Introduction: 

              We have already discussed the method of multiplication of fractions. As we are aware that the fractions are of different types, proper, improper and mixed fractions, let us discuss the method of multiplying these fractions with mixed fractions. 

How to multiply fractions with mixed numbers:

Proper Fractions: Proper fractions are those which are of the form, a/b, where a and b are integers and a < b.
Improper Fractions: Improper fractions are those which are of the form a/b, where a and b are integers and a > = b.
Mixed Numbers: Mixed fractions are those which are combination of a whole number > 0 and a proper fraction.

For example, $4\frac{5}{8}$, $5\frac{2}{3}$.


To multiply a proper/improper fraction with mixed numbers we should follow the following steps.
Step 1: Convert the mixed fraction to improper fraction.
Step 2: Multiply the numerators.
Step 3: Multiply the denominators.
Step 4: Divide the numerator and denominator by the Highest Common Factor (HCF)  if any.
Step 5: If the resulting fraction in step 4 is a proper fraction, highlight this as the final answer, else
            if the fraction is an improper fraction convert this into mixed fraction and highlight this as the final answer.

Example 1: Multiply $\frac{5}{9}\times 5\frac{4}{7}$

Solution: We have, $\frac{5}{9}\times 2\frac{4}{7}$

                                          = $\frac{5}{9}\times \frac{18}{7}$ [ converting the mixed fraction into improper fraction ]

                                          = $\frac{5\times 18}{9\times 7} $ [ grouping the numerators and the denominators ]

                                          =$\frac{90}{63} $ [ multiplying the numbers in the numerator and denominator separately ]

                                          = $\frac{90\div 9}{63\div 9}  $ Dividing the numerator and the denominator by the Highest Common Factor (HCF) ]

                                         = $\frac{10}{7}  $ Improper [Fraction in the simplest form ]

                                         = $1\frac{3}{7}  $ Final Answer in mixed fraction.

Example 2: $\frac{4}{5}\times 1\frac{2}{7}\times 4\frac{2}{3}$


Solution: We have $\frac{4}{5}\times 1\frac{2}{7}\times 4\frac{2}{3}$

                                            = $\frac{4}{5}\times \frac{9}{7}\times \frac{14}{3}$ [ converting mixed fractions into improper fractions ]

                                            = $\frac{4\times 9\times 14}{5\times 7\times 3} $ [combining numerators and denominators separately ]

                                            = $  \frac{504}{105} $ [ finding the products of the numerator and denominator ]

                                            = $\frac{504\div 21}{105\div 21}  $ [ dividing the numerator and the denominator by the HCF = 21  ]

                                            = $\frac{24}{5}  $ [ simplest form which is an improper fraction ]

                                            = $4\frac{4}{5}  $ Final answer in mixed form.

Working: To find the HCF of 504 and 105
Let us find the prime factors of 504 and 105.
                                          
fraction13
504 = 2 x 2 x 2 x 3 x 3 x 7
105 = 3 x 5 x 7
The common factors are 3 and 7.
Highest common factor = 3 x 7 = 21

Practice Questions:

Multiply the following fractions and express your answer in simplest form.

1. $\frac{2}{5}\times \frac{7}{8}\times 3\frac{1}{3}$

2. $7\frac{1}{5}\times 7\frac{1}{2}\times \frac{2}{5}$

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