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How to Multiply Fractions with Unlike Denominators








Introduction:

                 When the length of a rectangle is $2\frac{3}{4}$m and the width of the rectangle is $1\frac{1}{2}$m, we find the area of the rectangle by multiplying the length by the width. Here the two dimensions of the rectangle are given in mixed fraction. Hence we should know how to multiply these types of fractions to enable us to solve more problems of this kind.

Multiplication of fractions - Rules:

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While multiplying fractions of the form, $\frac{a}{b}$ x $\frac{c}{d}$, we use the following procedure.

Step 1: Convert the mixed fraction to improper fraction.
Step 2: Combine all the numerators and multiply them.
Step 3: Combine all the denominators and multiply them.
Step 4: Find the Highest Common Factor ( HCF ) of the numerator and the denominator of the fractions obtained from Steps 2 and 3.
Step 5: Divide the numerator and the Denominator by the HCF obtained from step 4.
Step 6: If the result obtained from step 5 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.

Example 1:$\frac{4}{15}\times \frac{3}{8}$


Solution: We have, $\frac{4}{15}\times \frac{3}{8}$

                                                = $\frac{4\times 3}{15\times 8} $ [ combining the numerator and the denominator separately ]

                                                =$\frac{12}{120}$ [ multiplying the numbers in the numerator and the denominator ]

                                                =$\frac{12\div 12}{120\div 12}$ [ dividing the numerator and the denominator by the Highest Common Factor (HCF) ]

                                                =$ \frac{1}{10}$ Final answer in the simplest form.

Working:
To find the HCF of 12 and 120.
Let us find the prime factors of 12 and 120.
fraction14
12 = 2 x 2 x 3
120 = 2 x 2 x 2 x 3 x 5
Common factors are 2,2,3
Highest Common Factor ( HCF ) = 2 x 2 x 3 = 12

Multiplication of fractions with unlike denominators :

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Multiplication of 3 fractions.
When we multiply more than 2 fractions we follow the same rule as that of the two fractions.
Let us discuss few examples

Example : Multiply $\frac{12}{28}\times \frac{21}{8}\times 1\frac{1}{4}$


Solution: We have, $\frac{12}{28}\times \frac{21}{8}\times 1\frac{1}{4}$

                                                = $\frac{12}{28}\times \frac{21}{8}\times \frac{5}{4}$ [ converting the mixed fraction into improper fraction ]

                                                = $\frac{12\times 21\times 5}{28\times 8\times 4}$ [combining the numerators and denominators separately ]

                                                = $\frac{1260}{896}$ [ multiplying the numerators and the denominators ]

                                                =$\frac{1260\div 28}{896\div 28}$ [ dividing the numerator and the denominator by the HCF ]

                                                =$\frac{45}{32}$ Improper Fraction in simplest form.

                                               =$1\frac{13}{32}$ Final Answer in mixed Fraction.

Working:
Let us find the HCF of 1260 and 896
Let us find the prime factors of these numbers.
fraction15
1260 = 2 x 2 x 3 x 3 x 5 x 7
896 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 7
Common factors are , 2,2,7
Highest Common Factor = 2 x 2 x 7 = 28

Practice Questions:

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Multiply the following and express your answer in simplest form.
1. $\frac{10}{21}\times \frac{7}{5}$

2. $2\frac{1}{3}\times 4\frac{2}{5}\times \frac{3}{11}$

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