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.
Find the reciprocal of 3 ¾
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\(\frac{1}{2}\) ÷ \(\frac{4}{7}\)
Related Concept
● Representation of a Fraction
● Properties of Equivalent Fractions
● Comparison of Like Fractions
● Comparison of Fractions having the same Numerator
● 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
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