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.

Division of Fractions

Find the reciprocal of 3 ¾

The reciprocal of 3 ¾ is 4/15.

Division of Fractions Reciprocal

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?

\(\frac{10}{15}\) ÷ \(\frac{5}{1}\)

= \(\frac{10}{15}\) × \(\frac{1}{5}\)

= \(\frac{2 × \not 5 × 1}{3 × \not 5 × 5}\)

= \(\frac{2}{15}\)

Prime Factors of 10, 5 and 3

              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\(\frac{1}{3}\)

Prime Factors of 16, 27, 9 and 4

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

Related Concept

Fraction of a Whole Numbers

Representation of a Fraction

Equivalent Fractions

Properties of Equivalent Fractions

Like and Unlike Fractions

Comparison of Like Fractions

Comparison of Fractions having the same Numerator

Types of Fractions

Changing Fractions

Conversion of Fractions into Fractions having Same Denominator

Conversion of a Fraction into its Smallest and Simplest Form

Addition of Fractions having the Same Denominator

Subtraction of Fractions having the Same Denominator

Addition and Subtraction of Fractions on the Fraction Number Line




4th Grade Math Activities

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