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How to Multiply Fractions with Different Denominators

Introduction:

We are aware of multiplication of whole numbers and integers. For multiplication of whole numbers, the meaning of the the addition,
say 2 + 2+ 2 + 2 + 2 = 5 times 2 = 5 x 2 = 10. We follow the same procedure in multiplication of integers also.
For example,  (-7) + (-7) = (-7) x 2 = - 14. The integers( or whole numbers) are the rational numbers whose denominators are 1. (i.e) when the denominators are 1 we multiply the numbers directly. In this section let us discuss with the multiplication of fractions with different denominators.

Types of Fractions:

We are aware that the fractions are rational numbers of the form a/b, where b ≠ 0.
Proper Fractions: Proper fractions are those where the numerator is smaller then the denominator. (i.e) a < b.
For example, $\frac{2}{3} , \frac{11}{20}$

Improper Fractions: Improper fractions are those where the numerator is greater (bigger) than the denominator. (i.e) a > b.
For example, $\frac{12}{7} , \frac{31}{20}$

Mixed Fraction: Mixed fractions are those which are combination of whole number and the proper fraction.
For example, $3\frac{2}{7} , 4\frac{5}{12}$

Multiplication of fractions with different denominators:

Steps for multiplication of two or more fractions:

Step 1 :  Multiply the numerators.
Step 2 : Multiply the denominators.
Step 3 : Reduce or simplify the fraction obtained from steps1 and 2 by dividing the numerator and the denominator by the highest common factor.
Step 4:  If the result in step 3 is a proper fraction, highlight this as the proper fraction else
convert it into mixed fraction and highlight this as the final answer.

(i.e) $\frac{a}{b}\times \frac{c}{d}\times \frac{e}{f}$ = $\frac{a\times c\times e}{b\times d\times f}$

Multiplication of proper and improper fractions:

Example 1: Multiply, $\frac{2}{5}\times \frac{45}{2}$

Solution: We have, $\frac{2}{5}\times \frac{45}{2}$

= $\frac{2\times 45}{5\times 2}$ [ grouping the numerator and the denominator with multiplication sign ]

= $\frac{90}{10}$ [ multiplying the numerator and the denominator ]

= $\frac{90\div 10}{10\div 10}$ [ Dividing the numerator and the denominator by the Highest Common Factor, HCF ]

= $\frac{9}{1}$ Final answer in the simplest form.

Working:
Let us express 90 and 10 as product of prime factors.

90 = 2 x 3 x 3 x 5
10 = 2 x 5
The common factors are 2 , 5
Highest Common Factor, HCF is  2 x 5 = 10

Multiplication of mixed fraction with proper or improper fraction:

Steps to follow to multiply mixed fraction and proper or improper fraction.

Step 1: Convert the mixed fraction into improper fraction.
Step 2: Multiply the numerators.
Step 3: Multiply the denominators.
Step 4: Reduce or simplify the fraction obtained from steps 2 and 3 by dividing the numerator and the denominator by the highest common factor.
Step 5: If the result in step 3 is a proper fraction, highlight this as the proper fraction else
convert it into mixed fraction and highlight this as the final answer.

Example : $2\frac{2}{5}\times \frac{7}{9}\times \frac{3}{7}$

Solution: We have $2\frac{2}{5}\times \frac{7}{9}\times \frac{3}{7}$

= $\frac{12}{5}\times \frac{7}{9}\times \frac{3}{7}$ [ converting the mixed fraction into improper fraction ]

= $\frac{12\times 7\times 3}{5\times9\times 7}$
[ grouping the numerators and the denominators as per the sign ]

= $\frac{252}{315}$ [ multiplying the numerators and the denominators ]

= $\frac{252\div 63}{315\div 63}$ [ Dividing the numerator and the denominator by the HCF ]

= $\frac{4}{5}$ Final answer in the simplest form.
Working:
Let us express 252 and 315 as product of prime factors.

252 = 2 x 2 x  3 x 3 x 7
315 = 3 x 3 x  3 x  5 x 7
Common factors are 3, 3, 7
Highest common factor = 3 x 3 x 7 = 63

Practice Questions:

1. $\frac{5}{3}\times \frac{9}{10}$
2. $2\frac{1}{7}\times \frac{14}{5}$