Introduction:

              We are already familiar with the simplification of fraction. We simplify a fraction by dividing the numerator and the denominator by the common factors. It is possible to apply the operations such as addition, subtraction, multiplication and division. Let us discuss with the operation addition and multiplication of fractions in this section.

Multiplication of Improper Fractions

Rule for multiplication of fractions: 
                    $\frac{a}{b}\times \frac{c}{d} = \frac{a\times c}{b\times d}=\frac{ac}{bd}$
Add and Multiply fraction:
Let us observe the following diagram. The fraction part of each whole is 1/3. We have 2 fractions from two wholes.

From the above diagram
Fraction1

We see that the sum of the two fractions 1/3 and 1/3 is 2/3.
As we add 1/3 two times, we can also write this as $\frac{1}{3}\times 2 = \frac{2}{3}$
Similarly when we have
     $\frac{2}{7}+\frac{2}{7}+\frac{2}{7}= \frac{2}{7}\times 3=\frac{2\times 3}{7}=\frac{6}{7}$

Multiplication of proper fractions: When we take a part from a whole we denote it in fraction form. When we take a part of the part, we multiply the two fractions by a multiplication signs.

Fraction2
We can observe that in the first figure 1/2 of the whole is shaded and in the next figure the half of the first half is shaded.
Hence for the whole, 1/2 of 1/2 is shaded .
Hence $\frac{1}{2}\times \frac{1}{2}$  = $\frac{1}{4}$

Let us discuss with other examples: 
         $\frac{2}{3}\times \frac{4}{5}$ = $\frac{2 \times 4}{3 \times 5}$ = $\frac{8}{15}$

We can follow the similar method for more than 2 fractions.

Multiplication of different types of fractions:

 1. Multiplication of Mixed Fractions: Product of two or more mixed fraction is a mixed fraction.

 Example :     $3\frac{1}{2}\times 1\frac{11}{21}$

 Solution: We have $3\frac{1}{2}\times 1\frac{11}{21}$
Converting the mixed fraction to improper fraction, we get
                     $ \frac{7}{2}\times \frac{32}{21}$
                                            = $\frac{7 \times  32}{2\times 21}$

                                            = $\frac{7 \times  16\times 2}{2\times3\times7}$
 
                                            = $\frac{16}{3}$
2. Multiplication of Improper fractions: Product of two or more improper fractions is an improper fraction.

Example : $\frac{12}{5}\times \frac{25}{18}$

Solution: We have $\frac{12}{5}\times \frac{25}{18}$
                          
                  = $\frac{12\times 25}{5\times 18}$

                  =$\frac{2\times 3\times 2\times 5\times 5}{5\times 2\times 3\times 3}$
 
                  = $\frac{2\times 5}{3}$
                                                       
                  = $\frac{10}{3}$                  


Simplification of Fractions:

Simplify the following:

Example 1. $\frac{6}{7}+\frac{2}{3}\times \frac{15}{14}$

Solution: We have, $\frac{6}{7}+\frac{2}{3}\times \frac{15}{14}$

                           = $\frac{6}{7}+\frac{2\times 3\times 5}{3\times 2\times 7}$

                           = $\frac{6}{7}+\frac{5}{7}$

                           = $\frac{11}{7}$

Example 2. $\left ( \frac{6}{7}+\frac{2}{3} \right )\times \frac{15}{14}$

Solution: We have $\left ( \frac{6}{7}+\frac{2}{3} \right )\times \frac{15}{14}$

                                                = $\frac{6\times 3}{7\times 3}+\frac{2\times 7}{3\times 7}\times \frac{15}{14}$

                                               = $\frac{18}{21}+\frac{14}{21}\times \frac{15}{14}$

                                               = $\frac{32}{21}\times \frac{15}{14}$

                                               = $\frac{32\times15}{21\times14}$

                                               = $\frac{2\times16\times3\times5}{3\times7\times2\times7}$

                                              = $\frac{16\times5}{7\times7}$

                                              = $\frac{80}{49}$

                                              = $1\frac{31}{49}$

Practice Questions:

Simplify the following as directed:
1. $\frac{5}{7}+\frac{5}{7}+\frac{5}{7}+\frac{5}{7}+\frac{5}{7}$

2. $4\times \frac{19}{24}$

3. $\frac{6}{11}\times \frac{22}{15}+\frac{3}{5}$

4.. $\frac{6}{11}\times (\frac{22}{15}+\frac{3}{5})$


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