on each continent. We drew a bar graph with vertical bars (columns), rather than horizontal bars, to be more indicative of mountains. But instead of rectangles, we could instead use triangles, which look a bit more like mountains. And we can add a splash of colour as well. Here is the result.

While this infographic might look more appealing and suggestive at first glance, it does have some issues. The goal of our bar graph earlier was to represent the heights of various mountains – using bars of the appropriate heights but the same widths. The purpose of using the same widths was to make it clear that we are only comparing heights. However, in this infographic, the taller triangles are also wider! Are taller mountains always wider? The infographic implies additional information that may be misleading and may or may not be correct. Sometimes going for more appealing pictures can also accidentally mislead.

Taking this idea further, and to make the picture even more visually stimulating and suggestive, we can further change the shapes of the mountains to make them look even more like mountains, and add other details, while attempting to preserve the heights. For example, we can create an imaginary mountain range that contains all these mountains. Is the infographic below better than the column graph with rectangular columns of equal width? The mountains look more realistic, but is the picture accurate? For example, Everest appears to be twice as tall as Elbrus.

What is 5642 × 2?
While preparing visually appealing presentations of data, we also need to be careful that the pictures we draw do not mislead us about the facts. In general, it is important to be careful when making or reading infographics, so that we do not mislead our intended audiences and so that we, ourselves, are not misled.

Class 6 Maths Notes

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Class 6 Maths Chapter 5 Notes Prime Time

Class 6 Maths Notes Chapter 5 – Class 6 Prime Time Notes

→ If a number is divisible by another, the second number is called a factor of the first. For example, 4 is a factor of 12 because 12 is divisible by 4 (12 ÷ 4 = 3).

→ Prime numbers are numbers like 2, 3, 5, 7, 11, … that have only two factors, namely 1 and themselves.

→ Composite numbers are numbers like 4, 6, 8, 9, … that have more than 2 factors, i.e., at least one factor other than 1 and themselves. For example, 8 has the factor 4, and 9 has the factor 3, so 8 and 9 are both composite.

→ Every number greater than 1 can be written as a product of prime numbers. This is called the number’s prime factorization. For example, 84 = 2 × 2 × 3 × 7.

→ There is only one way to factorize a number into primes, except for the ordering of the factors.

→ Two numbers that do not have a common factor other than 1 are said to be co-prime.

→ To check if two numbers are co-prime, we can first find their prime factorizations and check if there is a common prime factor. If there is no common prime factor, they are co-prime, and otherwise, they are not.

→ A number is a factor of another number if the prime factorization of the first number is included in the prime factorization of the second number.

Common Multiples and Common Factors Class 6 Notes

Idli-Vada Game
Children sit in a circle and play a game of numbers. One of the children starts by saying ‘1’. The second player says ‘2’, and so on. But when it is the turn of 3, 6, 9, … (multiples of 3), the player should say ‘idli’ instead of the number. When it is the turn of 5, 10, … (multiples of 5), the player should say ‘vada’ instead of the number. When a number is both a multiple of 3 and a multiple of 5, the player should say ‘idli-vada’! If a player makes any mistake, they are out.

The game continues in rounds till only one person remains. For which numbers should the players say ‘idli’ instead of saying the number? These would be 3, 6, 9, 12, 18, … and so on. For which numbers should the players say ‘vada’? These would be 5, 10, 20, … and so on. Which is the first number for which the players should say, ‘idli-vada’? It is 15, which is a multiple of 3, and also a multiple of 5. Find out other such numbers that are multiples of both 3 and 5.

Let us now play the ‘idli-vada’ game with different pairs of numbers:
a. 2 and 5,
b. 3 and 7,
c. 4 and 6.
We will say ‘idli’ for multiples of the smaller number, ‘vada’ for multiples of the larger number, and ‘idli-vada’ for common multiples. Draw a figure if the game is played up to 60.

Jump Jackpot
Jumpy and Grumpy play a game.

• Grumpy places a treasure on some number. For example, he may place it on 24.
• Jumpy chooses a jump size. If he chooses 4, then he has to jump only on multiples of 4, starting at 0.
• Jumpy gets the treasure if he lands on the number where Grumpy placed it.

Which jump sizes will get Jumpy to land on 24?
If he chooses 4: Jumpy lands on 4 → 8 → 12 → 16 → 20 → 24 → 28 → ……
Other successful jump sizes are 2, 3, 6, 8 and 12.

What about jump sizes 1 and 24? Yes, they also will land on 24. The numbers 1, 2, 3, 4, 6, 8, 12, 24 all divide 24 exactly. Recall that such numbers are called factors or divisors of 24. Grumpy increases the level of the game. Two treasures are kept in two different numbers. Jumpy has to choose a jump size and stick to it. Jumpy gets the treasures only if he lands on both the numbers with the chosen jump size. As before, Jumpy starts at 0.

Grumpy has kept the treasures on 14 and 36. Jumpy chooses a jump size of 7. Will Jumpy land on both the treasures?
Starting from 0, he jumps to 7 → 14 → 21 → 28 → 35 → 42 …
We see that he landed on 14 but did not land on 36, so he does not get the treasure. What jump size should he have chosen?
The factors of 14 are 1, 2, 7, 14. So these jump sizes will land on 14.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. These jump sizes will land on 36.
So, the jump sizes of 1 or 2 will land on both 14 and 36. Notice that 1 and 2 are the common factors of 14 and 36.
The jump sizes using which both the treasures can be reached are the common factors of the two numbers where the treasures are placed.

Prime Numbers Class 6 Notes

Guna and Anshu want to pack fis (anjeer) that grow on their farm. Guna wants to put 12 figs in each box and Anshu wants to put 7 figs in each box.
How many arrangements are possible? Think and find out the different ways.

• Guna can arrange 12 figs in a rectangular manner.
• Anshu can arrange 7 figs in a rectangular manner.

Guna has listed out these possibilities. Observe the number of rows and columns in each of the arrangements. How are they related to 12?
In the second arrangement, for example, 12 fis are arranged in two columns of 6 each or 12 = 2 × 6.
Anshu could make only one arrangement: 7 × 1 or 1 × 7. There are no other rectangular arrangements possible.

In each of Guna’s arrangements, multiplying the number of rows by the number of columns gives the number 12. So, the number of rows or columns are factor of 12. We saw that the number 12 can be arranged in a rectangle in more than one way as 12 has more than two factors. The number 7 can be arranged in only one way, as it has only two factors 1 and 7.

Numbers that have only two factors are called prime numbers or primes. Here is the fist few primes 2, 3, 5, 7, 11, 13, 17, 19. Notice that the factors of a prime number are 1 and the number itself. What about numbers that have more than two factors? They are called composite numbers. The first few composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20. What about 1, which has only one factor? The number 1 is neither a prime nor a composite number.

Can we list all the prime numbers from 1 to 100?
Here is an interesting way to find prime numbers. Just follow the steps given below and see what happens.

• Step 1: Cross out 1 because it is neither prime nor composite.
• Step 2: Circle 2, and then cross out all multiples of 2 after that, i.e., 4, 6, 8, and so on.
• Step 3: You will find that the next uncrossed number is 3. Circle 3 and then cross out all the multiples of 3 after that, i.e., 6, 9, 12, and so on.
• Step 4: The next uncrossed number is 5. Circle 5 and then cross out all the multiples of 5 after that, i.e., 10, 15, 20, and so on.
• Step 5: Continue this process till all the numbers in the list are either circled or crossed out.

All the circled numbers are prime numbers. All the crossed-out numbers, other than 1, are composite. This method is called the Sieve of Eratosthenes. This procedure can be carried on for numbers greater than 100 also. Eratosthenes was a Greek mathematician who lived around 2200 years ago and developed this method of listing primes.

Guna and Anshu started wondering how this simple method can find prime numbers! Think about how this method works. Read the steps given above again and see what happens after each step is carried out.

Co-prime Numbers for Safekeeping Treasures Class 6 Notes

Which pairs are safe?
Let us go back to the treasure-finding game. This time, treasures are kept on two numbers. Jumpy gets the treasures only if he can reach both numbers with the same jump size. There is also a new rule a jump size of 1 is not allowed. Where should Grumpy place the treasures so that Jumpy cannot reach both the treasures?

Will placing the treasure on 12 and 26 work? No! If the jump size is chosen to be 2, then Jumpy will reach both 12 and 26. What about 4 and 9? Jumpy cannot reach both using any jump size other than 1. So, Grumpy knows that the pair 4 and 9 are safe. Check if these pairs are safe:
a. 15 and 39
b. 4 and 15
c. 18 and 29
d. 20 and 55
What is special about safe pairs? They don’t have any common factor other than 1. Two numbers are said to be co-prime to each other if they have no common factor other than 1.

Example: As 15 and 39 have 3 as a common factor, they are not co-prime. But 4 and 9 are co-prime.

Prime Factorisation Class 6 Notes

Checking if two numbers are co-prime
Teacher: Are 56 and 63 co-prime?
Anshu and Guna: If they have a common factor other than 1, then they are not co-prime. Let us check.
Anshu: I can write 56 = 14 × 4 and 63 = 21 × 3. So, 14 and 4 are factors, of 56. Further, 21 and 3 are factors of 63. So, there are no common factors. The numbers are co-prime.
Guna: Hold on. I can also write 56 = 7 × 8 and 63 = 9 × 7. We see that 7 is a factor of both numbers, so, they are not co-prime.
Guna is right, as 7 is a common factor.
But where did Anshu go wrong?
Writing 56 = 14 × 4 tells us that 14 and 4 are both factors of 56, but it does not tell all the factors of 56. The same holds for the factors of 63.

Try another example: 80 and 63. There are many ways to factorize both numbers.
80 = 40 × 2 = 20 × 4 = 10 × 8 = 16 × 5 = ???
63 = 9 × 7 = 3 × 21 = ???
We have written ‘???’ to say that there may be more ways to factorize these numbers. But if we take any of the given factorizations, for example, 80 = 16 × 5 and 63 = 9 × 7, then there are no common factors.
Can we conclude that 80 and 63 are co-prime?
As Anshu’s mistake above shows, we cannot conclude that there may be other ways to factorize the numbers. What this means is that we need a more systematic approach to check if two numbers are co-prime.

Prime Factorisation
Take a number such as 56. It is composite, as we saw that it can be written as 56 = 4 × 14. So, both 4 and 14 are factors of 56. Now take one of these, say 14. It is also composite and can be written as 14 = 2 × 7. Therefore, 56 = 4 × 2 × 7. Now, 4 is composite and can be written as 4 = 2 × 2. Therefore, 56 = 2 × 2 × 2 × 7. All the factors appearing here, 2 and 7, are prime numbers. So, we cannot divide them further.

In conclusion, we have written 56 as a product of prime numbers. This is called a prime factorization of 56. The individual factors are called prime factors. For example, the prime factors of 56 are 2 and 7. Every number greater than 1 has a prime factorization. The idea is the same: Keep breaking the composite numbers into factors till only primes are left. The number 1 does not have any prime factorization. It is not divisible by any prime number. What is the prime factorization of a prime number like 7? It is just 7 (we cannot break it down any further).

Let us see a few more examples. By going through different ways of breaking down the number, we wrote 63 as 3 × 3 × 7 and as 3 × 7 × 3. Are they different? Not really! The same prime numbers 3 and 7 occur in both cases. Further, 3 appears two times in both and 7 appears once. Here, you see four different ways to get a prime factorization of 36. Observe that in all four cases, we get two 2s and two 3s.

Multiply back to see that you get 36 in all four cases. For any number, it is a remarkable fact that there is only one prime factorization, except that the prime factors may come in different orders. As we explain below, the order is not important. However, as we saw in these examples, there are many ways to arrive at the prime factorization!

Does the order matter?
Using this diagram, can you explain why 30 = 2 × 3 × 5, no matter which way you multiply 2, 3, and 5?

When multiplying numbers, we can do so in any order. The result is the same. That is why, when two 2s and two 3s are multiplied in any order, we get 36. In a later class, we shall study this under the names of commutativity and associativity of multiplication. Thus, the order does not matter. Usually, we write the prime numbers in increasing order.
For example, 225 = 3 × 3 × 5 × 5 or 30 = 2 × 3 × 5.

Prime factorization of a product of two numbers
When we find the prime factorization of a number, we first write it as a product of two factors. For example, 72 = 12 × 6. Then, we find the prime factorization of each of the factors.
In the above example, 12 = 2 × 2 × 3 and 6 = 2 × 3. Now, can you say what the prime factorization of 72 is?
The prime factorization of the original number is obtained by putting these together.
72 = 2 × 2 × 3 × 2 × 3
We can also write this as 2 × 2 × 2 × 3 × 3. Multiply and check that you get 72 back! Observe how many times each prime factor occurs in the factorization of 72. Compare it with how many times it occurs in the factorizations of 12 and 6 put together. Prime factorization is of fundamental importance in the study of numbers. Let us discuss two ways in which it can be useful.

Using prime factorization to check if two numbers are co-prime
Let us again take the numbers 56 and 63. How can we check if they are co-prime? We can use the prime factorization of both numbers.
56 = 2 × 2 × 2 × 7 and 63 = 3 × 3 × 7.
Now, we see that 7 is a prime factor of 56 as well as 63. Therefore, 56 and 63 are not co-prime.
What about 80 and 63? Their prime factorizations are as follows:
80 = 2 × 2 × 2 × 2 × 5 and 63 = 3 × 3 × 7.

There are no common prime factors. Can we conclude that they are co-prime? Suppose they have a common factor that is composite. Would the prime factors of this composite common factor appear in the prime factorization of 80 and 63? Therefore, we can say that if there are no common prime factors, then the two numbers are co-prime. Let us see some examples.

Example 1: Consider 40 and 231. Their prime factorizations are as follows:
40 = 2 × 2 × 2 × 5 and 231 = 3 × 7 × 11
We see that there are no common primes that divide both 40 and 231. Indeed, the prime factors of 40 are 2 and 5 while, the prime factors of 231 are 3, 7, and 11. Therefore, 40 and 231 are co-prime!

Example 2: Consider 242 and 195. Their prime factorizations are as follows:
242 = 2 × 11 × 11 and 195 = 3 × 5 × 13.
The prime factors of 242 are 2 and 11. The prime factors of 195 are 3, 5, and 13. There are no common prime factors. Therefore, 242 and 195 are co-prime.

Using prime factorization to check if one number is divisible by another
We can say that if one number is divisible by another, the prime factorization of the second number is included in the prime factorization of the first number. We say that 48 is divisible by 12 because when we divide 48 by 12, the remainder is zero. How can we check if one number is divisible by another without carrying out long division?

Example 1: Is 168 divisible by 12? Find the prime factorizations of both:
168 = 2 × 2 × 2 × 3 × 7 and 12 = 2 × 2 × 3.
Since we can multiply in any order, now it is clear that,
168 = 2 × 2 × 3 × 2 × 7 = 12 × 14
Therefore, 168 is divisible by 12.

Example 2: Is 75 divisible by 21? Find the prime factorizations of both:
75 = 3 × 5 × 5 and 21 = 3 × 7.
As we saw in the discussion above, if 75 was a multiple of 21, then all prime factors of 21 would also be prime factors of 75.
However, 7 is a prime factor of 21 but not a prime factor of 75. Therefore, 75 is not divisible by 21.

Example 3: Is 42 divisible by 12? Find the prime factorizations of both:
42 = 2 × 3 × 7 and 12 = 2 × 2 × 3.
All prime factors of 12 are also prime factors of 42. However, the prime factorization of 12 is not included in the prime factorization of 42. This is because 2 occurs twice in the prime factorization of 12 but only once in the prime factorization of 42. This means that 42 is not divisible by 12. We can say that if one number is divisible by another, then the prime factorization of the second number is included in the prime factorization of the first number.

Divisibility Tests Class 6 Notes

So far, we have been finding factors of numbers in different contexts, including to determine if a number is prime or not, or if a given pair of numbers is co-prime or not. It is easy to find factors of small numbers. How do we find factors of a large number?
Let us take 8560. Does it have any factors from 2 to 10 (2, 3, 4, 5, …, 9, 10)?
It is easy to check if some of these numbers are factors or not without doing long division. Can you find them?

Divisibility by 10
Let us take 10. Is 8560 divisible by 10? This is another way of asking if 10 is a factor of 8560. For this, we can look at the pattern in the multiples of 10.
The first few multiples of 10 are 10, 20, 30, 40, … Continue this sequence and observe the pattern.
Is 125 a multiple of 10? Will this number appear in the previous sequence? Why or why not?
Can you now answer if 8560 is divisible by 10?
Consider this statement: Numbers that are divisible by 10 are those that end with ‘0’. Do you agree?

Divisibility by 5
The number 5 is another number whose divisibility can easily be checked. How do we do it?
Explore by listing down the multiples: 5, 10, 15, 20, 25, …
What do you observe about these numbers? Do you see a pattern in the last digit?
What is the largest number less than 399 that is divisible by 5?
Is 8560 divisible by 5?
Consider this statement: Numbers that are divisible by 5 are those that end with either a ‘0’ or a ‘5’. Do you agree?

Divisibility by 2
The first few multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,…
What do you observe? Do you see a pattern in the last digit?
Is 682 divisible by 2? Can we answer this without doing the long division?
Is 8560 divisible by 2? Why or why not?
Consider this statement: Numbers that are divisible by 2 are those that end with ‘0’, ‘2’, ‘4, ‘6’ or ‘8’. Do you agree?
What are all the multiples of 2 between 399 and 411?

Divisibility by 4
Checking if a number is divisible by 4 can also be done easily!
Look at its multiples: 4, 8, 12, 16, 20, 24, 28, 32, ..…
Are you able to observe any patterns that can be used?
The multiples of 10, 5, and 2 have a pattern in their last digits which we can use to check for divisibility.
Similarly, can we check if a number is divisible by 4 by looking at the last digit?

It does not work! Look at 12 and 22. They have the same last digit, but 12 is a multiple of 4 while 22 is not. Similarly, 14 and 24 have the same last digit, but 14 is not a multiple of 4 while 24 is. Similarly, 16 and 26 or 18 and 28. What this means is that by looking at the last digit, we cannot tell whether a number is a multiple of 4.

Divisibility by 8
Interestingly, even checking for divisibility by 8 can be simplified. Can the last two digits be used for this?
We have seen that long division is not always needed to check if a number is a factor or not.
We have made use of certain observations to come up with simple methods for 10, 5, 2, 4, 8.
Do we have such simple methods for other numbers as well?
We will discuss simple methods to test divisibility by 3, 6, 7, and 9 in later classes!

Fun with Numbers Class 6 Notes

Special Numbers
There are four numbers in this box. Which number looks special to you? Why do you say so?

Look at what Guna’s classmates have to share:

• Karnawati says, “9 is special because it is a single-digit number whereas all the other numbers are 2-digit numbers.
• Gurupreet says, “9 is special because it is the only number that is a multiple of 3”.
• Murugan says, “16 is special because it is the only even number and also the only multiple of 4”.
• Gopika says, “25 is special as it is the only multiple of 5”.
• Yadnyikee says, “43 is special because it is the only prime number”.
• Radha says, “43 is special because it is the only number that is not a square”.

Class 6 Maths Notes

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Class 6 Maths Chapter 6 Notes Perimeter and Area

Class 6 Maths Notes Chapter 6 – Class 6 Perimeter and Area Notes

→ The perimeter of a polygon is the sum of the lengths of all its sides.

→ The perimeter of a rectangle is twice the sum of its length and width.

→ The perimeter of a square is four times the length of any one of its sides.

→ The area of a closed figure is the measure of the region enclosed by the figure.

→ Area is generally measured in square units.

→ The area of a rectangle is its length times its width. The area of a square is the length of any one of its sides multiplied by itself.

→ Two closed fiures can have the same area with different perimeters, or the same perimeter with different areas.

→ Areas of regions can be estimated (or even determined exactly) by breaking up such regions into unit squares, or into more generally shaped rectangles and triangles whose areas can be calculated.

Perimeter Class 6 Notes

Do you remember what the perimeter of a closed plane figure is? Let us refresh our understanding!
The perimeter of any closed plane figure is the distance covered along its boundary when you go around it once.
For a polygon, i.e., a closed plane figure made up of line segments, the perimeter is simply the sum of the lengths of its all sides, i.e., the total distance along its outer boundary.
The perimeter of a polygon = the sum of the lengths of its all sides.
Let us revise the formulas for the perimeter of rectangles, squares, and triangles.

Perimeter of a Rectangle
Consider a rectangle ABCD whose length and breadth are 12 cm and 8 cm, respectively. What is its perimeter?

The perimeter of the rectangle = Sum of the lengths of its four sides
= AB + BC + CD + DA
= AB + BC + AB + BC
Opposite sides of a rectangle are always equal. So, AB = CD and AD = BC
= 2 × AB + 2 × BC
= 2 × (AB + BC)
= 2 × (12 cm + 8 cm)
= 2 × (20 cm)
= 40 cm.
From this example, we see that
Perimeter of a rectangle = length + breadth + length + breadth.
The perimeter of a rectangle = 2 × (length + breadth).
The perimeter of a rectangle is twice the sum of its length and breadth.

Perimeter of a Square
Debojeet wants to put colored tape all around a square photo frame of side 1m as shown. What will be the length of the colored tape he requires? Since Debojeet wants to put the colored tape all around the square photo frame, he needs to find the perimeter of the photo frame.

Thus, the length of the tape required = perimeter of the square
= sum of the lengths of all four sides of the square
= 1 m + 1 m + 1 m + 1 m
= 4 m.
Now, we know that all four sides of a square are equal in length.
Therefore, in place of adding the lengths of each side, we can simply multiply the length of one side by 4.
Thus, the length of the tape required = 4 × 1 m = 4 m.
From this example, we see that
The perimeter of a square = 4 × length of a side.
The perimeter of a square is quadruple the length of its side.

Perimeter of a Triangle
Consider a triangle having three given sides of lengths 4 cm, 5 cm, and 7 cm. Find its perimeter.

The perimeter of the triangle = 4 cm + 5 cm + 7 cm = 16 cm.
The perimeter of a triangle = the sum of the lengths of its three sides.

Example 1.
Akshi wants to put lace all around a rectangular tablecloth that is 3 m long and 2 m wide. Find the length of the lace required.
Solution:

Length of the rectangular table cover = 3 m.
The breadth of the rectangular table cover = 2 m.
Akshi wants to put lace all around the tablecloth.
Therefore, the length of the lace required will be the perimeter of the rectangular tablecloth.
Now, the perimeter of the rectangular tablecloth = 2 × (length + breadth)
= 2 × (3 m + 2 m)
= 2 × 5 m
= 10 m.
Hence, the length of the lace required is 10 m.

Example 2.
Find the distance travelled by Usha if she takes three rounds of a square park of side 75 m.
Solution:

The perimeter of the square park = 4 × length of a side = 4 × 75 m = 300 m.
Distance covered by Usha in one round = 300 m.
Therefore, the total distance travelled by Usha in three rounds = 3 × 300 m = 900 m.

Matha Pachchi!

Each track is a rectangle.
Akshi’s track has a length of 70 m and a breadth of 40 m.
Running one complete round on this track would cover 220 m.
i.e., 2 × (70 + 40) m = 220 m.
This is the distance covered by Akshi in one round.

Deep Dive:
In races, usually, there is a common finish line for all the runners. Here are two square running tracks with an inner track of 100 m on each side and an outer track of 150 m on each side. The common finishing line for both runners is shown by the flags in the figure which are in the center of one of the sides of the tracks.

If the total race is of 350 m, then we have to find out where the starting positions of the two runners should be on these two tracks so that they both have a common finishing line after they run for 350 m. Mark the starting points of the runner on the inner track as ‘A’ and the runner on the outer track as ‘B’.

Estimate and Verify
Take a rough sheet of paper or a sheet of newspaper. Make a few random shapes by cutting the paper in different ways. Estimate the total length of the boundaries of each shape then use a scale or measuring tape to measure and verify the perimeter for each shape.

Akshi says that the perimeter of this triangle shape is 9 units. Toshi says it can’t be 9 units and the perimeter will be more than 9 units. What do you think?

This figure has lines of two different unit lengths. Measure the lengths of a red line and a blue line; are they the same? We will call the red lines – straight lines and the blue lines – diagonal lines. So, the perimeter of this triangle is 6 straight units + 3 diagonal units. We can write this in a short form as 6s + 3d units.

Perimeter of a Regular Polygon
Like squares, closed figures that have all sides and all angles equal are called regular polygons. We studied the sequence of regular polygons as ‘Shape Sequence’ #1 in Chapter 1. Examples of regular polygons are the equilateral triangle (where all three sides and all three angles are equal), regular pentagon (where all five sides and all five angles are equal), etc.

Perimeter of an Equilateral Triangle
We know that for any triangle its perimeter is the sum of all three sides.

Using this understanding, we can find the perimeter of an equilateral triangle.
The perimeter of an equilateral triangle = AB + BC + AC = AB + AB + AB = 3 times the length of one side.
The perimeter of an equilateral triangle = 3 × length of a side.

Split and Rejoin
A rectangular paper chit of dimension 6 cm × 4 cm is cut as shown into two equal pieces. These two pieces are joined in different ways.


For example, the arrangement a. has a perimeter of 28 cm.

Area Class 6 Notes

We have studied the areas of closed fields (regular and irregular) in previous grades. Let us recall some key points.
The amount of region enclosed by a closed figure is called its area.
Let’s see some real-life problems related to these ideas.

Example 1.
A floor is 5 m long and 4 m wide. A square carpet of sides 3 m is laid on the floor. Find the area of the floor that is not carpeted.
Solution:
Length of the floor = 5 m.
Width of the floor = 4 m.
Area of the floor = length × width = 5 m × 4 m = 20 sq m.
Length of the square carpet = 3 m.
Area of the carpet = length × length = 3 m × 3 m = 9 sq m.
Hence, the area of the floor laid with carpet is 9 sq m.
Therefore, the area of the floor that is not carpeted is an area of the floor minus the area of the floor laid with carpet = 20 sq m – 9 sq m = 11 sq m.

Example 2.
Four square flower beds each of side 4 m are in four corners on a piece of land 12 m long and 10 m wide. Find the area of the remaining part of the land.
Solution:
Length of the land (l) = 12 m.
Width of land (w) = 10 m.
Area of the whole land = l × w = 12 m × 10 m = 120 sq m.
The side length of each of the four square flower beds is (s) = 4 m.
Area of one flower bed = s × s = 4 m × 4 m = 16 sq m.
Hence, the area of the four flower beds = 4 × 16 sq m = 64 sq m.
Therefore, the area of the remaining part of the land is the area of the complete land minus the area of all four flower beds = 120 sq m – 64 sq m = 56 sq m.

Look at the figures below and guess which one of them has a larger area.

We can estimate the area of any simple closed shape by using a sheet of squared paper or graph paper where every square measures 1 unit × 1 unit or 1 square unit. To estimate the area, we can trace the shape onto a piece of transparent paper and place the same on a piece of squared or graph paper, and then follow the below conventions:

• The area of one full small square of the squared or graph paper is taken as 1 sq unit.
• Ignore portions of the area that are less than half a square.
• If more than half of a square is in a region, just count it as 1 sq unit.
• If exactly half the square is counted, take its area as 1/2 sq unit.

Let’s Explore!
Why is an area generally measured using squares?

Draw a circle on a graph sheet with a diameter (breadth) of length 3. Count the squares and use them to estimate the area of the circular region. As you can see, circles can’t be packed tightly without gaps in between. So, it is difficult to get an accurate measurement of area using circles as units. Here, the same rectangle is packed in two different ways with circles—the first one has 42 circles and the second one has 44 circles.

Area of a Triangle Class 6 Notes

Draw a rectangle on a piece of paper and draw one of its diagonals. Cut the rectangle along that diagonal and get two triangles. Check! whether the two triangles overlap each other exactly. Do they have the same area? Try this with more rectangles having different dimensions. You can check this for a square as well.

The area of triangle BAD is half of the area of the rectangle ABCD.

Area of triangle ABE = Area of triangle AEF + Area of triangle BEF.
Here, the area of triangle AEF = half of the area of rectangle AFED.
Similarly, the area of triangle BEF = half of the area of rectangle BFEC.
Thus, the area of triangle ABE = half of the area of rectangle AFED + half of the area of rectangle BFEC
= half of the sum of the areas of the rectangles AFED and BFEC
= half of the area of rectangle ABCD.

Making it ‘More’ or ‘Less’
Observe these two figures. Is there any similarity or difference between the two?

Using 9 unit squares (having an area of 9 sq units), we have made figures with two different perimeters—the first figure has a perimeter of 12 units and the second has a perimeter of 20 units. Arrange or draw different figures with 9 sq units to get other perimeters. Each square should align with at least one other square on at least one side completely and together all squares should form a single connected figure with no holes.

Let’s do something tricky now! We have a figure below having a perimeter of 24 units.

Without calculating all over again, observe, think, and find out what will change in the perimeter if a new square is attached as shown on the right. Experiment with placing this new square at different places and think what the change in perimeter will be. Can you place the square so that the perimeter: a) increases; b) decreases; and c) stays the same?

Class 6 Maths Notes

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NCERT Class 2 Maths Joyful Mathematics Worksheets Pdf Free Download

• Chapter 1 A Day at the Beach (Counting in Groups) Worksheet
• Chapter 2 Shapes Around Us (3D Shapes) Worksheet
• Chapter 3 Fun with Numbers (Numbers 1 to 100) Worksheet
• Chapter 4 Shadow Story (Togalu) (2D Shapes) Worksheet
• Chapter 5 Playing with Lines (Orientations of a Line) Worksheet
• Chapter 6 Decoration for Festival (Addition and Subtraction) Worksheet
• Chapter 7 Rani’s Gift (Measurement) Worksheet
• Chapter 8 Grouping and Sharing (Multiplication and Division) Worksheet
• Chapter 9 Which Season is it? (Measurement of Time) Worksheet
• Chapter 10 Fun at the Fair (Money) Worksheet
• Chapter 11 Data Handling Worksheet

NCERT Solutions for Class 2

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NCERT Class 1 Maths Joyful Mathematics Worksheets Pdf Free Download

• Chapter 1 Finding the Furry Cat! (Pre-number Concepts) Worksheet
• Chapter 2 What is Long? What is Round? (Shapes) Worksheet
• Chapter 3 Mango Treat (Numbers 1 to 9) Worksheet
• Chapter 4 Making 10 (Numbers 10 to 20) Worksheet
• Chapter 5 How Many? (Addition and Subtraction of Single Digit Numbers) Worksheet
• Chapter 6 Vegetable Farm (Addition and Subtraction up to 20) Worksheet
• Chapter 7 Lina’s Family (Measurement) Worksheet
• Chapter 8 Fun with Numbers (Numbers 21 to 99) Worksheet
• Chapter 9 Utsav (Patterns) Worksheet
• Chapter 10 How do I Spend My Day? (Time) Worksheet
• Chapter 11 How Many Times? (Multiplication) Worksheet
• Chapter 12 How Much Can We Spend? (Money) Worksheet
• Chapter 13 So Many Toys (Data Handling) Worksheet

NCERT Solutions for Class 1

The post Joyful Mathematics Class 1 Worksheets Pdf Free Download appeared first on Learn CBSE.

NCERT Class 1 English Mridang Worksheets Pdf Free Download

Mridang Workbook Class 1 Unit 1 My Family and Me

• Chapter 1 Two Little Hands Worksheet
• Chapter 2 Greetings Worksheet

Mridang Class 1 Worksheet Unit 2 Life Around Us

• Chapter 3 Picture Time Worksheet
• Chapter 4 The Cap-seller and the Monkeys Worksheet
• Chapter 5 A Farm Worksheet

Mridang NCERT Book Class 1 Worksheet Unit 3 Food

• Chapter 6 Fun with Pictures Worksheet
• Chapter 7 The Food We Eat Worksheet

Class 1 English Mridang Worksheet Unit 4 Seasons

• Chapter 8 The Four Seasons Worksheet
• Chapter 9 Anandi’s Rainbow Worksheet

NCERT Solutions for Class 1

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NCERT Class 2 English Mridang Worksheets Pdf Free Download

Class 2 Mridang Worksheet Unit 1 Fun with Friends

• Chapter 1 My Bicycle Worksheet
• Chapter 2 Picture Reading Worksheet

Mridang Workbook Class 2 Unit 2 Welcome to My World

• Chapter 3 It Is Fun Worksheet
• Chapter 4 Seeing without Seeing Worksheet

Mridang Class 2 Worksheet Unit 3 Going Places

• Chapter 5 Come Back Soon Worksheet
• Chapter 6 Between Home and School Worksheet
• Chapter 7 This is My Town Worksheet

Mridang NCERT Book Class 2 Worksheet Unit 4 Life Around Us

• Chapter 8 A Show of Clouds Worksheet
• Chapter 9 My Name Worksheet
• Chapter 10 The Crow Worksheet
• Chapter 11 The Smart Monkey Worksheet

Class 2 English Mridang Worksheet Unit 5 Harmony

• Chapter 12 Little Drops of Water Worksheet
• Chapter 13 We Are All Indians Worksheet

NCERT Solutions for Class 2

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NCERT Class 2 Hindi Sarangi Worksheets Pdf Free Download

Sarangi Class 2 Worksheet इकाई 1 परिवार

• Chapter 1 नीम की दादी Worksheet
• Chapter 2 घर Worksheet
• Chapter 3 माला की चाँदी की पायल Worksheet
• Chapter 4 माँ Worksheet
• Chapter 5 थाथू और मैं Worksheet
• Chapter 6 चींटा Worksheet
• Chapter 7 टिल्लू जी Worksheet

Class 2 Hindi Sarangi Worksheet इकाई 2 रंग ही रंग

• Chapter 8 तीन दोस्त Worksheet
• Chapter 9 दुनिया रंग-बिरंगी Worksheet
• Chapter 10 कौन Worksheet
• Chapter 11 बैंगनी जोजो Worksheet
• Chapter 12 तोसिया का सपना Worksheet

Sarangi Workbook Class 2 इकाई 3 हरी-भरी धरती

• Chapter 13 तालाब Worksheet
• Chapter 14 बीज Worksheet
• Chapter 15 किसान Worksheet
• Chapter 16 मूली Worksheet
• Chapter 17 बरसात और मेंढक Worksheet

Class 2 Sarangi Worksheet इकाई 4 मित्रता

• Chapter 18 शेर और चूहे की दोस्ती Worksheet
• Chapter 19 आउट Worksheet
• Chapter 20 छुप-छुपाई Worksheet
• Chapter 21 हाथी साइकिल चला रहा था Worksheet

Sarangi NCERT Book Class 2 Worksheet इकाई 5 आकाश

• Chapter 22 चार दिशाएँ Worksheet
• Chapter 23 चंदा मामा Worksheet
• Chapter 24 गिरे ताल में चंदा मामा Worksheet
• Chapter 25 सबसे बड़ा छाता Worksheet
• Chapter 26 बादल Worksheet

NCERT Solutions for Class 2

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NCERT Class 1 Hindi Sarangi Worksheets Pdf Free Download

Sarangi Class 1 Worksheet इकाई 1 परिवार

• Chapter 1 मीना का परिवार
• Chapter 2 दादा-दादी
• Chapter 3 रीना का दिन
• Chapter 4 रानी भी

Class 1 Hindi Sarangi Worksheet इकाई 2 जीव-जगत

• Chapter 5 मिठाई
• Chapter 6 तीन साथी
• Chapter 7 वाह, मेरे घोड़े!
• Chapter 8 खतरे में साँप

Sarangi Workbook Class 1 इकाई 3 हमारा खान-पान

• Chapter 9 आलू की सड़क
• Chapter 10 झूम-झूली
• Chapter 11 भुट्टे
• Chapter 12 फूली रोटी

Class 1 Sarangi Worksheet इकाई 4 त्योहार और मेले

• Chapter 13 मेला
• Chapter 14 बरखा और मेघा
• Chapter 15 होली
• Chapter 16 जन्मदिवस पर पेड़ लगाओ

Sarangi NCERT Book Class 1 Worksheet इकाई 5 हरी-भरी दुनिया

• Chapter 17 हवा
• Chapter 18 कितनी प्यारी है ये दुनिया
• Chapter 19 चाँद का बच्चा

NCERT Solutions for Class 1

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Class 6 Maths Chapter 7 Notes Fractions

Class 6 Maths Notes Chapter 7 – Class 6 Fractions Notes

→ Fraction as equal share: A fraction results when a whole number of units is divided into equal parts and shared equally.

→ Fractional Units: When one basic unit is divided into equal parts, each part is called a fractional unit.

→ Reading Fractions: In a fraction such as \(\frac {5}{6}\), 5 is called the numerator and 6 is called the denominator.

→ Mixed fractions contain a whole number part and a fractional part.

→ Number line: Fractions can be shown on a number line. Every fraction has a point associated with it on the number line.

→ Equivalent Fractions: When two or more fractions represent the same share/number, they are called equivalent fractions.

→ Lowest terms: A fraction whose numerator and denominator have no common factor other than 1 is said to be in lowest terms or its simplest form.

→ Brahmagupta’s method for adding fractions: When adding fractions, convert them into equivalent fractions with the same fractional unit (i.e., the same denominator), and then add the number of fractional units in each fraction to obtain the sum. This is accomplished by adding the numerators while keeping the same denominator.

→ Brahmagupta’s method for subtracting fractions: When subtracting fractions, convert them into equivalent fractions with the same fractional unit (i.e., the same denominator), and then subtract the number of fractional units. This is accomplished by subtracting the numerators while keeping the same denominator.

Recall that when a whole number of things are shared equally among some people, fractions tell us how much each share is.
Shabnam: Do you remember, if one roti is divided equally between two children, how much roti will each child get?

Mukta: Each child will get half a roti.

Shabnam:The fraction ‘one half’ is written as \(\frac{1}{2}\). We also sometimes read this as ‘one upon two.’

Mukta:If one roti is equally shared among 4 children, how much roti will one child get?

Shabnam: Each child’s share is \(\frac{1}{4}\) roti.

Mukta: And which is more \(\frac{1}{2}\) roti or \(\frac{1}{4}\) roti?

Shabnam: When 2 children share 1 roti equally, each child gets \(\frac{1}{2}\) roti. When 4 children share 1 roti equally, each child gets \(\frac{1}{4}\) roti. Since, in the second group more children share the same one roti, each child gets a smaller share. So, \(\frac{1}{2}\) roti is more than \(\frac{1}{4}\) roti.
\(\frac{1}{2}\) > \(\frac{1}{4}\)

Fractional Units and Equal Shares Class 6 Notes

Beni: Which fraction is greater? \(\frac{1}{5}\) or \(\frac{1}{9}\)?

Arvin: 9 is bigger than 5. So I would guess that \(\frac{1}{9}\) is greater than \(\frac{1}{5}\). Am I right?

Beni: No! That is a common mistake. Think of these fractions as shares.

Arvin: If one roti is shared among 5 children, each one gets a share of \(\frac{1}{5}\) roti. If one roti is shared among 9 children,
each one gets a share of \(\frac{1}{9}\) roti?

Beni: Exactly! Now think again – which share is higher?

Arvin: If I share with more people, I will get less. So \(\frac{1}{9}\) < \(\frac{1}{5}\).

Beni: You got it!
Oh, so \(\frac{1}{100}\) is bigger than \(\frac{1}{200}\)!

When one unit is divided into several equal parts, each part is called a fractional unit. These are all fractional units:
\(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \ldots, \frac{1}{10}, \ldots, \frac{1}{50}, \ldots, \frac{1}{100}\), etc.
We also sometimes refer to fractional units as ‘unit fractions.’

Fractions have been used and named in India since ancient times. In the Rig Veda, the fraction \(\frac{3}{4}\) is referred to as tripada. This has the same meaning as the words for \(\frac{3}{4}\) in many Indian languages today, e.g., ʻteen paavʼ in colloquial Hindi and ‘mukkaal’ in Tamil. Indeed, words for fractions used today in many Indian languages go back to ancient times.

Fractional Units as Parts of a Whole Class 6 Notes

The picture shows a whole chikki.

Below, is a picture of the chikki broken A whole chikki into 2 pieces is shown. How much of the original chikki is each piece?

We can see that the bigger piece has 3 pieces of \(\frac{1}{4}\) chikki in it. So, we can measure the bigger piece using the fractional unit \(\frac{1}{4}\). We see that the bigger piece is \(\frac{3}{4}\) chikki.

What is the fractional unit of chikki shown below?

Measuring Using Fractional Units Class 6 Notes

Take a strip of paper. We consider this paper strip to be one unit long.

Fold the strip into two equal parts and then open up the strip again. Taking the strip to be one unit in length, what are the lengths of the two new parts of the strip created by the crease?

What will you get if you fold the previously-folded strip again into two equal parts? You will now get four equal parts.

Fractional quantities can be measured using fractional units.
Let us look at another example,

We can describe how much the quantity is by collecting together the fractional units.

Reading Fractions
We usually read the fraction 34 as ‘three quarters’ or ‘three upon four’, but reading it as ‘3 times \(\frac{1}{4}\)’ helps us to understand the size of the fraction because it clearly shows what the fractional unit is (\(\frac{1}{4}\)) and how many such fractional units (3) there are. Recall what we call the top number and the bottom number of fractions. In the fraction \(\frac{5}{6}\), 5 is the numerator and 6 is the denominator.

Marking Fraction Lengths on the Number Line Class 6 Notes

We have marked lengths equal to 1, 2, 3, … units on the number line. Now, let us try to mark lengths equal to fractions on the number line. What is the length of the blue line? Write the fraction that gives the length of the blue line in the box.

The distance between 0 and 1 is one unit long. It is divided into two equal parts. So, the length of each part is \(\frac{1}{2}\) unit. So, this blue line is \(\frac{1}{2}\) unit long.

Mixed Fractions Class 6 Notes

Fractions greater than one
You marked some fractions on the number line earlier. Did you notice that the lengths of all the blue lines were less than one and the lengths of all the black lines were more than 1? Write down all the fractions you marked on the number line earlier. Now, let us classify these into two groups:

Did you notice something common between the fractions that are greater than 1?
In all the fractions that are less than 1 unit, the numerator is smaller than the denominator, while in the fractions that are more than 1 unit, the numerator is larger than the denominator.
We know that \(\frac{3}{2}\), \(\frac{5}{2}\) and \(\frac{7}{2}\) are all greater than 1 unit. But can we
see how many whole units they contain?

Writing fractions greater than one as mixed numbers
We saw that: \(\frac{3}{2}\) = 1 + \(\frac{1}{2}\)
We can similarly write other fractions. For example,

A mixed number/mixed fraction contains a whole number (called the whole part) and a fraction that is less than 1 (called the fractional part).

Equivalent Fractions Class 6 Notes

Using a fraction wall to find equal fractional lengths!
In the previous section, you used paper folding to represent various fractions using fractional units. Let us do some more activities with the same paper strips.

These are ‘equivalent fractions’ that denote the same length, but they are expressed in terms of different fractional units. Now, check whether \(\frac{1}{3}\) and \(\frac{2}{6}\) are equivalent fractions or not, using paper strips.
Make your fraction wall using such strips as given in the picture below!

We can extend this idea to make a fraction wall up to the fractional unit \(\frac{1}{10}\).

Understanding Equivalent Fractions Using Equal Shares

One roti was shared equally by four children. What fraction of the whole did each child get?
The adjoining picture shows the division of a roti among four children. The fraction of roti each child got is \(\frac{1}{4}\).

You can also express this event through division facts, addition facts, and multiplication facts.
The division fact is 1 ÷ 4 = \(\frac{1}{4}\)
The addition fact is 1 = \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\)
The multiplication fact is 1 = 4 × \(\frac{1}{4}\)

So, the share of each child is the same in both these situations!
Let us examine the shares of each child in the following situations.

• 1 roti is divided equally between 2 children.
• 2 rotis are divided equally among 4 children.
• 3 rotis are divided equally among 6 children.

Let us draw and share!
Did you notice that in each situation the share of every child is the same?
So, we can say that \(\frac{1}{2}=\frac{2}{4}=\frac{3}{6}\).

Fractions, where the shares are equal, are called ‘equivalent fractions’.
So, \(\frac{1}{2}\), \(\frac{2}{4}\), and \(\frac{3}{6}\) are all equivalent fractions.

In which group will each child get more chikki?
1 chikki divided between 2 children or 5 chikkis divided among 8 children.
Mukta: So, we must compare \(\frac{1}{2}\) and \(\frac{5}{8}\). Which is more?
Shabnam: Well, we have seen that \(\frac{1}{2}\) = \(\frac{4}{8}\); and clearly \(\frac{4}{8}\) < \(\frac{5}{8}\). So, the children for whom 5 chikkis are divided equally among 8 will get more than those children for whom 1 chikki is divided equally among 2. The children of the second group will get more chikki each.

What about the following groups? In which group will each child get more?
1 chikki divided between 2 children or 4 chikkis divided among 7 children.
Shabnam: The children of which group will get more chikki this time?
Mukta: We must compare \(\frac{1}{7}\) and \(\frac{4}{7}\).
Now \(\frac{1 \times 4}{2 \times 4}=\frac{4}{8}\) so, \(\frac{1}{2}=\frac{4}{8}\).
Shabnam: But why did you multiply the numerator and denominator by 4 again?
Mukta: You will see!


Now, decide in which of the two groups will each child get a larger share:

• Group 1: 3 glasses of sugarcane juice divided equally among 4 children.
• Group 2: 7 glasses of sugarcane juice divided equally among 10 children.
• Group 1: 4 glasses of sugarcane juice divided equally among 7 children.
• Group 2: 5 glasses of sugarcane juice divided equally among 7 children.

Which groups were easier to compare? Why?

Shabnam: To compare the first two groups, we have to find fractions equivalent to the fractions \(\frac{3}{4}\) and \(\frac{7}{10}\).
Mukta: How about \(\frac{6}{8}=\frac{3}{4}\) and \(\frac{21}{30}=\frac{7}{10}\)?
Shabnam: There is a condition. The fractional unit used for the two fractions has to be the same! Like \(\frac{2}{6}\) and \(\frac{3}{6}\) both use the same fractional unit \(\frac{1}{6}\) (i.e., the denominators are the same).
But \(\frac{6}{8}\) and \(\frac{21}{30}\) do not use the same fractional units (they have different denominators).
Mukta: Okay, so let us start making equivalent fractions then:
\(\frac{3}{4}=\frac{6}{8}=\frac{9}{12}=\frac{12}{16}=\frac{15}{20}\)… But when do I stop?
Shabnam: Got it! How about we go on till 4 × 10 = 40.
Mukta: You mean the product of the two denominators?
Sounds good! We have \(\frac{3}{4}\) and \(\frac{7}{10}\). The product of the two denominators (4 and 10) is 40.

Shabnam: So, fractions equivalent to \(\frac{3}{4}\) and \(\frac{7}{10}\) with the same fractional unit (same denominators) are \(\frac{30}{40}\) and \(\frac{28}{40}\), or \(\frac{15}{20}\) and \(\frac{14}{20}\).
Since clearly \(\frac{30}{40}\) > \(\frac{28}{40}\), we conclude that \(\frac{3}{4}\) > \(\frac{7}{10}\).

Expressing a Fraction in the Lowest Terms (or in its Simplest Form)
In any fraction, if its numerator and denominator have no common factor except 1, then the fraction is said to be in the lowest terms or its simplest form. In other words, a fraction is said to be in the lowest terms if its numerator and denominator are as small as possible. Any fraction can be expressed in the lowest terms by finding an equivalent fraction whose numerator and denominator are as small as possible. Let’s see how to express fractions in the lowest terms.

Example: Is the fraction \(\frac{16}{20}\) in lowest terms?
No, 4 is a common factor of 16 and 20. Let us reduce \(\frac{16}{20}\) to lowest terms.
We know that both 16 (numerator) and 20 (denominator) are divisible by 4. So, \(\frac{16 \div 4}{20 \div 4}=\frac{4}{5}\).
Now, there is no common factor between 4 and 5. Hence, \(\frac{16}{20}\) expressed in lowest terms is \(\frac{4}{5}\). So, \(\frac{4}{5}\) is called the simplest form of \(\frac{16}{20}\), since 4 and 5 have no common factor other than 1.

Expressing a fraction in the lowest terms can also be done in steps.
Suppose we want to express \(\frac{36}{60}\) in lowest terms.
First, we notice that both the numerator and denominator are even. So, we divide both by 2, and see that \(\frac{36}{60}=\frac{18}{30}\).
Both the numerator and denominator are even again, so we can divide them each by 2 again; we get \(\frac{18}{30}=\frac{9}{15}\)
We now notice that 9 and 15 are both multiples of 3, so we divide both by 3 to get \(\frac{9}{15}=\frac{3}{5}\)
Now, 3 and 5 have no common factor other than 1, so, \(\frac{36}{60}\) in lowest terms is \(\frac{3}{5}\).
Alternatively, we could have noticed that in \(\frac{36}{60}\), both the numerator and denominator are multiples of 12: we see that 36 = 3 × 12 and 60 = 5 × 12. Therefore, we could have concluded that \(\frac{36}{60}=\frac{3}{5}\) straight away.
Either method works and will give the same answer! But sometimes it can be easier to go in steps.

Comparing Fractions Class 6 Notes

Which is greater, \(\frac{4}{5}\) or \(\frac{7}{9}\)?
It can be difficult to compare two such fractions directly. However, we know how to find fractions equivalent to two fractions with the same denominator.

Let us see how we can use it:
\(\frac{4}{5}=\frac{4 \times 9}{5 \times 9}=\frac{36}{45}\)
\(\frac{7}{9}=\frac{7 \times 5}{8 \times 5}=\frac{35}{45}\)
Clearly, \(\frac{36}{45}>\frac{35}{45}\)
So, \(\frac{4}{5}>\frac{7}{9}\)!
Let us try this for another pair: \(\frac{7}{9}\) and \(\frac{17}{21}\).
63 is a common multiple of 9 and 21. We can then write:
\(\frac{7}{9}=\frac{7 \times 7}{9 \times 7}=\frac{49}{63}, \frac{17}{21}=\frac{17 \times 3}{21 \times 3}=\frac{51}{63}\)
Clearly, \(\frac{49}{63}<\frac{51}{63}\). So, \(\frac{7}{9}<\frac{17}{21}\)!

Let’s Summarise!
Steps to compare the sizes of two or more given fractions:

• Step 1: Change the given fractions to equivalent fractions so that they all are expressed with the same denominator/same fractional unit.
• Step 2: Now, compare the equivalent fractions by simply comparing the numerators, i.e., the number of fractional units each has.

Addition and Subtraction of Fractions Class 6 Notes

Meena’s father made some chikki. Meena ate \(\frac{1}{2}\) of it and her younger brother ate \(\frac{1}{4}\) of it. How
much of the total chikki did Meena and her brother eat together?


We can arrive at the answer by visualising it. Let us take a piece of chikki and divide it into two halves first like this. Meena ate \(\frac{1}{2}\) of it as shown in the picture.

Let us now divide the remaining half into two further halves as shown. Each of these pieces is \(\frac{1}{4}\) of the whole chikki.

Meena’s brother ate \(\frac{1}{4}\) of the whole chikki, as is shown in the picture.

The total chikki eaten is \(\frac{1}{2}\) (by Meena) and \(\frac{1}{4}\) (by her brother)
The total chikki eaten is = \(\frac{1}{2}+\frac{1}{4}\)
= \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\)
= 3 × \(\frac{1}{4}\)
= \(\frac{3}{4}\)
How much of the total chikki is remaining?

Adding Fractions with the Same Fractional Unit or Denominator
Example: Find the sum of \(\frac{2}{5}\) and \(\frac{1}{5}\).
Let us represent both using the rectangular strips. In both fractions, the fractional unit is the same \(\frac{1}{5}\), so, each strip will be divided into 5 equal parts. So \(\frac{2}{5}\) will be represented as—

And \(\frac{1}{5}\) will be represented as—

Adding the two given fractions is the same as finding out the total number of shaded parts, each of which represents the same fractional unit \(\frac{1}{5}\).
In this case, the total number of shaded parts is 3. Since each shaded part represents the fractional unit \(\frac{1}{5}\), we see that the 3 shaded parts together represent the fraction \(\frac{3}{5}\).

Therefore, \(\frac{2}{5}+\frac{1}{5}=\frac{3}{5}\)!

Example: Find the sum of \(\frac{4}{7}\) and \(\frac{6}{7}\).
Let us represent both again using the rectangular strip model. Here in both fractions, the fractional unit is the same, i.e., \(\frac{1}{7}\), so each strip will be divided into 7 equal parts. Then \(\frac{4}{7}\) will be represented as—

and \(\frac{6}{7}\) will be represented as—

In this case, the total number of shaded parts is 10, and each shaded part represents the fractional unit \(\frac{1}{7}\), so, the 10 shaded parts together represent the fraction \(\frac{10}{7}\) as seen here.

Adding Fractions with different fractional Units or Denominators
Example: Find the sum of \(\frac{1}{4}\) and \(\frac{1}{3}\).
To add fractions with different fractional units, first convert the fractions into equivalent fractions with the same denominator/fractional unit. In this case, the common denominator can be made 3 × 4 = 12, i.e., we can find equivalent fractions with fractional unit \(\frac{1}{12}\).
Let us write the equivalent fraction for each given fraction.
\(\frac{1}{4}=\frac{1 \times 3}{4 \times 3}=\frac{3}{12}, \quad \frac{1}{3}=\frac{1 \times 4}{3 \times 4}=\frac{4}{12}\)
Now, \(\frac{3}{12}\) and \(\frac{4}{12}\) have the same fractional unit, i.e., \(\frac{1}{12}\).
Therefore, \(\frac{1}{4}+\frac{1}{3}=\frac{3}{12}+\frac{4}{12}=\frac{7}{12}\).
This method of addition, which works for adding any number of fractions, was first explicitly described in general by Brahmagupta in the year 628 CE! We will describe the history of the development of fractions in more detail later in the chapter. For now, we simply summarise the steps in Brahmagupta’s method for addition of fractions.

Brahmagupta’s method for adding fractions

• Find equivalent fractions so that the fractional unit is common for all fractions. This can be done by finding a common multiple of the denominators (e.g., the product of the denominators, or the smallest common multiple of the denominators).
• Add these equivalent fractions with the same fractional units. This can be done by adding the numerators and keeping the same denominator.
• Express the result in the lowest terms if needed.

Let us carry out another example of Brahmagupta’s method.
Example 1: Find the sum of \(\frac{2}{3}\) and \(\frac{1}{5}\).
The denominators of the given fractions are 3 and 5. The lowest common multiple of 3 and 5 is 15. Then we see that \(\frac{2}{3}=\frac{2 \times 5}{3 \times 5}=\frac{10}{15}, \frac{1}{5}=\frac{1 \times 3}{5 \times 3}=\frac{3}{15}\)
Therefore, \(\frac{2}{3}+\frac{1}{5}=\frac{10}{15}+\frac{3}{15}=\frac{13}{15}\).

Example 2: Find the sum of \(\frac{1}{6}\) and \(\frac{1}{3}\).
The smallest common multiple of 6 and 3 is 6.
\(\frac{1}{6}\) will remain \(\frac{1}{6}\).
\(\frac{1}{3}=\frac{1 \times 2}{3 \times 2}=\frac{2}{6}\)
Therefore, \(\frac{1}{6}+\frac{1}{3}=\frac{1}{6}+\frac{2}{6}=\frac{3}{6}\).
The fraction \(\frac{3}{6}\) can now be re-expressed in lowest terms, if desired. This can be done by dividing both the numerator and denominator by 3 (the biggest common factor of 3 and 6):
\(\frac{3}{6}=\frac{3 \div 3}{6 \div 3}=\frac{1}{2}\)
Therefore, \(\frac{1}{6}+\frac{1}{3}=\frac{1}{2}\).

Subtraction of Fractions with the same Fractional Unit or Denominator
Brahmagupta’s method also applies when subtracting fractions!
Let us start with the problem of subtracting \(\frac{4}{7}\) from \(\frac{6}{7}\), i.e., what is \(\frac{6}{7}\) – \(\frac{4}{7}\)?
To solve this problem, we can again use the rectangular strips. In both fractions, the fractional unit is the same i.e. \(\frac{1}{7}\). Let us first represent the bigger fraction using a rectangular strip model as shown:

Each shaded part represents \(\frac{1}{7}\). Now, we need to subtract \(\frac{4}{7}\). To do this let us remove 4 of the shaded parts:

So, we are left with 2 shaded parts, i.e., \(\frac{6}{7}-\frac{4}{7}=\frac{2}{7}\).
Try doing this same exercise using the number line.

Subtraction of Fractions with Different Fractional Units or Denominators
Example: What is \(\frac{3}{4}\) – \(\frac{2}{3}\)?
As we already know the procedure for subtraction of fractions with the same fractional units, let us convert each of the given fractions into equivalent fractions with the same fractional units.

Brahmagupta’s method for subtracting two fractions:

• Convert the given fractions into equivalent fractions with the same fractional unit, i.e., the same denominator.
• Carry out the subtraction of fractions having the same fractional units. This can be done by subtracting the numerators and keeping the same denominator.
• Simplify the result into the lowest terms if needed.

A Pinch of History Class 6 Notes

Do you know what a fraction was called in ancient India? It was called binna in Sanskrit, which means ‘broken’. It was also called bhaga or ansha meaning ‘part’ or ‘piece’. The way we write fractions today, globally, originated in India. In ancient Indian mathematical texts, such as the Bakshali manuscript (from around the year 300 CE), when they wanted to write \(\frac{1}{2}\), they wrote it as \(\frac{1}{2}\) which is indeed very similar to the way we write it today! This method of writing and working with fractions continued to be used in India for the next several centuries, including by Aryabhata (499 CE), Brahmagupta (628 CE), Sridharacharya (c. 750 CE), and Mahaviracharya (c. 850 CE), among others. The line segment between the numerator and denominator in ‘\(\frac{1}{2}\)’ and in other fractions was later introduced by the Moroccan mathematician Al-Hassar (in the 12th century). Over the next few centuries, the notation then spread to Europe and around the world.

Fractions had also been used in other cultures such as the ancient Egyptian and Babylonian civilizations, but they primarily used only fractional units, that is, fractions with a 1 in the numerator. More general fractions were expressed as sums of fractional units, now called ‘Egyptian fractions’. Writing numbers as the sum of fractional units, e.g., \(\frac{19}{24}=\frac{1}{2}+\frac{1}{6}+\frac{1}{8}\), can be quite an art and leads to beautiful puzzles. We will consider one such puzzle below.

General fractions (where the numerator is not necessarily 1) were first introduced in India, along with their rules of arithmetic operations like addition, subtraction, multiplication, and even division of fractions. The ancient Indian treatises called the ‘Sulbasutras’ show that even during Vedic times, Indians had discovered the rules for operations with fractions. General rules and procedures for working with and computing with fractions were first codified formally and in a modern form by Brahmagupta.

Brahmagupta’s methods for working with and computing with fractions are still what we use today. For example, Brahmagupta described how to add and subtract fractions as follows:
“By the multiplication of the numerator and the denominator of each of the fractions by the other denominators, the fractions are reduced to a common denominator. Then, in case of addition, the numerators (obtained after the above reduction) are added. In case of subtraction, their difference is taken.’’ (Brahmagupta, Brahmasphutasiddhanta, Verse 12.2, 628 CE)

The Indian concepts and methods involving fractions were transmitted to Europe via the Arabs over the next few centuries and they came into general use in Europe around the 17th century and then spread worldwide.

It is easy to add up fractional units to obtain the sum 1, if one uses the same fractional unit,
e.g., \(\frac{1}{2}+\frac{1}{2}=1, \frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1, \frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=1\), etc.
However, can you think of a way to add fractional units that are all different to get 1?
It is not possible to add two different fractional units to get 1.
The reason is that \(\frac{1}{2}\) is the largest fractional unit, and \(\frac{1}{2}\) + \(\frac{1}{2}\) = 1.
To get different fractional units, we would have to replace at least one of the \(\frac{1}{2}\)’s with some smaller fractional unit – but then the sum would be less than 1! Therefore, two different fractional units can’t add up to 1. We can try to look instead for a way to write 1 as the sum of three different fractional units.

Can you find three different fractional units that add up to 1?
It turns out there is only one solution to this problem (up to changing the order of the 3 fractions)! Can you find it? Try to find it before reading further.

Here is a systematic way to find the solution. We know that \(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\) = 1. To get the fractional units to be different, we will have to increase at least one of the \(\frac{1}{3}\)’s and decrease at least one of the other \(\frac{1}{3}\)’s to compensate for that increase. The only way to increase \(\frac{1}{3}\) to another fractional unit is to replace it with \(\frac{1}{2}\). So \(\frac{1}{2}\) must be one of the fractional units.

Now \(\frac{1}{2}+\frac{1}{4}+\frac{1}{4}\) = 1. To get the fractional units to be different, we will have to increase one of the \(\frac{1}{2}\)’s and decrease the other \(\frac{1}{4}\) to compensate for that increase. Now the only way to increase \(\frac{1}{4}\) to another fractional unit, that is different from \(\frac{1}{2}\), is to replace it with \(\frac{1}{3}\). So two of the fractions must be \(\frac{1}{2}\) and \(\frac{1}{3}\)! What must be the third fraction then, so that the three fractions add up to 1? This explains why there is only one solution to the above problem.

What if we look for four different fractional units that add up to 1?

————————————————————————————-

A fraction is a number representing a part of a whole. The whole may be a single object or a group of objects. The other way round, we can say that a fraction is an operation on a number.

A Fraction
A fraction means a part of a group or of a region. \(\frac { 3 }{ 4 }\) is a fraction. We read it as “Three-fourths”.
Here, 4 stands for the number of equal parts into which the whole has been divided and 3 stands for the number of equal parts which have been taken out. It is to be noted that while expressing a situation of counting parts to write a fraction, all parts must be equal. Here, 3 is called the numerator and 4 is called the denominator. Numerator and denominator can be identified for any fraction.

Fractions can be represented on a number line. Any fraction has a point associated with it on the number line.

We know that a fraction essentially has a numerator and a denominator which can be identified for any fraction under consideration. If the numerator is less than the denominator, then the fraction is called a proper fraction.
For example:
\(\frac { 1 }{ 2 }\), \(\frac { 3 }{ 4 }\), \(\frac { 5 }{ 8 }\), \(\frac { 0 }{ 2 }\), etc.

A fraction, whose numerator is bigger than the denominator, is called an improper fraction.
For example: \(\frac { 5 }{ 4 }\), \(\frac { 7 }{ 2 }\), \(\frac { 11 }{ 3 }\), etc.
Improper fractions can be written as a combination of a whole and a part, and are then called mixed fractions.
For example: \(\frac { 5 }{ 4 }\) when written as 1\(\frac { 1 }{ 4 }\) is called a mixed fraction.

To express an improper fraction as a mixed fraction, we divide the numerator by denominator to get the quotient and the remainder. Then the mixed fraction can be written as
Quotient \(\frac { Remainder }{ Divisor }\)

To express a mixed fraction as an improper fraction, we multiply the whole with the denominator and add the numerator to it. Then the mixed fraction can be written as
\(\frac { (Whole\quad \times \quad Denominator)+Numerator }{ Denominator }\)

Two fractions are said to be equivalent if they represent the same quantity. Each proper or improper fraction has infinitely many equivalent fractions. To find an equivalent fraction of a given fraction, we multiply or divide both the numerator and the denominator of the given fraction by the same nonzero number. If two fractions are equivalent, then the product of the numerator of the first and the denominator of the second is equal to the product of the denominator of the first and the numerator of the second. This rule proves to be useful to find equivalent fractions.

The simplest form of a Fraction
A fraction is said to be in the simplest (or lowest) form if its numerator and denominator have no common factor except 1, i.e., they are coprime natural numbers. Thus, to find the equivalent fraction in the simplest form, we find the HCF of the numerator and the denominator. Then, we divide both of them by their HCF and get the equivalent fraction in the simplest form.

Like Fractions
Fractions, whose denominators are the same, are called like fractions.
For example: \(\frac { 1 }{ 9 }\), \(\frac { 4 }{ 9 }\), \(\frac { 7 }{ 9 }\) are all like fractions whereas \(\frac { 1 }{ 19 }\) and \(\frac { 1 }{ 20 }\) are not like fractions. The latter are called unlike fractions as their denominators are different.

Comparing Fractions
For fractions with the same numerators, the smaller the denominator, the greater the fraction.
For fractions with the same denominators, the greater the numerator, the greater the fraction.

Comparing like fractions
Since the like fractions are fractions with the same denominator, therefore, to compare two like fractions, it is just sufficient to compare their numerators. Greater the numerator, greater the fraction.

We know that in two fractions with the same denominator, one fraction with a bigger numerator is larger. Since in such cases the denominators are the same, i.e., these are like fractions, therefore only numerators are to be compared, so these comparisons are easy to make.

Comparing unlike fractions
We know that unlike fractions are fractions with different denominators. Now if their numerators are the same, then smaller the denominator, greater the fraction. However, if the two fractions have different numerators, then we find equivalent fractions of both and choose one from each such that their denominators are the same.
Then for comparison, greater the numerator, greater the fraction. Thus, we can find the greater of the two given fractions with different numerators and different denominators.

In daily life situations, sometimes we have to add two fractions and sometimes we have to subtract one from the other. This is respectively known as addition and subtraction of fractions.
An example of the addition of fractions is as follows:
Manish bought 2\(\frac { 1 }{ 2 }\) kg sugar whereas Preeti bought 1\(\frac { 1 }{ 2 }\) kg sugar. Find the total amount of sugar bought by them.
An example of subtraction of fractions is as follows:
The teacher taught \(\frac { 2 }{ 5 }\) of the book. Yash revised \(\frac { 2 }{ 5 }\) more on his own. How much does he still have to revise?

Adding or Subtracting like fractions
Addition of two or more like fractions can be carried out as follows:
Step 1. Add the numerators.
Step 2. Retain the denominator (common).

Subtraction of two fractions can be carried out as follows:
Step 1. Subtract the smaller numerator from the bigger numerator.
Step 2. Retain the denominator (common).

Adding and Subtracting Fractions
To add and subtract fractions that do not have the same denominator, we convert them into equivalent fractions with the same denominators and then proceed as above.

Class 6 Maths Notes

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एषः कः ? एषा का ? एतत् किम् ? Class 6th Sanskrit Deepakam Chapter 2 Question Answer

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१. उदाहरणं दृष्ट्वा रिक्तस्थानानि पूरयन्तु ।

(उदाहरण देखकर रिक्त स्थानों को पूरा करें।)
यथा- बालकः – बालकौ – बालकाः
(क) चषक: ………… ………..
(ख) ……… ………… देवा:
(ग) सैनिक: …….. ……….
(घ) ……. रजकौ ………..
(ङ) तन्त्रज्ञ: …… ………..

उत्तरम्:

२. उदाहरणानुसारं पट्टिकातः पदानि चित्वा रिक्त स्थानेषु संयोजयन्तु।

(उदाहरण के अनुसार पट्टिका से पदों को चुनकर रिक्त स्थानों में जोड़ें)

यथा – सः शुकः। ………… ……….
सः ………… ……….. …………
सा ………… ……….. …………
तत् …….. ……….. ………….
शुक: स्यूतः अजा फलम् पुष्पम् महिषी वृक्ष: कुक्कुरः पार्वती पुस्तकम्
उत्तरम्:

३. चित्राणि दृष्ट्वा संस्कृतपदानि लिखन्तु ।

(चित्र देखकर संस्कृत पदों को लिखें।)

(क)

……………
उत्तरम्:
पुस्तकम्

(ख)

……………
उत्तरम्:
बालकौ

(ग)

……………
उत्तरम्:
सैनिक:

(घ)

……………
उत्तरम्:
वातायनम्

(ङ)

उत्तरम्:
(ङ) गायिका

(च)

……………
उत्तरम्:
त्रिशूल:

४. उदाहरणानुसारं पट्टिकात पदानि चित्वा रिक्तस्थानेषु लिखन्तु |

(उदाहरणानुसार पट्टिका से पदों को चुनकर रिक्त स्थानों में लिखो ।)

बालकाः वृक्षा: शिक्षिका: पुष्पम् फलम् अजा:
मापिकाः गायिका रजक: फलानि सः लेखनी
एकवचनम् बहुवचनम्
यथा- पुष्पम् बालकाः
(क) ………….. …………..
(ख) ………….. ………….
(ग) ………….. ………….
(घ) ………….. …………..
उत्तरम्:
एकवचनम् – बहुवचनम्
(क) फलम् – वृक्षा:
(ख) रजक: – शिक्षिका :
(ग) गायिका – अजा:
(घ) सः – मापिकाः
(ङ) लेखनी – फलानि

५. पट्टिकायां कानिचन पदानि सन्ति, तानि पदानि उचिते घटे परयन्तु।

(पट्टिका में कुछ पद हैं, उन पदों को उचित घड़े में पूरा करें।)
सैनिक: कुक्कुरः गायिका महिषी
जलम् चषक: द्वारम् फलम्
वृद्धा वृक्ष: अजा वातायनम्
अङ्कनी पुस्तकम् शुक:

उत्तरम्:

६. उदाहरणानुसारं पट्टिकातः उपयुक्तपदं चित्वा रिक्तस्थानानि पूरयन्तु।

(उदाहरणानुसार पट्टिका (पट्टी) से उपयुक्त पद चुनकर रिक्त स्थानों को पूरा करें)
माला युवकौ पाठशाला मुखम् कमले
भवनानि बिडालः छात्रा: लेखन्यौ माला:

उत्तरम्-

७. निम्नलिखितानि वाक्यानि अधिकृत्य उदाहरणानुसारं प्रश्ननिर्माणं कुर्वन्तु ।

(निम्नलिखित वाक्यों को अधिकृत कर उदाहरणानुसारं प्रश्ननिर्माण करें।)
यथा सः गजः । – सः कः ?
(क) सः बालकः । – …………..
(ख) सा लता। – …………..
(ग) सा नदी । – ………….
(घ) तत् फलम्। – ……….
(ङ) सः वृक्षः। – ………….
उत्तरम्:
(क) सः बालकः । – सः कः ?
(ख) सा लता। – सा का?
(ग) सा नदी । – सा का?
(घ) तत् फलम्। – तत् किम्?
(ङ) सः वृक्षः। – सः कः ?

८. परस्परं सम्बद्धानि पदानि संयोजयन्तु, रिक्तस्थानेषु पूरयन्तु च ।

(परस्पर संबंध पदों को मिलाएँ और रिक्त स्थानों में भरें।)

उत्तरम्:

९. कोष्ठकेभ्यः उचितानि पदानि चित्वा वाक्यानि रचयन्त।

(कोष्ठकों से उचित पद चुनकर वाक्य-रचना करें।)
यथा – कः / का लिखति ? लेखकः लिखति । छात्रः लिखति । सा लिखति ।
(क) कः / का धावति ? (बालकाः, बालिका, शुनकः) ……….। ……….। ……….।
(ख) कः / का पठति ? ( सुरेश, जानकी, नलिनी) ……….। ……….। ……….।
(ग) किं पतति ? (फलम् जलम्, कुसुमम्) ……….। ……….। ……….।
(घ) का / कः गच्छति ? (शिक्षिका, बालिका, तन्त्रज्ञः) ……….। ……….। ……….।
(ङ) का / क: / किम् अस्ति ? (माता, पिता, वाहनम्) ……….। ……….। ……….।
उत्तरम्:
(क) कः / का धावति ? (बालकाः, बालिका, शुनकः) शुनकः धावति । बालिका धावति । सा धावति ।
(ख) कः / का पठति ? ( सुरेश, जानकी, नलिनी) सुरेशः पठति । जानकी पठति । नलिनी पठति ।
(ग) किं पतति ? (फलम् जलम्, कुसुमम्) फलम् पतति । जलम् पतति । कुसुमम् पतति ।
(घ) का / कः गच्छति ? (शिक्षिका, बालिका, तन्त्रज्ञः) शिक्षिका गच्छति । बालिका गच्छति । तन्त्रज्ञः गच्छति ।
(ङ) का / क: / किम् अस्ति ? (माता, पिता, वाहनम् ) माता अस्ति । पिता अस्ति । वाहनम् अस्ति ।

योग्यताविस्तरः

अकारान्तः शब्दः
(अकारान्त शब्द)

यस्य शब्दस्य अन्ते ‘अ’ वर्णः अस्ति सः अकारान्तः शब्दः भवति ।
(जिस शब्द के अन्त में ‘अ’ वर्ण है वह अकारान्त होता है।)
(एतादृश: शब्द: पुंलिङ्गे अथवा नपुंसकलिङ्गे भवति ।)
(इस तरह के शब्द पुंल्लिङ्ग अथवा नपुंसकलिङ्ग होता है)
यथा- बालक् + अ हस्त् + अ अज् + अ
फल् + अ ज्ञान् + अ गृह अ

आकारान्तः शब्दः
(आकारान्त शब्द)

यस्य शब्दस्य अन्ते ‘आ’ वर्णः अस्ति सः आकारान्तः शब्दः भवति ।
(जिस शब्द के अन्त में ‘आ’ वर्ण है वह आकारान्त शब्द होता है।)
(एतादृश: शब्द: प्रायश: स्त्रीलिङ्गे भवति ।)
(इस तरह के शब्द प्राय: स्त्रीलिङ्ग में होता है)

यथा बालिक् + आ रम् + आ लत् + आ

ईकारान्तः शब्दः
(ईकारान्त शब्द)

यस्य शब्दस्य अन्ते ‘ई’ वर्णः अस्ति सः ईकारान्तः शब्दः भवति ।
(जिस शब्द के अन्त में ‘ई’ वर्ण है वह ईकारान्त शब्द होता है।)
(ईकारान्तः शब्दः प्रायश: स्त्रीलिंगे भवति ।)
(ईकारान्त शब्द प्राय: स्त्रीलिङ्ग में होता है।)

यथा- नद् + ई लेखन् + ई घट् + + ई

तत् एतत् किम्

अनुवाद –

परियोजनाकार्यम्
(परियोजना कार्य )

१. गृहजनानां मित्राणां वा साहाय्येन पञ्चदश (१५) गृहवस्तुनामानि लिङ्गविभागेन लिखन्तु ।

(घर के लोगों अथवा मित्रों की सहायता से 15 घर की वस्तुओं के नाम लिङ्गविभाग द्वारा लिखें।)
उत्तरम्:

1. दूरदर्शनम्
2. दूरभाषम्
3. शीतकम्
4. आसन्दिकाः
5. फलकम्
6. कपाटिका (अलमारी)
7. घटिका
8. संगणकम्
9. पात्राणि
10. वस्त्राणि
11. पर्यंकम्
12. जवनिका
13. वातायनम्
14. कपाटम्
15. भोज्यपदार्थानि

२. अधः केचन व्यावहारिकाः शब्दाः सन्ति । तेषाम् अर्थं शिक्षकस्य साहाय्येन अन्विष्य लिखन्तु ।

(नीचे कुछ व्यावहारिक शब्द हैं। उनके अर्थ शिक्षक की सहायता से जानकर लिखें)


उत्तरम्:
छात्रा स्वयं करिष्यन्ति ।

अतिरिक्तः अभ्यासः

१. अधोलिखितानाम् एकवचनरूपाणि लिखत- (निम्नलिखित शब्दों के एकवचन रूप लिखें।)

(क) बालकाः
उत्तरम्:
बालकः

(ख) वृक्षा:
उत्तरम्:
वृक्ष:

(ग) गजा:
उत्तरम्:
गजः

(घ) बालिकाः
उत्तरम्:
बालिका

(ङ) वैद्या:
उत्तरम्:
वैद्या

(च) अजा:
उत्तरम्:
अज

२. अधोलिखितानाम् द्विवचनरूपाणि लिखत-

(निम्नलिखित के द्विवचन रूप लिखें)

(क) पत्रम्
उत्तरम्:
पत्रे

(ख) चक्रम्
उत्तरम्:
चक्रे

(ग) पुस्तकम्
उत्तरम्:
पुस्तके

(घ) पठति
उत्तरम्:
पठतः

(ङ) पिबति
उत्तरम्:
पिबतः

(च) खादति
उत्तरम्:
खादतः

अतिरिक्तः अभ्यासः

१. अधोलिखितानाम् एकवचनरूपाणि लिखत-

(निम्नलिखित शब्दों के एकवचन रूप लिखें।)

(क) बालकाः
उत्तरम्:
बालक:

(ख) वृक्षा:
उत्तरम्:
वृक्षः

(ग) गजा:
उत्तरम्:
गजः

(घ) बालिकाः
उत्तरम्:
बालिका

(ङ) वैद्या:
उत्तरम्:
वैद्या

(च) अजा
उत्तरम्:
अज

२. अधोलिखितानाम् द्विवचनरूपाणि लिखत-

(निम्नलिखित के द्विवचन रूप लिखें)

(क) पत्रम्
उत्तरम्:
पत्रे

(ख) चक्रम्
उत्तरम्:
चक्रे

(ग) पुस्तकम्
उत्तरम्:
पुस्तके

(घ) पठति
उत्तरम्:
पठतः

(ङ) पिबति
उत्तरम्:
पिबतः

(च) खादति
उत्तरम्:
खादतः

बहुविकल्पीय प्रश्नाः

अधोलिखितेषु विकल्पेषु समुचितम् उत्तरं चित्वा लिखत-

(निम्नलिखित विकल्पों में से उचित उत्तर चुनकर लिखें।)

१. पुंलिङ्गः कः ?
(क) शिक्षक:
(ख) शिक्षिका
(ग) बालिका
(घ) महिला
उत्तरम्:
(क) शिक्षक:

२. स्त्रीलिङ्गः कः ?
(क) सः
(ख) सा
(ग) तत्
(घ) बालकाः
उत्तरम्:
(ख) सा

३. नपुंसकलिङ्गः कः ?
(क) चन्द्रयानम्
(ख) अजा:
(ग) बालिका
(घ) लता
उत्तरम्:
(क) चन्द्रयानम्

४. यस्य शब्दस्य अन्ते ‘अ’ वर्णः अस्ति सः …….शब्दः भवति ।
(क) अकारान्तः
(ख) आकारान्तः
(ग) ईकारान्तः
(घ) ऋकारान्तः
उत्तरम्:
(क) अकारान्तः

५. यस्य शब्दस्य अन्ते ‘आ’ वर्णः अस्ति सः ………. शब्दः भवति ।
(क) अकारान्तः
(ख) आकारान्तः
(ग) ईकारान्तः
(घ) इकारान्तः
उत्तरम्:
(ख) आकारान्तः

६. यस्य शब्दस्य अन्ते ‘ई’ वर्ण । अस्ति सः शब्द भवति-
(क) अकारान्तः
(ख) आकारान्तः
(ग) ईकारान्तः
(घ) इकारान्तः
उत्तरम्:
(ग) ईकारान्तः

७. ईकारान्तः शब्दः प्रायशः ……… भवति ।
(क) स्त्रीलिङ्गे
(ख) पुंलिङ्गे
(ग) नपुंसकलिङ्गे
(घ) बहुवचने
उत्तरम्:
(क) स्त्रीलिङ्गे

८. आकारान्तः शब्दः प्रायशः ……. भवति ।
(क) पुंलिङ्गे
(ख) स्त्रीलिङ्गे
(ग) नपुंसकलिङ्गे
(घ) एकवचने
उत्तरम्:
(ख) स्त्रीलिङ्गे

९. अकारान्तः शब्दः भवति-
(क) पुंलिङ्गे अथवा नपुंसकलिङ्गे
(ख) स्त्रीलिङ्गे
(ग) द्विवचने
(घ) बहुवचने
उत्तरम्:
(क) पुंलिङ्गे अथवा नपुंसकलिङ्गे

१०. पुष्पम् ……….
(क) चालयति
(ख) गायति
(ग) विकसति
(घ) नृत्यति
उत्तरम्:
(ग) विकसति

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