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Chapter 8: Working With Fractions

    Multiplication of Fractions

    Division of Fractions

    Some Problems Involving Fractions

    Fractional Relations

    Summary

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Chapter 8: Working With Fractions > Division of Fractions

Division of Fractions

What is 12 ÷ 4? You know this already. But can this problem be restated as a multiplication problem? What should be multiplied by 4 to get 12? That is,

4 × = 12

We can use this technique of converting division into multiplication problems to divide fractions. What is 1 ÷ 23 ?

Let us rewrite this as a multiplication problem 23 × ? = 1

What should be multiplied by 23 to get the product 1? If we somehow cancel out the 2 and the 3, we are left with 1.

So, 1 ÷ 23 =

Let us try another problem: 3 ÷ 23

This is the same as: 23 × ? = 3

Can you find the answer?

We know what to multiply 23 by to get 1. We just need to multiply that by 3 to get 3. So,

So, 3 ÷ 23 = 32 × 3 =

What is 15 ÷ 12 ?

Rewriting it as a multiplication problem, we have:

12 × ? = 15

How do we solve this?

So, 15 ÷ 12 = 2 × 15 =

What is 23 ÷ 35 ?

Rewriting this as multiplication, we have:

35 × ? = 23

How will we solve this?

So, 23 ÷ 35 = × 23 =

Discussion

In each of the division problems above, observe how we found the answer. Can we frame a rule that tells us how to divide two fractions? Let us consider the previous problem.

In every division problem we have a dividend, divisor and quotient.

The technique we have been using to get the quotient is:

  1. First, find the number which gives 1 when multiplied by the divisor.

We see that the resulting number is a fraction whose numerator is the divisor’s denominator and denominator is the divisor’s numerator.

For the divisor 35, this fraction is .

We call 53 the reciprocal of 35.

When we multiply a fraction by its reciprocal, we get .

So, the first step in our technique is to find the divisor’s reciprocal.

2. We then multiply the dividend with this reciprocal to get the quotient.

Summarising, to divide two fractions:

• Find the reciprocal of the divisor • Multiply this by the dividend to get the quotient.

So,

ab ÷ cd = dc × ab = d×ac×b

This can be rewritten as:

As with methods and formulas for addition, subtraction, and multiplication of fractions that you learnt earlier, this method and formula for division of fractions, in this general form, was first explicitly stated by Brahmagupta in his Brāhmasphuṭasiddhānta (628 CE).

So, to evaluate, for example, 23 ÷ 35 using Brahmagupta’s formula above, we write:

2/3 ÷ 3/5 = 23 × = =

Dividend, Divisor and the Quotient

When we divide two whole numbers, say 6 ÷ 3, we get the quotient .

Here the quotient is than the dividend i.e. 6 ÷ 3 = 2, 2 6

But what happens when we divide 6 by 14? 6 ÷ 14 = 24

Here the quotient is greater than the dividend!

What happens when we divide 18 by 14?

18 ÷ 14 = 12

Here too the quotient is greater than the dividend!

When do you think the quotient is less than the dividend and when is it greater than the dividend?

Is there a similar relationship between the divisor and the quotient?

Use your understanding of such relationships in multiplication to answer the questions above.