A Rational Number is a number that can be written in the form of p/q  where p, q are integers, and q ≠ 0. You can learn about the General Properties of Rational Numbers like Closure, Commutative, Associative, Distributive, Identity, Inverse, etc. here. Not just the regular properties we have all listed all the properties that we know regarding Rational Numbers.

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property
• Identity and Inverse Properties of Rational Numbers

Closure Property

For two rational numbers x, y the addition, subtraction, multiplication results always yield a rational number. The Closure Property isn’t applicable for the division as division by zero isn’t defined. In other words, we can say that closure property is applicable for division too other than zero.

4/7 + 2/3 =26/21

4/3 – 2/4 = 6/12

3/5. 2/3 = 6/15

Commutative Property

Considering two rational numbers x, y the addition and multiplication are always commutative. Subtraction doesn’t obey commutative property. You can get a clear idea of this property by having a look at the solved examples.

Commutative Law of Addition: x+y = y+x Ex: 1/3+2/3 = 3/3

Commutative Law of Multiplication: x.y = y.x Ex: 1/2.2/3 =2/3.1/2 =2/6

Subtraction x-y≠y-x Ex: 4/3-1/3 = 3/3 whereas 1/3-4/3=-3/3

Division isn’t commutative x/y ≠y/x Ex: 3/9÷1/2=6/9 whereas 1/2 ÷3/9 =9/6

Associative Property

Rational Numbers obey the Associative Property for Addition and Multiplication. Let us assume x, y, z to be three rational numbers then for Addition, x+(y+z)=(x+y)+z

whereas for Multiplication x(yz)=(xy)z

Ex: 1/3 + (1/4 + 3/3) = (1/3+ 1/4) + 3/3

⇒19/12 =19/12

Distributive Property

Let us consider three rational numbers x, y, z then x . (y+z) = (x . y) + (x . z). We will prove the property by considering an example.

Ex: 1/3.(1/4+2/5) =(1/3.1/4)+(1/3.2/5)

1/3.(17/20)= 1/12+2/10

17/60 =17/60

Thus, L.H.S = R.H.S

Identity and Inverse Properties of Rational Numbers

Identity Property: We know 0 is called Additive Identity and 1 is called Multiplicative Identity of Rational Numbers.

Ex: 1/4+0 = 1/4(Additive Identity)

5/3.1 = 5/3(Multiplicative Identity)

Inverse Property: For a Rational Number x/y additive inverse is -x/y and multiplicative inverse is y/x.

Ex: Additive Inverse of 2/3 is -2/3

Multiplicative Inverse of 4/5 is 5/4

There are few other properties that you need to be aware of Rational Numbers and they are explained below.

Property 1:

If a/b is a rational number and m is a non-zero integer then a/b =(a*m)/(b*m).

In other words, we can say that the rational number remains unaltered if we multiply both the numerator and denominator with the same integer.

Ex: 2/3 = 2*2/3*2 = 4/6, 2*3/3*3 = 6/9, 2*4/3*4 = 8/12….

Property 2:

If a/b is a rational number and m is a common divisor then a/b = (a÷m)/(b÷m)

On dividing the numerator and denominator of a rational number with a common divisor the rational number remains unchanged.

Ex: 36/42 =36÷6/42÷6 = 6/7

Property 3:

Consider a/b, c/d to be two rational numbers.

Then a/b = c/d ⇒ a*d = b*c

Ex: 2/4 =4/8 ⇒ 2.8=4.4

Property 4:

For each and every Rational Number n, any of the following conditions hold true.

(i) n>0, (ii) n=0, (iii) n<0

Ex: 3/4 is greater than 0.

0/5 is equal to 0.

-3/4 is less than 0.

Property 5:

For any two rational numbers a, b any one condition is true

(i) a>b, (ii) a=b, (iii) a<b

Ex: 2/3 and 2/5 are two rational numbers and 2/3 is greater than 2/5

If 4/8 and 8/16 are two rational numbers then 4/8 = 8/16

If -4/7 and 3/4 are two rational numbers then -4/7 is less than 3/4

Property 6:

In the case of three rational numbers a > b, b > c then a>c

If 4/5, 16/30, -8/15 are three rational numbers then 4/5 >16/30 and 16/30 is greater than -8/15 then 4/5 is also greater than -8/15.

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Rational Numbers are said to be in the lowest form if both the numerator and denominator have no other common factor other than 1. In other words, we can say that a Rational Number a/b is said to be in its simplest form if the HCF of a, b is 1 and a, b are relatively prime. To help you understand the concept better we have illustrated a few examples of how to convert Rational Numbers to their respective lowest form.

Example: 5/3 is a Rational Number in the Lowest Form or Simplest Form as it has no other common factors other than 1 Whereas, 14/21 is not in the simplest form as it has common factor 7 along with 1. Thus, it is not in the simplest form.

How to Convert a Rational Number to Lowest or Simplest Form?

Go through the below steps to change a Rational Number into its Lowest or simplest form. The Guidelines are along the lines

Step 1: Firstly, note down the rational number a/b given.

Step 2: Find the HCF of a, b

Step 3: If you get the HCF(a, b) i.e. k = 1 then they are in the simplest form.

Step 4: If at all the HCF(a,b) i.e. k ≠ 1 the simply divide both the numerator and denominator by k i.e. (a÷k)/(b÷k) to obtain the lowest form of rational number a/b. 

Solved Examples

1.  Determine whether the following rational numbers are in their lowest form or not?

(i)14/81 (ii) 72/24

Solution:

1/81 is in its lowest form as the HCF is 1 i.e. both the numerator and denominator have no common factors other than 1.

(ii) 72/24

72 = 2 × 2 × 2 × 3 × 3, 24 = 2 × 2 × 2 × 3

HCF of 72, 24 is 24

Therefore, the rational number 72/24 is not in the lowest form.

2. Express 45/30 in its simplest form?

Solution: 

We have 45/30

Find the HCF of 45, 30 initially

HCF(45, 30) = 15

As the HCF is not equal to 1 simply divide the numerator and denominator of the given rational number with 15.

45/30 = (45÷15)/(30÷15) = 3/2

Therefore, 45/30 in its simplest form is 3/2.

3. Express the Rational Number 36/24 into its Lowest Form?

Solution:

Given Rational Number 36/24

Figure out the HCF(36, 24) at first

HCF(36, 24) = 12

Since HCF is not equal to 1 divide both the numerator and denominator with the HCF obtained.

36/24 = (36÷12)/ (24÷12)

= 3/2

Therefore, 36/24 expressed into its lowest form is 3/2.

4. Reduce 3/15 to its Lowest Form?

Solution:

Given Rational Number is 3/15

Find out the HCF(3,15)

HCF(3,15) = 3

As the HCF isn’t equal to 1 divide the numerator and denominator of rational number with HCF to change it to the lowest form.

3/15 = (3÷3)/(15÷3)

= 1/5

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Standard Form of a Rational Number: A rational number is said to be in standard form if the common factor between numerator and denominator is only 1 and the denominator is always positive. Furthermore, the numerator can have a positive sign. Such Numbers are called Rational Numbers in Standard Form. Check out a few examples that illustrate the procedure of expressing Rational Number in Standard Form to be familiar with the concept even better.

What is the Standard Form of a Rational Number?

Usually, a rational number a/b is said to be in standard form if it has no common factors other than 1 between the numerator and denominator alongside the denominator b should be positive.

How to Convert a Rational Number into Standard Form?

Go through the below-listed guidelines to express a Rational Number into Standard Form. The Detailed Procedure is explained for better understanding and they are along the lines

Step 1: Have a look at the given rational number.

Step 2: Firstly, find whether the denominator is positive or not. If it is not positive multiply or divides numerator and denominator with -1 so that the denominator no longer remains negative.

Step 3: Determine the GCD of the absolute values of both numerator and Denominator.

Step 4: Divide the numerator and denominator with the GCD obtained in the earlier step. Thereafter, the rational number obtained is the standard form of the given rational number.

Solved Examples

1.  Determine whether the following Rational Numbers are in Standard Form or Not?

(i) -8/23 (ii) -13/-39

Solution:

-8/23 is said to be in Standard Form since both the numerator and denominator doesn’t have any common factors other than 1. In fact, the denominator is also positive. Thus, the given rational number -8/23 is said to be in its Standard Form.

-13/-39 is not in standard form since it has common factor 13 along with 1. Moreover, the denominator is not positive. Thus we can say the given rational number is not in standard form.

2. Express the Rational Number 18/45 in Standard Form?

Solution:

Given Rational Number 18/45

Check for the denominator in the given rational number. Since it is positive you need not do anything.

Later find the GCD of the absolute values of numerator 18, denominator 45

GCD(18, 45) = 9

Thus, to convert the given rational number 18/45 to standard form simply divide both the numerator and denominator by 9

18/45 = (18÷9)/(45÷9)

= 2/5

Therefore, 18/45 expressed in standard form is 2/5.

3. Find the Standard Form of 12/-18?

Solution:

Given Rational Number is 12/-18

Check for the denominator in the given Rational Number

Since the denominator, -18 has a negative sign multiply both numerator and denominator with -1 to make it positive.

12/-18 = 12*(-1)/-18*(-1)

= -12/18

Find the GCD of absolute values of both numerator and denominator

GCD(12, 18) = 6

To convert a given rational number to its standard form multiply and divide both numerator and denominator by 6.

-12/18 = ((-12)÷6)/(18÷6)

= -2/3

Thus, the standard form of Rational Number 12/-18 is -2/3.

4. Reduce 3/15 to Standard Form?

Solution:

Given Rational Numbers is 3/15

Since the denominator is positive you need not do anything to change it to positive.

Find the GCD of absolute values of numerator and denominator of the given rational number.

GCD(3, 15) = 3

Divide numerator and denominator with GCD obtained.

3/15 = (3÷3)/(15÷3)

= 1/5

Therefore, 3/15 Reduced to Standard Form is 1/5.

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In this article of ours, we tried covering everything about the concept of Equality of Rational Numbers using Standard Form. You will find all about how to determine whether two rational numbers are equal or not. However, there are various methods to know whether given rational numbers are equal or not but we will employ the standard form method here.

How to determine the Equality of Rational Numbers using Standard Form?

To check whether the given rational numbers are equal or not you need to find out the standard form of both of them individually. If the standard form of both the rational numbers is equal then the rational numbers are equal or else not equal.

Solved Examples

1. Determine whether the Rational Numbers 4/-9 and -16/36 equal or not using a standard form?

Solution:

Given Rational Numbers are 4/-9, -16/36

Check for the denominators in both the rational numbers if they aren’t positive change them to positive.

4/-9 = 4*(-1)/-9*(-1)

=-4/9

GCD(4,9) = 1, thus -4/9 is the standard form.

-16/36 since it has a positive denominator it remains unchanged

Find the GCD of absolute values of the numerator and denominator for the rational numbers.

GCD(16, 36) = 4

To reduce the rational number to standard form divide both numerator and denominator with GCD obtained.

-16/36 = (-16÷4)/(36÷4)

= -4/9

-4/9 is the standard form of -16/36

Since the standard forms of rational numbers are equal both the given rational numbers 4/-9 and -16/36 are equal.

2. Determine whether the Rational Numbers 2/3 and 5/7 equal or not using a Standard Form?

Solution:

Given Rational Numbers are 2/3 and 5/7

Since both the denominators are positive you need not multiply or divide to make them positive.

Find the GCD of absolute values of numerator and denominator in the given rational numbers.

GCD(2, 3) =1

GCD(5, 7) = 1

Since both the rational numbers have GCD 1 and the numbers are relatively prime. Both the Rational Numbers are in Standard Form.

2/3 is not equal to 5/7

Therefore, Rational Numbers 2/3 and 5/7 are not equal.

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You will learn about the Equality of Rational Numbers with Common Denominator from this article. Check out how to determine whether given Rational Numbers are equal or not using the Common Denominator. Apart from the procedure, we have even listed a few examples for better understanding.

How to determine Equality of Rational Numbers with Common Denominator?

There are many methods to check the equality of rational numbers but here in this article, we are going to discuss the method of the same denominator. We have listed the procedure on how to make denominators equal for the given rational numbers and they are in the following fashion.

Step 1: Check the given Rational Numbers.

Step 2: Multiply the numerator and denominator of the first number with the denominator of the second number.

Step 3: On the other hand, multiply the numerator and denominator of the second number with the denominator of the first number.

Step 4: Later check the numerators of the numbers obtained in steps 2, 3. If the numerators are equal then the given rational numbers are equal or else they are not equal.

Solved Examples

1. Are the Rational Numbers -7/5 and -5/3 equal?

Solution:

Given Rational Numbers are -7/5, -5/3

Multiply the second number numerator and denominator with the denominator of the first number i.e.

-5*5/3*5

= -25/15

Multiply the first number numerator and denominator with the denominator of the second number i.e.

-7*3/5*3

= -21/15

Check the numerators obtained in the earlier steps and see whether they are equal or not.

Since 21, 25 aren’t equal both the given rational numbers aren’t equal.

Therefore, -7/5 and -5/3 are not equal.

2. Show that the Rational Numbers -6/8 and -9/12  are equal?

Solution:

Given Rational Numbers are -6/8, -9/12

Multiply both numerator and denominator of the second number with the denominator of the first number.

= (-9/12)*8

= -72/96

Multiply both numerator and denominator of the first number with the denominator of the second number.

= (-6/8)*12

= -72/96

Therefore, Rational Numbers -6/8 and -9/12 are equal.

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Let us learn about Equality of Rational Numbers using Cross Multiplication in detail here. Check out the Procedure to determine whether Rational Numbers are Equal or not using the Cross Multiplication Technique. Have a glance at the solved examples explaining the concept in detail so that you can solve related problems.

How to determine Equality of Rational Numbers using Cross Multiplication?

There are numerous methods to check the equality of rational numbers. But here we are using the Cross Multiplication Method to check whether the given rational numbers are equal or not. Follow the guidelines to check the equality of rational numbers.

Let us consider two rational numbers a/b, c/d

a/b = c/d

⇔ a × d = b × c

⇔ The Numerator of First × The Denominator of Second = The Denominator of First × The Numerator of the Second

Solved Examples

1.  Determine whether the following pair of Rational Numbers are Equal or Not?

8/4 and 6/3

Given Rational Numbers are 8/4 and 6/3

⇔ We know a × d = b × c

Multiplying Numerator of First × The Denominator of Second = The Denominator of First × The Numerator of the Second we get

8*3 = 6*4

24 = 24

Therefore, the given rational numbers 8/4 and 6/3 are equal.

2. If -8/6 = k/30, find the value of k?

Solution:

-8/6 = k/30

Cross multiplying we get

-8*30 = k*6

Performing basic math we get the value of k

(-8*30)/6 =k

k=-40

Therefore, the value of k is -40.

3. If 5/m = 40/16 determine the value of m?

Solution:

5/m = 40/16

Cross multiplying we get 5*16 = m*40

Separating m to get the value of it.

m= (5*16)/40

= 80/40

= 2

Therefore, the value of m is 2.

4. Fill in the Blank -7/10 = …/120?

Solution:

In order to express -7 as a denominator with 120, we first need to find out the number which when multiplied by 10 gives 120.

Thus, the integer is 120÷ 10 = 12

Multiplying the numerator and denominator of a given rational number with 12 we get

-7/10 = (-7*12)/(10*12)

= -84/120

Thus, the required number is -84/120.

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We are all familiar with the concept of comparing two integers or two fractions and determining which is smaller or greater. Let us go a step ahead and compare Two Rational Numbers. We know fact that every positive integer is greater than 0, and a negative integer is less than 0. By knowing this fundamental rule we can infer some facts about how to compare rational numbers. They are listed below

1. Every Positive Rational Number is Greater than Zero.
2. Every Negative Rational Number is Less than Zero.
3. Comparison of Positive and Negative Rational Number is quite obvious i.e. Positive Rational Number is greater than a negative rational number.
4. Every Rational Number represented on a number line is greater than every other rational number represented present to its left.
5. Every Rational Number represented on a number line is less than every other rational number represented present to its right.

How to Compare Rational Numbers?

In order to compare any two rational numbers, you can go through the below-mentioned steps. They are as under

Step 1: Check the given rational numbers

Step 2: Write down the given rational numbers in a way that they have their denominators the same.

Step 3: Determine the Least Common Multiple of the Positive Denominators you obtained in the earlier step.

Step 4: Express rational numbers obtained in the second step using the LCM obtained as Common Denominator.

Step 5: Compare the numerators of rational numbers obtained and declare the one having a greater numerator as a greater rational number.

Solved Examples

1. Of the two rational numbers which is greater 2/3 or 5/7?

Solution: 

Given Rational Numbers are 2/3, 5/7

LCM of 3, 7 is 21

Expressing the rational numbers with the same denominator using the LCM obtained we get

Therefore, we get 2/3 = (2*7)/(3*7) = 14/21

5/7 = (5*3)/(7*3) = 15/21

See the numerators of both the rational numbers obtained i.e. 14/21, 15/21

Since 15 is greater the rational number 5/7 is greater.

Therefore, of the two rational numbers, 2/3 and 5/7,  5/7 is greater.

2. Which of the two rational numbers 2/5 and -3/4 is greater?

Solution:

Given Rational Numbers are 2/5 and -3/4

We clearly know between a positive rational number and a negative rational number positive rational number is always greater.

Therefore, 2/5 is greater than -3/4.

3. Which is greater among -1/2 and – 1/5?

Solution: 

Given rational numbers are -1/2 and -1/5

LCM of 2, 5 is 10

Expressing the rational numbers with the same denominator using the LCM obtained.

-1/2 = (-1*5)/(2*5)= -5/10

-1/5 = (-1*2)/(5*2) = -2/10

-2 > -5

Therefore, – 1/5 is greater than -1/2.

4. Which of the numbers 3/4 and -3/4 are greater?

Solution:

We know that every positive rational number is greater than a negative rational number. Therefore, 3/4 is greater than -3/4.

The post Comparison of Rational Numbers appeared first on Learn CBSE.

Let us learn in detail how to arrange Rational Numbers in Ascending Order. Have a look at the general method to arrange the Rational Numbers in Increasing Order. To help you get a better idea of the concept we even listed the solved examples provided step by step.

Procedure to arrange from Smallest to Largest Rational Numbers

Go through the below-listed guidelines in order to arrange Rational Numbers from smallest to largest. They are along the lines

Step 1: Express the given rational number in terms of a positive denominator.

Step 2: Determine the Least Common Multiple of the positive denominators obtained.

Step 3: Express each rational number with the LCM acquired as the common denominator.

Step 4: The number which has the smaller numerator is the smaller rational number.

Solved Examples for Rational Numbers in Ascending Order

1.  Write the following rational numbers in Ascending Order -3/5, -1/5, -2/5

Solution:

Since all the numbers have a common denominator the one with a smaller numerator is the smaller rational number. However, when it comes to negative numbers the higher one is the smaller one.

Therefore arranging the given rational numbers we get -3/5, -2/5, -1/5

2.  Arrange the rational numbers 1/2, -2/9, -4/3 in Ascending Order?

Solution:

Find the LCM of the denominators 2, 9, 3

LCM of 2, 9, 3 is 18

Express the given rational numbers with the LCM in terms of common denominator.

1/ 2= 1*9/2*9 = 9/18

-2/9 = -2*2/9*2 = -4/18

-4/3 = -4*6/3*6 = -24/18

Check the numerators of all the rational numbers expressed with a common denominator.

Since -24 is less than the other two we can arrange the given rational numbers in Ascending Order.

-4/3, -2/9, 1/2 is the Ascending Order of Given Rational Numbers.

3. Arrange the Rational Numbers 5/8, 4/-6, 3/5 in Ascending Order?

Solution:

Firstly, express the rational numbers with positive denominators by multiplying with -1

4/-6 = 4*(-1)/-6*(-1) = -4/6

So, find the LCM of the denominators 8, 6, 5

LCM of 8, 6, 5 is 120

5/8 = 5*15/8*15 = 75/120

-4/6 = -4*20/6*20 = -80/120

3/5 = 3*24/5*24 = 72/120

Check the numerator of the rational numbers having common denominators.

since -80 is the smallest that itself is the smallest rational number.

Therefore, 4/-6, 3/5, 5/8 are in Ascending Order.

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Learn how to arrange Rational Numbers in Descending Order or Decreasing Order. In order to make you familiar with the concept of Rational Numbers in Decreasing Order we even listed examples explaining the step by step process. Check out the general method to arrange rational numbers from Largest to Smallest easily.

Procedure to arrange Rational Numbers from Largest to Smallest

Follow the easy guidelines on how to arrange Rational Numbers in Decreasing Order. They are as follows

Step 1: Express the given rational number in terms of the positive denominator.

Step 2: Find out the Least Common Denominator of the Positive Denominators.

Step 3: Express the given rational numbers using the LCM as Common Denominator.

Step 4: Compare the numerators and the one having the highest numerator is the largest one.

Solved Examples on Rational Numbers in Decreasing Order

1. Arrange the numbers 5/-3, 10/-7, -5/8 in Descending Order?

Solution:

Given Rational Numbers are 5/-3, 10/-7, -5/8

Express the Rational Numbers with Positive Denominators

5/-3 = 5*(-1)/-3*(-1) = -5/3

10/-7 = 10*(-1)/-7*(-1) = -10/7

-5/8 already has a positive denominator

Find the LCM of Positive Denominators

LCM of 3, 7, 8 is 168

Express the Rational Numbers with Common Denominator with the LCM obtained.

-5/3 = -5*56/3*56 = -280/ 168

-10/7 = -10*24/7*24 = -240/168

-5/8 = -5*21/8*21 = -105/168

Check the numerators of the rational numbers. Since all of them are negative numbers the lesser one is the highest fraction.

Therefore, Rational Numbers in Descending Order are -5/8, 10/-7, 5/-3.

2. Arrange the Rational Numbers 4/9, 5/6, 7/12 in Descending Order?

Solution: 

Given Rational Numbers are 4/9, 5/6, 7/12

Find the LCM of the Positive Denominators

LCM of 9, 6, 12 is 36

Express the Rational Numbers in terms of Common Denominator using the LCM obtained earlier.

4/9 = 4*4/9*4 = 16/36

5/6 = 5*6/6*6 = 30/36

7/12 = 7*3/12*3 = 21/36

Check the numerators of the rational numbers and the one having highest numerator is the highest rational number.

5/6, 7/12, 4/9 is the Descending Order of Rational Numbers.

3. Arrange the Rational Numbers 3/8, 5/7, 2/9 in Descending Order?

Solution:

Given Rational Numbers are 3/8, 5/7, 2/9

Determine the LCM of Positive Denominators.

LCM of 8, 7, 9 is 504.

Express the rational numbers with common denominator using the LCM obtained.

3/8 = 3*63/8*63 = 189/504

5/7 = 5*72/7*72 = 360/504

2/9 = 2*56/9*56 = 112/504

Check the numerators of the rational numbers and arrange the ones from highest to lowest.

Therefore Rational Numbers arranged in Descending Order is 5/7, 3/8, 2/9.

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To calculate the profit we need to know “What is Profit?” The term Profit refers to the amount which we gained after selling a product. This means the “Selling price” should be more than the “Actual price” or “Cost Price.”
Now coming to the calculation of the profit we have a formula i.e

Profit(P)= Selling price(SP) – Cost Price (CP)

Here we can see the new terms “Selling Price” and “Actual Price” or “Cost Price.” Let us discuss what they refer to.

Selling Price: The price of the product which was sold by the shopkeeper to the customer for a certain price is known as the selling price. The Selling price can also be written as SP. To calculate the Selling price, we have a formula i.e.

Selling Price(SP)= Profit(P) + Cost Price(CP)

Cost Price: It is also called Actual price, which means the actual cost of a product or original cost of a product or an item that was bought from the merchant or retailer. To calculate Cost Price, we have a formula i.e,
Cost price(CP)= Selling price(SP) – Profit (P).

In the above, we have discussed the related terms. Let’s discuss more on the concept of Profit with some examples.

Solved Examples on Profit

1. Find the Profit if
a) SP= 100 and CP= 60
b) SP= 125 and CP= 100
c) SP= 90 and CP= 75

Solution:
a) SP= 100 and CP= 60
Profit(P)= Selling Price(SP) – Cost Price (CP).
= 100-60
= 40.
Profit= 40.

b) SP= 125 and CP= 100
Profit(P)= Selling Price(SP) – Cost Price (CP).
= 125-100
= 25.
Profit= 25.

c) SP= 90 and CP= 75
Profit(P)= Selling price(SP) – Cost price (CP).
= 90-75
= 15.
Profit= 15.

2. The cost price of chocolate is Rs.10 and the selling price is Rs. 15. Find the profit?

Solution:
CP of the chocolate= Rs.10
SP of the chocolate= Rs.15
Profit= Selling price(SP) – Cost price(CP).
= 15 – 10
= 5.
Profit= 5.

3. Mark bought 4 dozens of apples at $15 a dozen and sold at $20 a dozen. Find the profit?

Solution:

The cost price of apples= $15
The selling price of apples= $20
Profit(P)= Selling price(SP) – Cost price (CP).
= $20-$15
= $5.
Profit= $5.

4. Mr.Singh bought a table for Rs 15,000 and spent Rs 500 on transportation. He sold the table for Rs 17,000. Find his profit?

Solution:
Cost price of the table= Rs 15,000.
Transportation cost= Rs 500.
So, Total Cost Price= Rs 15,000+Rs 500
= Rs 15,500The selling price of the table= Rs 17,000.
Profit(P)= Selling Price(SP) – Cost Price (CP).
= Rs 17,000- Rs 15,500
= Rs 1500
Profit= Rs 1500.

Solved Examples on Selling Price

1. Find the Selling Price if
a) P= 20 and CP= 100
b) P= 32 and CP= 150
c) P= 5 and CP= 22

Solution:

a) P= 20 and CP= 100
Selling price(SP)= Profit(P) + Cost price(CP).
= 20+100
= 120.
Selling price= 120.

b) P= 32 and CP= 150
Selling price(SP)= Profit(P) + Cost price(CP).
= 32+150
= 162.
Selling price= 162.

c) P= 5 and CP= 22
Selling price(SP)= Profit(P) + Cost price(CP).
= 5+22
= 27.
Selling Price= 27.

2. The cost price of a dining set is Rs 8000 and the shopkeeper got a profit of Rs 2000. Find the selling price of the dining set?

Solution:
The cost price of the dining set= Rs 8000.
The profit that the shopkeeper got= Rs 2000.
Selling price(SP)= Profit(P) + Cost price(CP).
= Rs 2000+ Rs 8000
= Rs 10,000.
The selling price of the dining set is Rs 10,000.

Solved Examples on Cost Price

1.  Find the Cost price if
a) SP= 200 and P= 20.
b) SP= 250 and P= 50.
c) SP= 125 and P= 25.

Solution:
a)SP= 200 and P= 20.
Cost price(CP)= Selling price(SP) – Profit (P).
= 200-20.
= 180.
Cost Price= 180.

b) SP= 250 and P= 50.
Cost Price(CP)= Selling Price(SP) – Profit (P).
= 250-50
= 50.

c) SP= 125 and P= 25.
Cost price(CP)= Selling price(SP) – Profit (P).
= 125-25
= 100.

2. A shopkeeper sold a chair for Rs 2000 and he got a profit of Rs 500. What is the actual price of the chair?

Solution:
The selling price of the chair= Rs 2000.
The profit that the shopkeeper got= Rs 500.
Cost price of the chair =
Cost price(CP)= Selling price(SP) – Profit (P).
= Rs 2000- Rs 500
= Rs 1500.
So the actual price of the chair is Rs. 1500.

How to Calculate the Profit %?

The profit percentage is the percentage which is calculated with the Cost Price in the base. To calculate Profit Percent we need to know profit and cost price. The formula of profit percent is
Profit % = Profit/CP × 100.
To know how to calculate profit %, refer to the solved examples below for better understanding and solve the problems on your own.

Solved Examples on Profit %

1. Find Profit % if,
a) SP= 140 and CP= 100
b) SP= 220 and CP= 200
c) SP= 70 and CP= 50

Solution:

a) SP= 180 and CP= 140
Profit(P)= Selling price(SP) – Cost Price (CP).
= 180-140
= 40.
Profit = 40
Profit % = Profit/CP × 100.
= (40/100)×100
= 40%

b) SP= 220 and CP= 200
Profit(P)= Selling Price(SP) – Cost Price (CP).
= 220-200
= 20
Profit= 20
Profit% = profit/CP × 100.
= (20/200)×100
= (1/10)×100
= 10%.

c) SP= 60 and CP= 50
Profit(P)= Selling Price(SP) – Cost Price (CP).
= 60-50
= 10
Profit= 10
Profit%= profit/CP × 100.
= (10/50)×100
= (1/5)×100
= 20.

2. A man bought 50 bulbs for Rs 125 each. And sells them at Rs 150 each. Find the profit and profit percent?

Solution:

The selling price of the bulb= Rs 150
The cost price of the bulb= Rs 125.
So, Profit (P)= Selling Price(SP)- Cost Price(CP)
= Rs 150- Rs 125
= Rs 25
Profit = Rs 25.
Now to find the Profit percentage, apply the formula
Profit% = Profit/CP × 100.
= (25/125)×100
= (1/5)×100
= 20%.

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To calculate the Loss we need to know “What is Loss?” When we sold an item or a product less than it’s actual price, then Loss occurs which means the product is sold for a low price than the actual price. Now coming to the formula to calculate Loss we have
Loss(L)= Cost Price (CP) – Selling Price(SP)

Here we can see the new terms “Selling price” and “Actual Price” or “Cost Price.” Let us discuss what they refer to.
Selling Price: The price of the product which was sold by the shopkeeper to the customer for a certain price is known as the selling price. Selling Price can also be written as SP. To calculate the Selling Price, we have a formula i.e.

Selling Price(SP)= Cost Price(CP) – Loss(L)

Cost Price: It is also called Actual Price, which means the actual cost of a product or original cost of a product or an item that was bought from the merchant or retailer. To calculate Cost Price, we have a formula i.e,
Cost Price(CP)= Selling Price(SP) + Loss(L)
In the above, we have discussed the related terms. Let’s discuss more about Loss with some examples covering every detail.

Solved Examples on Loss

1.  Find the Loss if
a) SP= 80 and CP= 100
b) SP= 120 and CP= 150
c) SP= 200 and CP= 300

Solution:

a) SP= 80 and CP= 100

We know the formula to calculate the Loss as below
Loss(L)= Cost price (CP) – Selling price(SP).
Substituting the input data in the formula and doing basic math we get
= 100- 80
= 20
Loss= 20.

b) SP= 120 and CP= 150

Formula to calculate the Loss is as under
Loss(L)= Cost price (CP) – Selling price(SP).
Apply the input data you have in the formula and perform basic math
= 150-120
= 30
Loss= 30.

c) SP= 200 and CP= 300

Formula to calculate the Loss is as below
Loss(L)= Cost price (CP) – Selling price(SP).
Substitute the input information we have to find out the Loss
= 300- 200
= 100
Loss= 100.

2. A shopkeeper bought 5 dozens bananas for Rs 300 and sold them at Rs 250. How much he loses?

Solution:
The cost price of bananas= Rs 300
The selling price of bananas= Rs 250
Formula for Loss(L)= Cost Price (CP) – Selling Price(SP).
= Rs 300- Rs 250
= Rs 50.
Loss= Rs 50.

How to Calculate the Loss Percent?

Loss percent is the percent which is expressed as the percentage of the cost price. The formula of Loss Percent is
Loss %= (Loss/ Cost Price)×100. Cost Price is always considered for reference to determine whether you got Loss or Profit. We have listed few examples explaining the process on how to find the Loss Percentage. They are as such

1.  Find Loss % if
a) SP= 80 and CP= 100
b) SP= 120 and CP= 150
c) SP= 200 and CP= 300

Solution:

a) SP= 80 and CP= 100
Formula to calculate the Loss(L)= Cost price (CP) – Selling price(SP).
= 100- 80
= 20
Loss= 20.
After finding the Loss we can determine the Loss % easily
Loss %= (Loss/ Cost Price)×100.
= (20/100) × 100
= 20%

b) SP= 120 and CP= 150
Formula to calculate the Loss(L)= Cost price (CP) – Selling price(SP).
= 150-120
= 30
Loss= 30.
Substitute the Loss value in th Loss % formula we have the equation as such
Loss%= (Loss/ Cost price)×100.
= (30/150)×100
= (1/5)×100
= 20%

c) SP= 100 and CP= 200

Apply the given input data in the formula for
Loss(L)= Cost price (CP) – Selling price(SP).
= 200- 100
= 100
Loss = 100.
Substitute the Loss in the Loss % we have the equation as such
Loss %= (Loss/ Cost Price)×100.
= (100/200)×100
= (1/2)×100
= 50.

2. A man purchased a scooter for Rs 50,000 and after two years he sold it for Rs 30,000. Find Loss and Loss percent.

Solution:

The cost price of a scooter = Rs 50,000
The selling price of a scooter= Rs 30,000
Loss(L)= Cost price (CP) – Selling Price(SP).
Substitute the input values in the formula of Loss we have
= Rs 50,000 – Rs 30,000
= Rs 20,000.

Apply the Loss in the Loss Formula we have
Loss %= (Loss/ Cost price)×100.
= (20,000/50,000)×100
= (2/5)×100
= 2×20
= 40%.

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The concept of Profit and Loss is used in our daily life, which is like we will buy some essentials goods from the shopkeeper and the shopkeeper will buy them either from the manufacturer or from wholesalers. After that, the shopkeeper will sell those goods for a higher price than they bought so that he can earn some profit.

We can see many business people follow these tactics to earn money by buying and selling goods. If the Selling Price is higher than the Actual Price or Cost Price, then he will get a profit. And if the Cost Price is higher than the Selling Price, he will get a loss. Here in this article, we will guide you on how to solve problems on Profit and Loss. In addition, you can get the Formulas for Profit and Loss along with Solved Examples.

Basics of Profit and Loss

Before stepping into the concept of Profit and Loss you need to know the fundamentals involved to calculate the Profit and Loss. They are Selling Price and Cost Price.

Cost Price(CP): Amount you usually pay to buy a product or commodity is called the Cost Price. It is in short represented as CP. In fact, CP is subdivided into two categories namely Fixed Cost and Variable Cost. The major difference between Fixed Cost and Variable Cost is that Fixed Cost remains unchanged under any circumstances whereas variable cost changes depending on the units.

Selling Price(SP): Amount for which the product is sold is referred to as Selling Price. It is represented as SP and is also called as Sale Price.

Profit: The amount gained by selling a product or commodity more than its actual price or cost price.

Loss: The Amount the Seller gets after selling a product less than its actual price.

• Calculate Profit and Profit Percent
• Calculate Loss and Loss Percent

Profit and Loss Formulas

The Formula for Profit and Loss with reference to Cost Price and Selling Price is given as such.

Profit(P)= Selling Price(SP)- Cost Price(CP)
Loss(L)= Cost Price(CP) – Selling Price(SP)
On the other hand formulas to determine the Profit %, Loss % is as such
Profit % = (Profit/Cost Price)*100
Loss % = (Loss/Cost Price)*100

Examples on Profit and Loss

1. The actual price of the book is Rs 60 and the shopkeeper sold the book for Rs 100. Find the profit that shopkeeper earned?

Solution:
The cost price of the book= Rs 60
The selling price of the book= Rs 100
Profit(P)= Selling price(SP) – Cost price (CP)
Applying the S.P and C.P of the book in the Profit Formula we have
= Rs 100 – Rs 60
= Rs 40.
Profit= Rs 40.
Therefore, Profit Gained by the Shokeeper on selling the book for 100 is Rs.40/-

2.  A man bought a cycle for Rs 5000. After a year he sold it for Rs 4000. How much did he lose?

Solution:
The cost price of the cycle= Rs 5000.
The selling price of the cycle= Rs 4000.
Substituting the Cost Price and Selling Price of the cycle we get
Loss(L)= Cost Price(CP) – Selling Price(SP).
= Rs 5000- Rs 4000
= Rs 1000.
Loss = Rs 1000.
Therefore, the man had a loss of Rs. 1000/- on selling the cycle for Rs. 4000/-.

3. Consider a Shopkeeper bought 1kg of Apples for Rs. 80 and Sold it for Rs. 100 per kg. How much is the profit gained by him?

Solution:

Cost Price of Apples = Rs. 80

Selling Price of Apples = Rs. 100

Profit = S.P – C.P

= 100 – 80

= 20

Therefore, the shopkeeper gained a profit of Rs. 20/- on selling the Apples at Rs. 100/- per Kg.

4. Calculate the Profit% gained by the Shopkeeper for the above example?

Solution:
We know Profit Percentage = (Profit/Cost Price)*100
Profit = Rs. 20/-
Apply the Profit and Cost Price Values in the Formula for Profit %
Profit % = (20/80)*100
= 25
Therefore, the shopkeeper gained a 25 % Profit Percentage.

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Let us learn how to represent Integers on a Number Line at first. To represent Integers on Number Line consider a Point called Zero. The Points to the right of 0 are denoted by + Sign and are Positive Integers. The Points to the left of 0 are denoted by – Sign and are Negative Integers. Now that you are aware of Representation of Integers on the Number Line representing Rational Numbers on the number line would be much easier to understand.

How to Represent Rational Numbers on the Number Line?

Let us discuss how to represent Rational Numbers on the Number Line. Similar to Positive Integers Positive Rational Numbers would be placed to the right of 0 and Negative Rational Numbers would be marked to the left of 0.

For instance which side you would mark -1/3 on the number line. The Answer is quite clear as it is negative you need to place it on the left of 0. While marking integers on the number line successive integers will be placed at equal intervals, i.e. 1 and -1 will be equidistant. The same is with 2, -2 and 3, -3 and so on.

This would be the case with Rational Numbers 1/2, -1/2 would be equidistant from 0.

Solved Examples

1.  Represent 1/2 and -1/2 on the Number Line?

Solution:

Draw a line and mark point 0 at the center. Set off units to the right and left of 0 as they are equidistant.

Mark the rational number 1/2 between 0 and 1.

So is the case with -1/2 and place it between 0 and -1.

2. Represent 4/3, 5/3, 6/3, 7/3 on the Number Line?

Solution:

Draw a line and mark point 0 at the center. The line extends indefinitely on both sides.

Split the number line into 3 equal parts between 0 and 1. The first point of the division is given by 1/3 and the second point of the division is 2/3, third point 3/3, fourth point as 4/3, 5th point as 5/3, sixth point as 6/3, seventh point as 7/3.

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To help you better understand the concept of Representing Rational Numbers on the Number Line we have taken a few examples. Check out the detailed explanation provided and understand the concept better. After going through this article, you will learn how to solve related problems on your own.

In general, Positive Rational Numbers are always represented on the right of the 0 on the number line. Negative Rational Numbers are represented to the left of 0 on the number line.

Solved Examples on Rational Numbers on the Number Line

1. Represent 1/8 and 3/8 on the Number Line?

Solution: 

Firstly, draw the number line mark 0 on it. Split the number line into 8 equal parts and then mark the first division to the right i.e. 1/8. The Second division is 2/8 and the third division is 3/8. The fourth division is 4/8 and so on.

2. Write the rational number for each point labeled with a letter below?

Solution:

We know the positive rational numbers are represented to the right of 0. As all of the rational numbers given are successive. Therefore the Rational Numbers Labelled with Letters A, B, C, D, E are 1/5, 4/5, 5/5, 8/5, 9/5.

Thus, All the Rational Numbers 0/5, 1/5, 2/5, 3/5, 4/5, 5/5, 6/5, 7/5, 8/5, 9/5, 10/5, 11/5, 12/5 will be adjacent to each other on the Number Line.

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Let us learn about the Addition of Rational Numbers with Same Denominator from here. Get to know the detailed procedure on how to add rational numbers with the same denominator. Have a look at the few examples to better understand the concept of Rational Numbers Addition having the Same Denominator.

How to Add Rational Numbers with the Same Denominator?

Below are easy guidelines listed on how to Add Rational Numbers with the Same Denominator. Follow them while adding rational numbers having the common denominator. They are along the lines

Step 1: Determine the numerators of two given rational numbers and their common denominator.

Step 2: Add the Numerator of the Rational Numbers obtained in the earlier step.

Step 3: Make a note of the Rational Number whose numerator is the sum of rational numbers obtained in step 2 and keep the denominator unaltered. Simplify the Rational Number if Required.

From the above steps, we can infer that a/b, c/b are two rational numbers then the Addition of Two Rational Numbers with the Same Denominator is given by (a+b)/c.

Solved Examples on Addition of Rational Numbers with Same Denominator

1. Find the Sum of 3/9 and -7/9?

Solution:

Given Rational Numbers are 3/9 and -7/9

= 3/9+(-7)/9

= (3-7)/9

= -4/9

Sum of 3/9, -7/9 is -4/9.

2. Find the Sum of 8/-10 and 4/10?

Solution:

Given Rational Numbers are 8/-10, 4/10

Since the rational numbers don’t have a common denominator. Firstly, make them into a positive denominator.

8/-10 = 8*(-1)/-10*(-1) = -8/10

Adding -8/10 and 4/10

= -8/10+4/10

= (-8+4)/10

= -4/10

3. Add 3/5 and 17/5?

Solution:

Given Rational Numbers are 3/5 and 17/5

Adding the given Rational Numbers we get

= 3/5+17/5

= (3+17)/5

= 20/5

On Simplifying we get the following Rational Number

= 4/1

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Let us discuss in detail How to Add Rational Numbers with Different Denominators. To find the Sum of Rational Numbers you can have a look at the below examples provided. Check the detailed procedure to add rational numbers not having the common denominator.

How to Add Rational Numbers with Different Denominators?

Go through the below-mentioned steps to add rational numbers having different denominators. They are along the lines

Step 1: Check the Rational Numbers and see whether the denominators are positive or not. If any of the denominators is negative rearrange them to become positive.

Step 2: List out the denominators of rational numbers in the earlier step.

Step 3: Determine the LCM of the denominators of the given Rational Numbers.

Step 4: Express the Rational Numbers in a way that they have the Common Denominator using the LCM obtained.

Step 5: Write down the numerator as the sum of numerators of rational numbers obtained in the earlier step and denominators is the LCM obtained in Step 3.

Step 6: Rational Number obtained in Step 5 is the required sum and simplify if required.

Solved Examples on Adding Rational Numbers with Different Denominator

1.  Add Rational Numbers -8/21 and 3/9?

Solution:

Given Rational Numbers are -8/21, 3/9

= -8/21+3/9

Firstly, find the Least Common Multiple of the Denominators i.e. 21, 9

LCM(21, 9) = 63

Express the Rational Numbers with Common Denominator using the LCM Obtained.

-8/21 = -8*3/21*3 = -24/63

3/9 = 3*7/9*7 = 21/63

Adding them we get

-24/63+21/63

= (-24+21)/63

= -3/63

Simplifying further we get -1/21

Therefore the Sum of -8/21 and 3/9 is -1/21.

2. Add 3/4 and 5?

Solution:

Given Rational Numbers are 3/4 and 5/1

= 3/4+5/1

Find the LCM of 4, 1

LCM(4, 1) = 4

Express the Given Rational Numbers in terms of Common Denominator using the LCM obtained.

3/4 = 3*1/4*1 = 3/4

5/1 = 5*4/1*4 = 20/4

Add the obtained rational numbers numerators with a common denominator

= (3+20)/4

= 23/4

Therefore, the Sum of 3/4 and 5 is 23/4.

3. Simplify 5/-12+7/-4?

Solution:

Given Rational Numbers are 5/-12 and 7/-4

Since the rational numbers are having a negative denominator rearrange them to have a positive denominator.

5/-12 = 5*(-1)/-12*(-1) = -5/12

7/-4 = 7*(-1)/-4*(-1) = -7/4

Find the LCM of Denominators of the Rational Numbers.

LCM(12, 4) = 12

Express the Rational Numbers with Common Denominator using the LCM obtained.

-5/12 = -5*1/12*1 = -5/12

-7/4= -7*3/4*3 = -21/12

Add the Obtained Rational Numbers Numerators and keep the denominator unchanged.

= (-5-21)/12

= -26/12

Thus, 5/-12+7/-4 is -26/12.

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In Maths, Rational Numbers sound similar to Fractions and they are expressed in the form of p/q where q is not equal to zero. Any fraction that has non zero denominators is called a Rational Number. Thus, we can say 0 also a rational number as we can express it in the form of 0/1, 0/2 0/3, etc. However, 1/0, 2/0 aren’t rational numbers as they give infinite values.

Continue reading further modules to learn completely about Rational Numbers. Get to know about Types of Rational Numbers, Difference Between Rational and Irrational Numbers, Solved Examples, and learn how to Identify Rational Numbers, etc. In order to represent Rational Numbers on a Number Line firstly change them into decimal values.

Definition of Rational Number

Rational Number in Mathematics is defined as any number that can be represented in the form of p/q where q ≠ 0. On the other hand, we can also say that any fraction fits into the category of Rational Numbers if bot p, q are integers and the denominator is not equal to zero.

• Class 8 Rational Numbers NCERT Solutions
• Rational Numbers Class 8 Extra Questions
• Rational Numbers Class 8 Notes
• NCERT Exemplar Class 8 Maths Rational Numbers
• Class 7 Rational Numbers NCERT Solutions
• Rational Numbers Class 7 Extra Questions
• Rational Numbers Class 7 Notes
• Introduction to Rational Numbers
• What are Rational Numbers?
• Is Every Rational Number a Natural Number?
• Is Zero a Rational Number?
• Is Every Rational Number an Integer?
• Is Every Rational Number a Fraction?
• Positive Rational Number
• Negative Rational Number
• Equivalent Rational Numbers
• Equivalent Form of Rational Numbers
• Rational Numbers in Different Forms
• Properties of Rational Numbers
• Lowest Form of a Rational Number
• Standard Form of a Rational Number
• Equality of Rational Numbers using Standard Form
• Equality of Rational Numbers with Common Denominator
• Equality of Rational Numbers using Cross Multiplication
• Comparison of Rational Numbers
• Rational Numbers in Ascending Order
• Rational Numbers in Descending Order
• Representation of Rational Numbers on the Number Line
• Rational Numbers on the Number Line
• Addition of Rational Numbers with Same Denominator
• Addition of Rational Numbers with Different Denominator
• Addition of Rational Numbers
• Properties of Addition of Rational Numbers

How to Identify Rational Numbers?

You need to check the following conditions to know whether a number is rational or not. They are as follows

• It should be represented in the form of p/q, where q ≠ 0.
• Ratio p/q can be further simplified and expressed in the form of a decimal value.

The set of Rational Numerals include positive, negative numbers, and zero. It can be expressed as a Fraction.

Examples of Rational Numbers

Types of Rational Numbers

You can better understand the concept of sets by having a glance at the below diagram.

• Real numbers (R) include All the rational numbers (Q).
• Real numbers include the Integers (Z).
• Integers involve Natural Numbers(N).
• Every whole number is a rational number as every whole number can be expressed in terms of a fraction.

Standard Form of Rational Numbers

A Rational Number is said to be in its standard form if the common factors between divisor and dividend is only one and therefore the divisor is positive.

For Example, 12/24 is a rational number. It can be simplified further into 1/2. As the Common Factors between divisor and dividend is one the rational number 1/2 is said to be in its standard form.

Positive and Negative Rational Numbers

Arithmetic Operations on Rational Numbers

Let us discuss how to perform basic operations i.e. Arithmetic Operations on Rational Numbers. Consider p/q, s/t as two rational numbers.

Addition: Whenever we add two rational numbers p/q, s/t we need to make the denominator the same. Thus, we get (pt+qs)/qt.

Ex: 1/3+3/4 = (4+3)/12 = 7/12

Subtraction: When it comes to subtraction between rational numbers p/q, s/t we need to make the denominator the same and then subtract.

Ex: 1/2-4/3 = (3-8)/6 = -5/6

Multiplication: While Multiplying Rational Numbers p/q, s/t simply multiply the numerators and the denominators of the rational numbers respectively. On multiplying p/q with s/t then we get (p*s)/(q*t)

Ex: 1/3*4/2=4/6

Division: Division of p/q & s/t is represented as (p/q)÷(s/t) = pt/qs

Ex: 1/4÷4/3 =1*3/4*4 = 3/16

Properties of Rational Numbers

• If we add a zero to a Rational Number you will get the Rational Number Itself.
• Addition, Subtraction, Multiplication of a Rational Number yields in a Rational Number.
• Rational Number remains the same on multiplying or dividing both the numerator and denominator with the same factor.

There are few other properties of rational numbers and they are given as under

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property
• Identity Property
• Inverse Property

Representation of Rational Numbers on a Number Line

Number Line is a straight line diagram on which each and every point corresponds to a real number. As Rational Numbers are Real Numbers they have a specific location on the number line.

Rational Numbers Vs Irrational Numbers

There is a difference between Rational Numbers and Irrational Numbers. Fractions with non zero denominators are called Rational Numbers. All the numbers that are not Rational are Called Irrational Numbers. Rational Numbers can be Positive, Negative, or Zero. To specify a negative Rational Number negative sign is placed in front of the numerator.

When it comes to Irrational Numbers you can’t write them as simple fractions but can represent them with a decimal. You will endless non-repeating digits after the decimal point.

Pi (π) = 3.142857…

√2 = 1.414213…

Solved Examples

Example 1.

Identify whether Mixed Fraction 1 3/4 is a Rational Number or Not?

Solution: The Simplest Form of Mixed Number 1 3/4 is 7/4

Numerator = 7 which is an integer

Denominator = 4 which is an integer and not equal to 0.

Thus, 7/4 is a Rational Number.

Example 2.

Determine whether the given numbers are rational or irrational?

(a) 1.45 (b) 0.001 (c) 0.15 (d) 0.9 (d) √3

Solution:

Given Numbers are in Decimal Format and to find out whether they are rational or not we need to change them into fraction format i.e. p/q. If the denominator is non zero then the number is rational or else irrational.

FAQs on Rational Numbers

1. How to Identify a Rational Number?

If the Number is expressed in the form of p/q where p, q are integers and q is non zero then it called a Rational Number.

2. Is 5 a Rational Number?

Yes, 5 is a Rational Number as it can be expressed in the form of 5/1.

3. What do we get on adding zero to a Rational Number?

On Adding Zero to a Rational Number, you will get the Same Rational Number.

4. What is the difference between Rational and Irrational Numbers?

Rational Numbers are terminating decimals whereas Irrational Numbers are Non-Terminating Decimals.

The post Rational Numbers | Definition, Types, Properties, Standard Form of Rational Numbers appeared first on Learn CBSE.

In this article, you will learn about the Addition Operation and how to perform the Addition of Rational Numbers. Usually, the Addition of Rational Numbers is much similar to the Addition of Fractions. The First and Foremost Step to keep in mind when it comes to the Rational Numbers Addition is the denominators should be positive. If the denominators aren’t positive simply rearrange them to make them positive.

The two subcategories that you encounter while dealing with the Addition Operation in Rational Numbers are listed below. You can check out the solved examples explained step by step for reference and get a grip on the concept.

Rational Numbers with the Same Denominator

Let suppose a/b, c/b be two rational numbers having the common denominator b then Addition of Rational Numbers is given by the summation of numerators leaving the common denominator unchanged i.e. (a+c)/b.

Solved Examples

(i) Add 4/7 and 13/7?

Solution:

Given Rational Numbers are 4/7 and 13/7

Adding them we get 4/7+13/7

= (4+13)/7

= 17/7

Therefore, the Sum of 4/7 and 13/7 is 17/7.

(ii) Add 5/12 and -3/12?

Solution:

Given Rational Numbers are 5/12, -3/12

Adding them we get 5/12+(-3)/12

= (5-3)/12

= -2/12

Rational Numbers with the Different Denominator

In the case of Rational Numbers with Different Denominators, we need to find the LCM of Denominators. After that, express the given rational numbers with a common denominator and then add the numerators of the obtained rational numbers while keeping the denominator unchanged.

Solved Examples 

(i) Add 4/5 and 7/9?

Solution:

Clearly, the denominators are different and we need to figure out the LCM of Denominators.

LCM(5,9) is 45

Express the Given Rational Numbers with a Common Denominator using the LCM obtained.

4/5 = 4*9/5*9 = 36/45

7/9 = 7*5/9*5 = 35/45

Add the Numerators of the Rational Numbers while keeping the Denominator unchanged to get the sum of Rational Numbers.

= 36/45+35/45

= 71/45

Therefore the sum of 4/5 and 7/9 is 71/45.

(ii) Find the Sum of -7/6 and 5/12?

Solution: 

Given Rational Numbers are -7/6 and 5/12

Since the denominators aren’t equal find the LCM of them

LCM of 6, 12 is 12

Express the given rational numbers with a common denominator using the LCM acquired.

-7/6 = -7*2/6*2 = -14/12

5/12 = 5*1/12*1 = 5/12

Adding the Rational Numbers we get

= (-14+5)/12

=-9/12

Therefore, the Sum of -7/6 and 5/12 is -9/12.

The post Addition of Rational Numbers appeared first on Learn CBSE.

Learn the Properties of Rational Numbers such as Closure Property, Commutative Property, Identity Property, Associative Property, Additive Inverse Property, etc. We tried explaining each and every Property of Rational Numbers Addition in the following sections. Check them and learn the concepts easily. To help you understand the concept, even more, better we have provided a few examples.

Closure Property of Addition of Rational Numbers

Closure Property is applicable for the Addition Operation of Rational Numbers. The Sum of Two Rational Numbers always yields in a Rational Number. Let a/b, c/d be two rational numbers then (a/b+c/d) is also a Rational Number.

Examples

(i) Consider the Rational Numbers 5/4 and 1/3

= (5/4+1/3)

= (5*3 +1*4)/12

= (15+4)/12

= 19/12

Therefore, the Sum of Rational Numbers 5/4 and 1/3 i.e. 19/12 is also a Rational Number.

(ii) Consider the Rational Numbers -4/3 and 2/5

= -4/3+2/5

= (-4*5+2*3)/15

= (-20+6)/15

= -14/15 is also a Rational Number.

Commutative Property of Addition of Rational Numbers

Commutative Property is applicable for the Addition Operation of Rational Numbers. Two Rational Numbers can be added in any order. Let us consider two rational numbers a/b, c/d then we have

(a/b+c/d) = (c/d+a/b)

Examples

(i) 1/3+4/5

= (5+12)/15

= 17/15

and 4/5+1/3

= (12+5)/15

= 17/15

Therefore, (1/3+4/5) = (4/5+1/3).

(ii) -1/2+3/2

= (-1+3)/2

= 2/2

and 3/2+(-1/2)

= (3-1)/2

= 2/2

Therefore, (-1/2+3/2) = (3/2+-1/2)

Associative Property of Addition of Rational Numbers

While adding Three Rational Numbers you can group them in any order. Let us consider three Rational Numbers a/b, c/d, e/f we have

(a/b+c/d)+e/f = a/b+(c/d+e/f)

Example

Consider Three Rational Numbers 1/2, 3/4 and 5/6 then

(1/2+3/4)+5/6 = (2+3)/4+5/6

= 5/4+5/6

= (15+10)/12

= 25/12 and

1/2+(3/4+5/6) = 1/2+(9+10)/12

= 1/2+19/12

= (6+19)/12

=25 /12

Therefore, (1/2+3/4)+5/6 = 1/2+(3/4+5/6)

Additive Identity Property of Addition of Rational Numbers

0 is a Rational Number and any Rational Number added to 0 results in a Rational Number.

For every Rational Number a/b (a/b+0)=(0+1/b)= a/b and 0 is called the Additive Identity for Rationals.

Example

(i) (4/5+0) = (4/5+0/5) =(4+0)/5 =4/5 and similarly (0+4/5) = (0/5+4/5) = (0+4)/5 = 4/5

Therefore, (4/5+0) = (0+4/5) = 4/5

(ii)(-1/3+0) =(-1/3+0/3) =(-1+0)/3 = -1/3 and similarly (0-1/3) = (0/3-1/3) =(0-1)/3 = -1/3

Therefore, (-1/3+0) =(0+-1/3) = -1/3

Additive Inverse Property of Addition of Rational Numbers

For every Rational Number a/b there exists a Rational Number -a/b such that (a/b+-a/b)=0 and (-a/b+a/b)=0

Thus, (a/b+-a/b) = (-a/b+a/b) = 0

-a/b is called the Additive Inverse of a/b

Example

(4/3+-4/3) = (4+(-4))/3 = 0/3 = 0

Similarly, (-4/3+4/3) = (-4+4)/3 = 0/3 = 0

Thus, 4/3 and -4/3 are additive inverse of each other.

The post Properties of Addition of Rational Numbers appeared first on Learn CBSE.

If you are worried about How to Subtract Rational Numbers having the Same Denominator you have come the right way. To help you out we have given the Step by Step Process to follow while solving problems on Subtraction of Rational Numbers having the Common Denominator. Have a glance at the solved examples and better understand the concept.

How do you Subtract Rational Numbers with the Same Denominator?

Go through the easy guidelines listed to Subtract Rational Numbers with the Same Denominator. Following these steps, you will arrive at the solution easily. They are along the lines

Step 1: Firstly, obtain the numerators of given rational numbers along with the common denominator.

Step 2: Subtract the First Numerator from the Second One.

Step 3: Note down the Rational Number whose numerator is the difference of two given Rational Numbers in the earlier step and retain the common Denominator unchanged. Simplify the Rational Number Obtained if Required.

From the above-provided information, we can say that for two rational a/b and c/b having the common denominator b, a/b – c/b is equal to (a-c)/b.

Solved Examples on Subtraction of Rational Numbers with Common Denominator

1. Find the Difference Between 4/7 and 12/7?

Solution:

Given Rational Numbers are 4/7 and 12/7

Difference between Rational numbers = 12/7 -4/7

= (12-4)/7

= 8/7

Therefore 12/7-4/7 = 8/7.

2. Find the Difference between 3/4 and -5/4?

Solution:

Given Rational Numbers are 3/4, 5/4

=3/4-(-5/4)

= 8/4

=2/1

3. Subtract 2/17 from -6/17?

Solution:

Rational Numbers are 2/17, -6/17

= -6/17-2/17

= -8/17

4. Subtract Rational Number 5/3 from 17/3?

Solution:

Given Rational Numbers are 5/3 and 17/3

= 17/3-5/3

= 12/2

= 4/1

5. Find the Difference between 4/-5 and 2/5?

Solution:

Given Rational Numbers are 4/-5 and 2/5

Changing the Rational Number with negative Denominator to Positive Denominator we get

4/-5 = 4*(-1)/-5*(-1) = -4/5

= -4/5-2/5

= -6/5

6. Subtract Rational Number -2/3 from -4/3?

Solution:

Given Rational Numbers are -2/3 and -4/3

= -4/3-(-2/3)

= (-4+2)/3

= -2/3

The post Subtraction of Rational Numbers with Same Denominator appeared first on Learn CBSE.