Dividing Fractions

Dividing fractions is equal to the multiplication of a fraction by the reciprocal of another fraction. A fraction has a numerator and a denominator. When we divide one fraction by another, then we almost multiply the fractions with a twist.   

Suppose, ½ is divided by ⅔. A reciprocal of ⅔ is 3/2. Now, ½ is multiplied by 3/2 to get the required division value. Hence, the division of fractions includes an additional step that we follow while multiplying fractions.

What are Fractions?

A fraction is a part of a whole value or number. It is represented by p/q or a/b or m/n, etc. The upper part of a fraction is called the numerator and the lower part is the denominator. Examples of fractions are ½, ¼, ⅔, ⅗, etc.

All the arithmetic operations such as addition, subtraction, multiplication and division can be performed upon the fractions. Let us learn here how to divide a fraction by a fraction, by a whole number and by a mixed number with the help of examples, with simple steps.

What is Meant by Dividing Fractions?

Dividing fractions is nothing but multiplying the fractions by reversing one of the two fraction numbers or by writing the reciprocal of one of the fractions. By reciprocal we mean, if a fraction is given as a/b, then the reciprocal of it will b/a. Thus, interchanging the position of numerator and denominator with each other.

How to Divide Fractions?

The division of fractions can be classified into three different ways. They are

• Dividing fractions by a fraction
• Dividing fractions by whole number
• Dividing fractions by mixed fraction

Let us discuss all these three methods in a detailed way

Dividing Fraction by a Fraction

In three simple steps, we can solve the division of fractions by converting them into the multiplication of fractions. Let us learn one by one.

Step 1: Write the reciprocal of the second fraction number and multiply it with the first fraction number

Step 2: Multiply the numerators and denominators of both fractions

Step 3: Simplify the fraction number

In general, if a/b is a fraction which is divided by c/d. Then we can solve the division as;

• a/b ÷ c/d = a/b × d/c
• a/b ÷ c/d = a×d / b×c
• a/b ÷ c/d = ad/bc

You can see from the above expressions. The a/b is divided by c/d, then we can write it as a/b multiplied by d/c (reciprocal of c/d). And in the next step, we have to multiply both the numerator a & d and both the denominator, c & d. Hence, we can simplify the rest calculation.

Dividing Fraction by a Whole Number

While dividing the fractions with whole numbers, the process of division is very easy. Follow the procedure given below. 

Step 1: The whole number is converted into a fraction by applying the denominator value is 1

Step 2: Take the reciprocal of the number

Step 3: Now, multiply the fractional value by a given fraction

Step 4: Simplify the given expression

Example: Divide 6/5 by 10

Step 1: Convert 10 into a fraction: 10/1

Step 2: Take reciprocal: 1/10

Step 3: Multiply 6/5 and 1/10: (6/5)×(1/10)

Step 4: Simplify: 3/25

Dividing Fractions by a Mixed Fraction

The process of dividing fractions by a mixed fraction is almost similar to dividing fractions by a fraction. The steps to perform the division of a fraction by a mixed fraction are as follows:

Step 1: Convert the mixed fraction into the improper fraction

Step 2: Now, take the reciprocal for the improper fraction

Step 3: Multiply the obtained fraction by a given fraction

Step 3: Simplify the fractions

Example: Divide ⅖ by 3½.

Step 1: Convert 3½ into an improper fraction, we get 7/2

Step 2: Take reciprocal for improper fraction: 2/7

Step 3: Multiply ⅖ and 2/7

Step 4: Simplify: 4/35

Dividing Decimals as Fractions

We have learned to divide fractions using three simple steps. Now with the help of these steps let us learn how to divide decimals with examples.

Example: Divide 0.5 ÷ 0.2

Solution: To divide these decimal numbers, we have to convert both the decimal number into natural numbers by multiplying numerator and denominator by 10.

Therefore, 0.5 × 10 / 0.2 × 10

We get, 5/2 = 2.5

Also, we can use the dividing fractions method to solve the above problem.

We can write 0.5 and 0.2 as 5/10 and 2/10.

So for 5/10 ÷ 2/10, we can use the same steps fraction’s division.

5/10 × 10/2

= 5 × 10 / 10 × 2

= 50/20

= 5/2

= 2.5

Note: These are the simple method of dividing decimals. You can also use the direct division method to divide decimals. The only difference is to place the decimal into the right place of the quotient. Let us take an example of this.

Example: Divide 13.2 ÷ 2

Solution: 2) 13.2 (6.6

-12

—————

12

-12

—————

00

—————

Therefore, 13.2 ÷ 2 = 6.6

Dividing the natural numbers or whole numbers is an easy task but dividing the fractions is a little complex one. The operations performed on natural numbers and whole consist of simple calculations, which one can easily solve. But the operations performed on fractions are sometimes typical and also time-consuming. The simple division has four parts divisor, dividend, quotient and remainder. Also, know some of the divisibility rules for the whole number here.

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Solved Examples

Example 1: 

¼ ÷ ½

Solution:

Given, ¼ ÷ ½

Write the reciprocal of the second fraction and multiply it with the first fraction.

¼ × 2/1

Multiplying both numerators and denominators

1×2/4×1

2/4

Simplifying the fraction;

2/4 = ½ = 0.5

Example 2:

⅗ ÷ ⅔

Solution:

Following the same steps:

1. ⅗ ÷ ⅔ = ⅗ × 3/2
2. 3 × 3/ 5 × 2
3. 9/10=0.9

One more method to divide fractions is to make the denominator equal and then divide it.

Example 3:

¾ ÷ 3/2

Solution:

Given, ¾ ÷ 3/2

By making the denominator equal we get,

¾ ÷ 6/4

Now the denominators are the same, we can cancel both the denominator and write it as;

3/6 = ½ = 0.5

Practice Questions

Solve the following problems:

1. ⅔ ÷ ⅓ = ?
2. 3 ÷ ⅓ = ?
3. 5 ¾ ÷ ¼ = ?
4. ⅔ ÷ ⅙ = ?
5. 5 ½ ÷ 4 ⅓ = ?
6. ⅜ ÷ ⅞ = ?
7. ¼ ÷ 1/7 = ?
8. 7/11 ÷ ⅙ = ?
9. ⅖ ÷ ½ = ?
10. ⅓ ÷ 6/9 = ?

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