Introduction:

          We are quiet aware of what a fraction is and how to  convert a fraction from one form to another. The term simplest form means, we express the fraction in a form such that the numerator and the denominator are co-prime numbers. whenever we multiply or divide fractions we always express the final result in the simplest form. In this section let us see how we multiply and divide fractions and express the result in simplest form.

Multiplication of Fractions:

When we multiply fractions of the form,$ \frac{a}{b}\times \frac{c}{d}$, we multiply the numerators and the denominators and express the resulting fraction in simplest form.

Example : Multiply : $\frac{3}{5}\times \frac{15}{21}$

Solution: $\frac{3}{5}\times \frac{15}{21}$

                             = $\frac{3\times 15}{5\times 21}$ [ grouping the numbers in the numerator and denominator ]

                            = $\frac{45}{105}$ [ multiplying the the numbers in the numerator and denominator ]

                            = $\frac{45\div 15}{105\div 15}$ [ Dividing the numerator and denominator by the Highest Common Factor 15 ]

                            = $\frac{3}{7}$ Final Answer in Simplest form.

Working :
To find the Highest Common Factor of 45 and 105 let us express them as product of primes.

fraction20
45 = 3 x 3 x 5
105 = 3 x 5 x 7
Common factors are 3, 5
Highest Common Factor = 3 x 5 = 15

Division of Fractions:

When we divide two fractions of the form $\frac{a}{b}\div  \frac{c}{d}$, we multiply the first fraction by the reciprocal of the second.
We multiply the two fractions and express it in simplest form.

Example : $\frac{16}{21}\div \frac{4}{9}$


Solution: We have,  $\frac{16}{21}\div \frac{4}{9}$

                                                   = $\frac{16}{21}\times \frac{9}{4}$ [ multiplying the first fraction by the reciprocal of the second fraction ]

                                                   = $\frac{16\times 9}{21\times 4}$ [ grouping the numbers in the numerator and denominator ]

                                                   = $\frac{144}{84}$ [ multiplying the numbers in the numerator and denominator ]

                                                   = $\frac{144\div 12}{84\div 12}$ [ dividing the numerator and denominator by the Highest Common Factor 12 ]

                                                   = $\frac{12}{7}$ Simplest form of the fraction.

                                                   = $ 1 \frac{5}{7} $ Final answer in mixed form.

Working:
To find the Highest Common Factor of 144 and 84, let us first find the prime factors of 144 and 84.
fraction21
144 = 2 x 2 x 2 x 2 x 3 x 3
84   = 2 x 2 x 3 x 7
Common factors are , 2,2,3
Highest Common Factor ( HCF ) = 2 x 2 x 3 = 12


Practice Questions:

1. Multiply and express the result in simplest form.
   $\frac{11}{49}\times \frac{14}{33}$

2. Divide and express the result in simplest form.
    $\frac{4}{5}\div \frac{12}{25}$

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