Introduction:

              We are aware that the fractions are the rational numbers which are of the form a/b, where a and b are integers and b is not equal to 0. Regarding the basics of fractions, we should be aware of framing the equivalent fractions and simplification of fractions before we apply the operation on fractions. In this section let us see how to multiply three fractions.

How to multiply Three Fractions:

When we multiply the fractions, we follow any one of the following steps.
Method I
Step 1: Multiply the numerators.
Step 2: Multiply the denominators.
Step 3: Reduce(simplify)  the fraction by dividing the numerator and the denominator by the Highest Common Factor (HCF).
Step 4: If the simplest form is a proper fraction, highlight this as the answer else
            if the simplest form is an improper fraction, convert it as mixed fraction and highlight this as the final answer.

Example : Simplify $\frac{3}{4}\times \frac{8}{15}\times \frac{5}{7}$


Solution: We have $\frac{3}{4}\times \frac{8}{15}\times \frac{5}{7}$

                                              =$\frac{3\times 8\times 5}{4\times 15\times 7}$ [ Multiplying the numerators and the Denominators ]

                                              = $\frac{120}{420}$

                                              = $\frac{120\div 60}{420\div 60}$ [ dividing the numerator and the denominator by the HCF  120, 420 ]

                                              = $\frac{2}{7}$ Final answer in the simplest form.

Working: To find the HCF of 120 and 420

Factors of 120 are 2 x 2 x 2 x 3 x 5
Factors of 420 are 2 x 2 x 3 x 5 x 7

Common factors are 2, 2,3,5
HCF  = 2 x 2 x 3 x 5 = 60

Method II


Step 1: Combine the numerators with a multiplication sign.
Step 2: Combine the denominators with a multiplication sign.
Step 3: Write each number as product of primes both in the numerator and the denominator.
Step 4: Cancel the common factors which occur in pairs in the numerator and the denominator, since for any integer a, a/a = 1.
Step 5: Rewrite the numbers in the numerator and the denominator and look for any other common factors. If there are no common factors multiply the
            numbers in the numerator and the denominator.
Step 6: If the result in step 5 is a proper fraction, highlight this as the final answer.
           If the result in step 5 is an improper fraction, convert this into mixed fraction and highlight this as the final answer.

Example : $\frac{6}{35}\times \frac{49}{18}\times \frac{10}{14}$


Solution: We have $\frac{6}{35}\times \frac{49}{18}\times \frac{10}{14}$

                                             = $\frac{6\times 49\times 10}{35\times 18\times 14}$
                                                                                 [ combining the numerator and the denominator with a multiplication signs between them ]

                                             = $\frac{2\times 3\times 7\times 7\times 2\times 5}{5\times 7\times 2\times 3\times 3\times 2\times 7}$
                                                                                 [ writing the numerator and the denominator by the product of primes ]

                                            = $\frac{1}{3}$ [ by cancelling the pairs of prime factors which are same in the numerator and the denominator ]
                                                                 [ common factors are 2,2,3,5,7,7 ]

                                            = $\frac{1}{3}$ Final answer in the simplest form.

Working:
Numbers  
 Product of Prime factors
    6
            2 x 3
   49             7 x 7
   10             2 x 5
   35             5 x 7
   18             2 x 3 x 3
   14             2 x 7
                                   

Practice Questions:

Multiply the three fractions given below:

1. $\frac{4}{15}\times \frac{5}{8}\times \frac{12}{13}$

2. $\frac{3}{7}\times \frac{3}{7}\times \frac{12}{13}$

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