Multiplication of a Fraction by a Fraction

This topic deals with multiplication of a fraction by another fraction. For eg : \(\frac{3}{7}\)  by  \(\frac{5}{11}\). Here both the numbers are in numerator and denominator form.

Here are few examples to show multiplication of fraction by another fraction

There are few steps or rules that are to be kept in mind while carrying out multiplication of fraction with another fraction.

Step 1:

First change both the fractions into improper fraction if they are mixed fraction


Step 2:

Now we will have to multiply numerator with numerator and denominator with denominator. Or, it can be said product of numerators and product of denominators


Step 3:

Then will have to reduce the fraction that is the numerator and denominator into lowest term


Step 4:

Then the final answer is expressed in whole number or mixed fraction (if it is not in proper fraction) or in proper fraction. The final answer is usually not left in improper fraction we have to either change it into mixed fraction or whole number (if possible)


Here are few examples to show multiplication of a fraction by another fraction:

Multiplication of proper fraction by another proper fraction

Multiplication of mixed fraction by a proper fraction

Multiplication of mixed fraction by another mixed fraction


1: Multiplying proper fraction by proper fraction.

\(\frac{5}{8}\) × \(\frac{3}{7}\)

= \(\frac{5 × 3}{8 × 7}\); [multiplying numerator by numerator and denominator by denominator]

= \(\frac{15}{56}\); [This is in proper fraction and hence cannot be changed into mixed fraction]


2. Multiplying mixed fraction with proper fraction

7\(\frac{1}{4}\) × \(\frac{6}{7}\)

= \(\frac{4 × 7 + 1}{4}\) × \(\frac{6}{7}\); [Changing the first fraction into improper factor as it is in mixed fraction]

= \(\frac{29}{4}\) × \(\frac{6}{7}\)

= \(\frac{29 × 6}{4 × 7}\); [multiplying numerator by numerator and denominator by denominator]

= \(\frac{174}{28}\)

= \(\frac{174 ÷ 2}{28 ÷ 2}\); [Changing into lowest terms]

= \(\frac{87}{14}\)

Changing it into mixed fraction as it is in improper fraction

= 6\(\frac{3}{14}\)


3. Multiplying mixed fraction with another mixed fraction

4\(\frac{1}{3}\) × 5\(\frac{1}{4}\)

= \(\frac{4 × 3 + 1}{3}\) × \(\frac{4 × 5 + 1}{4}\); [Changing into improper factor as it is in mixed fraction]

= \(\frac{13}{3}\) × \(\frac{21}{4}\)

= \(\frac{21 × 13}{4 × 3}\); [multiplying numerator by numerator and denominator by denominator]

= \(\frac{273}{12}\)

= 22\(\frac{9}{12}\)

Here to write into mixed fraction first we will have to consider the quotient and that has to be written in whole number form, then remainder divided by the divisor.

= 22\(\frac{3}{4}\); [Here \(\frac{9}{12}\)  changed into lowest term \(\frac{3}{4}\)]


4. Multiplying a proper fraction with mixed fraction

\(\frac{1}{8}\) × 7\(\frac{14}{15}\)

= \(\frac{1}{8}\) × \(\frac{7 × 15 + 14}{15}\); [Changing the second fraction into improper factor as it is in mixed fraction]

= \(\frac{1}{8}\) × \(\frac{119}{15}\)

= \(\frac{1 × 119}{15 × 8}\); [Multiplying numerator by numerator and denominator by denominator]

= \(\frac{119}{120}\)


The above problems are examples of multiplication of a fraction by another fraction. These fractions can be in any form whether mixed fraction or proper fraction.






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