Introduction:

        When we multiply the fractions of different kinds, we multiply them and then express them into their simplest form. Suppose a cloth of 12 meters is to be divided among few students and if each student require 1 and 1/2 meter of cloth, for how many students the cloth was distributed. If the problem is related to whole numbers we can divide it directly to find the number of students. But here fraction is involved. Let us discuss how we perforem division of fractions.

Division of Fractions:

Let us divide 3 pieces cakes and each is cut into one-fourth. Let us see how many pieces are obtained.
(i,e)  $3\div \left (  \frac{1}{4}\right )$

fraction16
In this above figure, we see that 12 pieces are obtained.
Let us evaluate this arithmetically,
$3\div \left (  \frac{1}{4}\right )$ = $3\times \frac{4}{1}$

                                             = $\frac{3\times 4}{1}$

                                             = 12

Rules to Divide Fractions:

We follow following steps when we divide fractions.
Step 1: Make sure that both the fractions are of either proper or improper fractions. If one of the fraction is a mixed fraction, convert it into improper fraction.

Step 2: Write the first fraction as it is, then replace the divided by sign ( $\div$ ) as a multiplication sign ( X ).

Step 3: Write the reciprocal of the fraction that is followed by the divided by sign after the multiplication sign.
Step 4: Multiply the Numerators.
Step 5: Multiply the Denominators.
Step 6: Find the Highest Common Factor of the fraction obtained from step 4 and 5.
Step 7: Divide the numerator and the denominator by the Highest Common Factor (HCF ) obtained from step 4  and 5.
Step 8: If the resulting fraction is a proper fraction or a whole number, highlight this as the final answer else
           convert it into mixed fraction and highlight this as the final answer.

Divide and Multiply Fractions - Example :

Example : Divide: $\frac{3}{4}\div \frac{9}{20}$


Solution: We have,$\frac{3}{4}\div \frac{9}{20}$

                                            =$\frac{3}{4}\times \frac{20}{9}$ [ multiplying by the reciprocal of the fraction followed by the division sign ]

                                            =$\frac{3\times 20}{4\times 9}$ [ grouping the numerators and denominators separately ]

                                            = $\frac{60}{36}$ [ multiplying the numbers in the numerator and the denominator ]

                                            = $\frac{60\div 12}{36\div 12}$ [ Dividing the numerator and the denominator by the HCF ]

                                           = $\frac{5}{3}$ [ simplest form of the fraction in improper fraction ]

                                           = $1 \frac{2}{3}$ Final answer in mixed fraction.
Working:
Let us find the HCF of 60 and 36.
Let us express each number as product of primes.

fraction17
60 = 2 x 2 x 3 x 5
36 = 2 x 2 x 3 x 3
Common factors are 2 , 2, 3
Highest Common Factor ( HCF ) = 2 x 2 x 3 = 12


Practice Questions:

Divide the following fractions as directed and express your answer in simplest form.
1. $\frac{5}{2}\div \frac{25}{8}$

2. $\frac{14}{3}\div \frac{35}{6}$

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