Introduction:

We know that the fractions are of three different types. They are proper fractions, improper fractions and mixed fractions.
Proper fractions are those which lie between 0 and 1 on the number line and the improper fractions are those which are equal to 1 or greater than 1. Improper fractions can be written as the combination of whole number and proper fraction. In this section let us see how we  multiply mixed fractions.


Multiplication of Mixed Fractions:

Let us study the following figure before we discuss the method of multiplying mixed fractions.


fraction9fraction9fraction9

In the above figure , we have

                   $1\frac{2}{3}$ + $1\frac{2}{3}$ + $1\frac{2}{3}$ = $1\frac{2}{3}\times 3$
 
                                                                                     =$\frac{5}{3}\times 3$

                                                                                     = $\frac{5\times 3}{3}$

                                                                                     = $ \frac{15}{3}$

                                                                                     = 5
If we count the total number of shaded region we see that there are 15 one-thirds, which is 5 whole.
This is shown in the following figure.
                     
fraction10
The following steps are followed when we multiply 2 or more mixed fractions:

Step 1: Convert the mixed fractions to improper fractions.
Step 2: Group the numerators and multiply them.
Step 3: Group the denominators and multiply them.
Step 4: Find the Highest Common factor of the numerator and denominator of the fraction obtained in steps 2 and 3.
Step 5: Divide the numerator and Denominator by the HCF obtained in step 4.
Step 6: The result obtained will be an improper fraction in the simplest form.
Step 7: Convert this into mixed fraction, which will be the final answer.
Let us discuss the following example:

Example :  $2\frac{1}{3}\times1\frac{5}{7} $

Solution: We have , $2\frac{1}{3}\times1\frac{5}{7} $

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

                                                 =$\frac{7\times 12}{5\times 7}$ [ grouping the numbers in the numerator and the denominator ]

                                                 =$\frac{84}{35}$ [ multiplying the numbers in the numerator and denominator separately ]

                                                 =$\frac{84\div 7}{35\div 7}$[ dividing the numerator and the denominator by the common factor]

                                                 = $\frac{12}{5}$ [ simplest form ]

                                                 =$2\frac{2}{5}$ Final Answer in mixed form.


Practice Questions:

Multiply the following  and express your answer in mixed fraction.

1. $2\frac{1}{5}\times 3\frac{10}{17}$

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

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