# Division of Fractions

Fractions are represented in a “p/q” form and to perform operations is easier in this form. We have studied the concept of Multiplication of fractions. We will study the operation of the division of fractions and we will study solved examples to help understand the concept easily. Let’s start.

## Division of Fractions

Division of fractions is a simple operation. You will be surprised to know that division of one fraction by other fraction is actually a multiplication. Multiply the first fraction with the reciprocal of the other fraction to obtain the final result.

### Reciprocal of a Fraction

The reciprocal of a number “a” is given as “1/a”. Reciprocal of a fraction basically means changing the places of the numerator and the denominator. Consider the fraction 3/4. Reciprocal of this fraction is obtained by changing the places of 3 and 4. So the reciprocal of 3/4 is 4/3. Similarly, reciprocal of 7/9 and 1/6 is 9/7 and 6/1 respectively.

Note that changing the places of the numerator and denominator changes the value of the fraction entirely. Simplification of the operation of division is the only aim here for which we do this. So, the steps for the division of fractions include, firstly change the second fraction to its reciprocal and then using the rules of multiplication obtain the final result. For example,

• 3/4 divided by 1/3. To solve this, we first obtain the reciprocal of 1/3. Interchanging the numerator and denominator we get 3/1. Now, multiplying the fractions we get 3/4 × 3/1 = 9/4.
• 3/4 divided by 1/2. Now again we take the reciprocal of the second fraction we get 2/1. Using rules of multiplication we get, 3/4 × 2/1. Canceling the common terms here we get 3/2.

## Division of Whole Numbers by Fractions and Vice Versa

The use of fractions to divide whole numbers or vice versa is done to obtain ratios.

• For example, if asked to divide 6 by 3/4, we convert 6  to a fraction as studied to 6/1. We take reciprocal of 3/4 i.e. 4/3. Using the rules of multiplication we get 6/1 × 4/3. Canceling the common term we get 2/4 which is equal to 1/2.
• Again consider an example of dividing 7/8 by 4. To do so, we first convert the whole number into a fraction i.e. 4/1. We take reciprocal of the second fraction, so we get 1/4. Using the rules of multiplication of fractions we get 7/8 ×1/4=7/32

So to divide a whole number by a fraction or vice versa we convert the whole number to a fraction, take the reciprocal of the second fraction, multiply both the fractions using rules of multiplication and obtain the results.

## Division of Mixed Fractions

Mixed fractions represented as xy/z can be converted to fractions using the formula (z.x+y)/z and then division can be carried out on these fractions as studied. Consider the example, divide 23/5  by 31/4.

We convert the mixed fractions into fractions. We get 2 fractions 13/5 and 13/4. Now we take reciprocal of the second fraction and multiply 13/5 × 4/13. Canceling common terms and we get 4/5.

## Solved Questions for You

Question. Divide 7 by 23/4

Solution: Now converting the whole number to a fraction and converting mixed fraction to regular fraction

= 7/1 and 11/4

taking reciprocal of the second fraction and multiplying,

7/1 × 4/11

= 28/11

= 2 6/11

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