Dividing Fractions

We will discuss here about dividing fractions by a whole number, by a fractional number or by another mixed fractional number.

First let us recall how to find reciprocal of a fraction, we interchange the numerator and the denominator.

 For example, the reciprocal of ¾ is 4/3.

Find the reciprocal of 3 ¾

 The reciprocal of 3 ¾ is 4/15.

I. Division of a Fraction by a Whole Number:

4 ÷ 2 = 2 means, there are two 2’s in 4.

6 ÷ 2 = 3 means, there are two 2’s in 6.

Similarly 5 ÷ $$\frac{1}{2}$$ means, how many halves are there in 5?

We know that $$\frac{1}{2}$$ + $$\frac{1}{2}$$ = 1

 $$\frac{1}{2}$$ + $$\frac{1}{2}$$   + $$\frac{1}{2}$$ + $$\frac{1}{2}$$   + $$\frac{1}{2}$$ + $$\frac{1}{2}$$   + $$\frac{1}{2}$$ + $$\frac{1}{2}$$   + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ 1      + 1      + 1      + 1      + 1 =   5

i.e. there are 10 halves in 5.

5 ÷ $$\frac{1}{2}$$ = 5 × $$\frac{2}{1}$$ = $$\frac{10}{1}$$ = 10

For Example:

1. $$\frac{7}{10}$$ ÷ 5 = $$\frac{7}{10}$$ ÷ $$\frac{5}{1}$$

= $$\frac{7}{10}$$ × $$\frac{1}{5}$$

= $$\frac{7 × 1}{10 × 5}$$

= $$\frac{7}{50}$$

 2. What is $$\frac{10}{15}$$ ÷ 5?$$\frac{10}{15}$$ ÷ $$\frac{5}{1}$$ = $$\frac{10}{15}$$ × $$\frac{1}{5}$$= $$\frac{2 × \not 5 × 1}{3 × \not 5 × 5}$$= $$\frac{2}{15}$$ 10 = 2 × 5              15 = 3 × 5                5 = 1 × 5

To divide a fraction by a number, multiply the fraction with the reciprocal of the number.

For example:

3. Divide 3/5 by 12

 Solution:3/5 ÷ 12 = 3/5 ÷ 12/1 = 3/5 × 1/12 = (3 × 1)/(5 × 12) = 3/60 = 1/20 Step I: Find the reciprocal of the whole number and multiply with the fractional number as usual. Step II: Express the product in its lowest terms.

4. Solve: 5/7 ÷ 10

 = 5/7 ÷ 10/1 = 5/7 × 1/10 = (5 × 1)/(7 × 10) = 5/70 Step I: Find the reciprocal of the whole number and multiply with the fractional number as usual. Step II: Express the product in its lowest terms.

II. Division of a Fractional Number by a Fractional Number:

For example:

1. Divide 7/8 by 1/5

 Solution:7/8 ÷ 1/5 = 7/8 × 5/1 = (7 × 5)/(8 × 1) = 35/8 = 4 3/8 Step I: Find reciprocal of 1/5. Step II: Multiply 7/8 by it. Step III: Express the product in its simplest form.

2. Divide: 5/9 ÷ 10/18

 Solution:5/9 ÷ 10/18 = 5/9 × 18/10 = (5 × 18)/(9 × 10) = 90/90 = 1 Step I: Find reciprocal of 1/5. Step II: Multiply 7/8 by it. Step III: Express the product in its simplest form.

Division of a Fraction by a Fraction:

3. Divide $$\frac{3}{4}$$ ÷ $$\frac{5}{3}$$

Step I: Multiply the first fraction with the reciprocal of the second fraction.

Reciprocal of $$\frac{5}{3}$$ = $$\frac{3}{5}$$

Therefore, $$\frac{3}{4}$$ ÷ $$\frac{5}{3}$$  = $$\frac{3}{4}$$ × $$\frac{3}{5}$$

= $$\frac{3 × 3}{4 × 5}$$

= $$\frac{9}{20}$$

Step II: Reduce the fraction to the lowest terms. (if necessary)

 4. Divide $$\frac{16}{27}$$ ÷ $$\frac{4}{9}$$Therefore, $$\frac{16}{27}$$ ÷ $$\frac{4}{9}$$ = $$\frac{16}{27}$$ × $$\frac{9}{4}$$; [Reciprocal of $$\frac{4}{9}$$ = $$\frac{9}{4}$$]                            = $$\frac{\not 2 × \not 2 × 2 × 2 × \not 3 × \not 3}{\not 3 × \not 3 × 3 × \not 2 × \not 2}$$                            = $$\frac{4}{3}$$                            = 1$$\frac{1}{3}$$ 16 = 2 × 2 × 2 × 2             9 = 3 × 3             27 = 3 × 3 × 3             4 = 2 × 2

III. Division of a Mixed Number by another Mixed Number:

For example:

1. Divide 2 ¾ by 1 2/3

 Solution:2 ¾ ÷ 1 2/3 = 11/4 ÷ 5/3 = 11/4 × 3/5 = (11 × 3)/(4 × 5) = 33/20 = 1 13/20 Express the mixed numbers as improper fractions and multiply as usual.

2. Divide: 2  4/17 ÷ 1  4/17

 Solution:2  4/17 ÷ 1  4/17 = 38/17 ÷ 21/17 = 38/17 × 17/21 = (38 × 17)/(17 × 21) = 646/357 = 38/21 = 1 17/21 Express the mixed numbers as improper fractions and multiply as usual.

Questions and Answers on Dividing Fractions:

I. Divide the following.

(i) $$\frac{2}{6}$$ ÷ $$\frac{1}{3}$$

(ii) $$\frac{5}{8}$$ ÷ $$\frac{15}{16}$$

(iii) $$\frac{5}{6}$$ ÷ 15

(iv) $$\frac{7}{8}$$ ÷ 14

(v) $$\frac{2}{3}$$ ÷ 6

(vi) 28 ÷ $$\frac{7}{4}$$

(vii) 2$$\frac{5}{6}$$ ÷ 34

(viii) 9$$\frac{1}{2}$$ ÷ $$\frac{38}{2}$$

(ix) 3$$\frac{1}{4}$$ ÷ $$\frac{26}{28}$$

(x) 7$$\frac{1}{3}$$ ÷ 1$$\frac{5}{6}$$

(xi) 2$$\frac{3}{5}$$ ÷ 1$$\frac{11}{15}$$

(xii) 1$$\frac{1}{2}$$ ÷ $$\frac{4}{7}$$

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