Division of Fractions

Division means sharing something equally. For example, if we divide 20 sweets equally among 5 children, each child will get 4 sweets. Similarly, Division of fractions means dividing a fraction into equal parts. Like if we want to divide 2/3 of the cake among 6 people, each person will get 2/3 ÷ 6 = 1/9 part of the cake. For division in fractions, we multiply the first fraction with the reciprocal of the second fraction.

In this article, we will learn about fractions, the reciprocal of fractions, the division of a fraction with whole numbers, another fraction, and decimals along with some solved examples for better understanding.

What is a Fraction?

A fraction is simply a portion of a whole. If we divide a whole into equal parts, each part is a fraction. A fraction consists of two parts: a numerator and a denominator. For example, 2/5 is a fraction, where 2 is a numerator and 5 is a denominator. A fraction is represented as:

Fraction = Numerator/Denominator

Here, the numerator represents the number of parts taken and the denominator represents the total number of parts. For example, 2/5 represents 2 parts out of a total of 5 parts.

Read More about Numerator and Denominator.

Definition of Fraction

A fraction is represented in the form of p/q, where p and q are whole numbers with a condition that q cannot be 0.

What is Division of Fractions?

Division of fractions means further dividing a part (or fraction) into equal parts or groups. For the division of fractions, we multiply the dividend with the multiplicative inverse of the divisor. First, we should learn what the multiplicative inverse of a fraction is.

Multiplicative Inverse of a Fraction

If the product of two numbers is 1 then, each number is called Multiplicative inverse or Reciprocal of the other. For example:

As 3/5 × 5/3 = 1,

3/5 is multiplicative inverse of 5/3 and vice-versa.

We can simply find the multiplicative inverse of a fraction by interchanging its numerator and denominator. Some facts about Multiplicative Inverse:

• 1 is the only positive number whose Multiplicative inverse is the number itself.
• The number 0 does not have any reciprocal as division of 0 is not possible.

How to do Division of Fraction?

Keep-Switch-Flip is the basic rule for dividing fractions, which means Keep the first number, Switch the division sign to the multiplication sign, and Flip the other number. We can also say that to divide a fraction, the following steps are involved:

• Find the multiplicative inverse of the second fraction (or number).
• Multiply the first fraction by the second fraction.
• Simplify the result if needed.

We can divide a fraction with another fraction, a mixed fraction, or a whole number. Further in this article, we will learn about the division of fractions with a whole number and, the division of fractions with other fractions or mixed fractions.

Division of a Fraction by a Whole Number

To find a division of a fraction with a whole number we can simply multiply the whole number with the denominator of the fraction or we can follow these steps:

Steps involved in the division of a fraction with a whole number

• First, find the multiplicative inverse of the whole number by taking 1 as its denominator.
• Reduce the fraction to its lowest term.
• Then multiply the fraction with the new divisor(i.e. the multiplicative inverse of the whole number).
• Lastly, reduce the result to its lowest term.

Let’s take an example for better understanding.

Question: Divide 7/5 by 14.

Solution:

Step 1: Find the multiplicative inverse of 14/1, which is 1/14.

Step 2: Reduce the fraction to its lowest term.

7/5 is already in its lowest term

Step 3: Find the product of 7/5 and 1/14:

7/5 × 1/14 = 7/70.

Step 4: Reduce the result to its lowest term

The lowest term of 7/70 is 1/10

Hence, 7/5 ÷ 14 = 1/10.

​Division of a Whole Number by a Fraction

While dividing a whole number with a fraction, we can follow these simple steps:

Steps involved in the division of a whole number with a fraction

• First, find the multiplicative inverse of the fraction.
• Reduce the fraction to its lowest term.
• Then multiply the whole number with the new divisor(i.e. the multiplicative inverse of the fraction).
• Lastly, reduce the result to its lowest term.

Question: Divide 8 by 4/14.

Solution:

Step 1: Find the multiplicative inverse of 4/14, which is 14/4.

Step 2: Reduce the fraction to its lowest term.

The lowest term of 14/4 is 7/2

Step 3: Find the product of 8 and 7/2:

8 × 7/2 = 56/2.

Step 4: Reduce the result to its lowest term

The lowest term of 56/2 is 28/1 = 28

Hence, 8 ÷ 4/14 = 28.

Division of a Fraction by Fraction

We can easily Divide a fraction with another fraction by following these simple steps:

Steps involved in the division of fraction by fraction

• First, find the multiplicative inverse of the divisor (or the second fraction).
• Reduce both fractions to their lowest term.
• Then, multiply the dividend with the new divisor.
• Lastly, reduce the result to its lowest term.

Question: Divide 3/5 by 15/6.

Solution:

Step 1: Find the reciprocal of 15/6, which is 6/15

Step 2: Reduce both fractions to their lowest term.

3/5 is already in its lowest term and the lowest term of 6/15 = 2/5

Step 3: Find the product of 3/5 and 6/15:

3/5 × 2/5 = 6/25.

Step 4: Reduce the result to its lowest term

6/25 is already in its lowest term.

Hence, 3/5 ÷ 15/6 = 6/25.

Division of a Fraction by a Mixed fraction

In this case, we first convert the mixed fraction to an improper fraction, the rest of the steps are the same as a division of a fraction with a fraction.

Steps involved in the division of a fraction by a mixed fraction

• First, convert the mixed fraction into an improper fraction.
• Then, find the multiplicative inverse of the divisor (or the second fraction).
• Reduce both fractions to their lowest term.
• Then, multiply the dividend with the new divisor.
• Lastly, reduce the result to its lowest term.

Question: Divide 2/5 by 1 5/6.

Solution:

Step 1: Convert mixed fraction to improper fraction

1 5/6 = 11/6

Step 2: Find the reciprocal of 11/6, which is 6/11

Step 3: Reduce both fractions to their lowest term.

2/5 and 6/11 are already in their lowest term

Step 4: Find the product of 2/5 and 6/11:

2/5 × 6/11 = 12/55.

Step 5: Reduce the result to its lowest term

12/55 is already in its lowest term

Hence, 2/5 ÷ 1 5/6 = 12/55.

Division Of Fractions by Decimal

For the division of decimals and fractions, we first convert the decimal number into a fraction and then multiply the reciprocal of the divisor with the dividend.

Steps involved in the division of a fraction with a decimal

• First, convert the decimal into a fraction.
• Then, find the multiplicative inverse of the divisor (or the second fraction).
• Reduce both fractions to their lowest term.
• Then, multiply the dividend with the new divisor.
• Lastly, reduce the result to its lowest term.

Question: Divide 6/20 by 0.12

Solution:

Step 1: Convert the decimal into a fraction

0.12 = 12/100

Step 2: Find the reciprocal of 12/100, which is 100/12

Step 3: Reduce both fractions to their lowest term.

The lowest term of 6/20 is 3/10 and the lowest term of 100/12 is 25/3.

Step 4: Find the product of 3/10 and 25/3:

3/10 × 25/3 = 75/30.

Step 5: Reduce the result to its lowest term

The lowest term of 75/30 is 5/2.

Hence, 6/20 ÷ 0.12 = 5/2.

Multiplication and Division of fraction

Multiplication of fractions is pretty simple, we just need to multiply all the numerators together, multiply all the denominators together and then simplify the result if required. We can multiply two or more fractions together. It can be explained by the following example:

Example: Multiply 3/4 by 1/12.

Solution:

we have 3/4 × 1/12

= (3 × 1)/ (4 × 12)

= 3/48

= 1/16 ( on simplifying)

Hence, 3/4 × 1/12 = 1/16

For the Division of Fractions, we use the same approach as the multiplication of fraction after flipping the second fraction. In division, we multiply the first fraction with the reciprocal of the other fraction. It can be explained by following example:

Example: Divide 3/4 by 1/12.

Solution:

we have 3/4 ÷ 1/12

= 3/4 × 12/1

= (3 × 12)/ (4 × 1)

= 36/4

= 9 ( on simplifying)

Hence, 3/4 ÷ 1/12 = 9.

Read more

• Addition of Fraction
• Subtraction of Fraction
• Multiplication of Fraction

Examples of Division of Fraction

Example 1: Divide 2 1/5 by 3 1/5.

Solution:

First, convert mixed fraction to improper fraction

2 1/5 = 11/5 and 3 1/5 = 16/5

Now we have, 11/5 ÷ 16/5

= 11/5 × 5/16

= 55/80

= 11/16 (by reducing in its lowest term).

Hence, 2 1/5 ÷ 3 1/5 = 11/16

Example 2: Divide 7/10 by 14/15.

Solution:

We have, 7/10 ÷ 14/15

= 7/10 × 15/14

= 105/140

= 3/4 (by reducing in its lowest term)

Hence, 7/10 ÷ 14/15 = 3/4.

Example 3: Divide 1.2 by 0.6.

Solution:

First converting 1.2 and 0.6 in fraction, we have, 12/10 and 6/10

Now, 12/10 ÷ 6/10

= 6/5 ÷ 3/5 (by reducing each in its lowest term)

= 6/5 × 5/3

= 30/15

=2 (by reducing in its lowest term)

Hence, 0.12 ÷ 0.6 = 2.

Example 4: Divide 3 by 9/10.

Solution:

We have, 3 ÷ 9/10

= 3 × 10/9

= 30/9

=10/3 (by reducing in its lowest term)

Hence, 3 ÷ 9/10 = 10/3.

Example 5: Divide 0.45 by 5.

Solution:

First, by converting 0.45 into a fraction, we get 45/100

Now we have,

45/100 ÷ 5

= 45/100 × 1/5

= 9/20 × 1/5 (by reducing in its lowest term)

= 9/100

= 0.09

Hence, 0.45 ÷ 5 = 0.09

Division of Fraction Word Problem

Example 1: A recipe requires 3/4th cup of flour for 15 cupcakes. How much flour is needed for one cupcake?

Solution:

Flour required for 15 cupcakes = 3/4th cup

Flour required for 1 cupcake = 3/4 ÷ 15

= 3/4 ÷ 15/1

= 3/4 × 1/15

= (3 × 1) / (4 × 15)

= 3/60

= 1/20

Hence, flour required for 1 cupcake is 1/20th cup.

Example 2: A car can travel 44 km using 11/4 litres of petrol. How much distance it can travel using 1 litre of petrol?

Solution:

Distance travelled by a car in 11/4 litres of petrol = 44 km

Distance travelled by a car in 1 litre of petrol = 44 ÷ 11/4

= 44/1 × 4/11

= (44 × 4)/ (1 × 11)

= 176/11

= 16

Hence, the car can travel 16 km in 1 litre of petrol.

Example 3: A jug contains 4/5 litre of orange juice, it is poured equally into 8 glasses. How much juice is in each glass?

Solution:

Quantity of orange juice in the jug = 4/5 litre

Number of glasses to be filled = 8

Quantity of juice in each glass = 4/5 ÷ 8

= 4/5 × 1/8

= (4 × 1) / (5 × 8)

= 4/40

= 1/10

Hence, each glass will have 1/10 litre of juice.

Example 4: The breadth of the rectangular park of area 5 1/4 m² is 1 2/5m. Find the length of the park.

Solution:

Area of rectangular park = 5 1/4 m² = 21/4 m²

Breadth of rectangular park = 1 2/5 m = 7/5 m

Length of rectangular park = 21/4 ÷ 7/5

= 21/4 × 5/7

= (21 × 5)/(4 × 7)

= 105/28

= 15/4

= 3 (3/4) (in mixed fraction)

Hence, the length of the park is 3 (3/4) m.

Example 5: The product of two numbers is 11/18, if one number is 2/9, find the other number.

Solution:

The product of two numbers = 11/18

The first number = 2/9

The other number = 11/18 ÷ 2/9

= 11/18 × 9/2

= (11 × 9) / (18 × 2)

= 99/36

= 11/4

Hence, the other number is 11/4.

Practice Problems on Division of Fraction

Problem 1: Divide the following:

• 15/9 ÷ 5/3
• 5/8 ÷ 4 1/2
• 3/5 ÷ 9/24
• 2 1/2 ÷ 3 1/3
• 0.2 ÷ 0.06
• 7/15 ÷ 0.50
• 3 ÷ 2/5
• 8/9 ÷ 9/4
• 3/8 ÷ 20
• 12 ÷ 2 1/5

Practice Word Problems on Division of Fraction

Problem 1: A vehicle can travel 5/9 km in 1/3 litres of petrol. Find the distance traveled using 1 litre of petrol.

Problem 2: The product of two numbers is 3 2/5. if one of the numbers is 5/6. Find the other number.

Problem 3: The cost of 6 2/5 kg of potatoes is Rs 320. Find the cost of 1 kg of potatoes.

Problem 4: The perimeter of a square is 3 1/5 m. Find the length of each side of the square.

Division of Fraction – FAQs

1. What is the Division of Fraction Rule?

The rule of division of fraction is Keep, Switch, Flip

• Keep the first number
• Switch the division sign with multiplication sign.
• Flip the second number and then multiply the first and second number.

2. How to Divide Fractions?

To perform division of fractions, follow the following steps:

• Find the multiplicative inverse of the second fraction ( or number).
• Multiply the first fraction with second fraction.
• Simplify the result if needed.

3. Why Division by Zero is Undefined?

Division simply means repeated subtraction. Division is finding out how many times a number is subtracted from another number until the result turns to 0. However, if we divide a number by 0, it remains undefined because its like trying to subtract 0 from a number over and over again, no matter how many times we subtract 0 from a number, the result will never turn to 0.

4. What is meant by the Lowest Term or Simple Form of a Fraction?

By lowest term or simplest form of a factor, we mean that the numerator and denominator are relatively prime i.e. 1 is the only common factor of the numerator and denominator.

5. What is a Proper Fraction?

A fraction is said to be a proper fraction if its value lies between 0 and 1 or we can say in a proper fraction the value of the numerator is always less than that of the denominator.

6. What is a Unit Fraction?

If the numerator of a fraction is 1 it is said to be a unit fraction,

7. What is the Reciprocal of 0?

0 (Zero) is the only number whose reciprocal does not exist.

8. Can the Value of a Fraction be less than Zero?

No the value of a fraction can never be less than zero i.e. a fraction is always positive.

9. What is the result of the Division of Equivalent Fractions?

If we divide two equivalent fractions the result will always be 1.