Introduction:

             When we multiply whole numbers and integers we use the multiplication tables and the rule of signs and multiply the numbers. When we come across the problems like " Find the area of a rectangle whose length is $5\frac{3}{4}$ cm and width $2\frac{1}{2}$ cm" , we need to multiply the mixed fractions. In this section let us see how we multiply these types of fractions.

Conversion of Mixed Fractions:

Mixed Numbers:  Mixed numbers are those which are the combination of whole number greater than zero and a proper fraction.
Before multiplying the mixed numbers we should convert them into improper fractions.

Conversion of Mixed Fractions into Improper Fractions:

               We follow the following procedure to convert a mixed fraction into its improper fraction.

For any fraction of the form, $k\frac{p}{q}$ , 'k' is the whole number, 'q' is the divisor and 'p' is the remainder.

           the improper fraction is $ \frac{\left ( k\times q \right )+p}{q}  $

Example : $5\frac{3}{4}$

Solution: $5\frac{3}{4}$

                         =$5\frac{3}{4} = \frac{\left ( 5\times 4 \right )+3}{4}$

                         = $\frac{20+3}{4}$

                         = $\frac{23}{4}$

Note: When we convert a mixed fraction into improper fractions we see that the numerator is greater than that of the denominator.



Multiplication of mixed numbers by fractions: 

When we multiply two or more mixed numbers we follow the following rules.
Step 1. Convert the mixed fractions into improper fractions of the form $\frac{a}{b}$
Step 2. Group the numerators and multiply them.
Step 3. Group the denominators and multiply them.
Step 4. Find the Highest Common Factor (HCF) of the fraction obtained from Steps 2 and 3.
Step 5. Divide the numerator and denominator by the HCF obtained from step 4.
Step 6. If the fraction obtained is is a proper fraction or a whole number highlight this as the final answer else
            convert the improper fraction into mixed fraction and highlight this as the answer.
Example : Multiply and express the answer in the simplest form
 $ 6\frac{3}{16}\times 2\frac{7}{9}\times \frac{2}{5} $


Solution: $6\frac{3}{16}\times 2\frac{7}{9}\times \frac{2}{5} $

                                = $\frac{99}{16}\times \frac{25}{9}\times \frac{2}{5}$ [ converting the mixed fraction into improper fraction ]

                                = $\frac{99\times 25\times 2}{16\times 9\times 5}$ [ grouping the numerators and the denominators separately ]

                                = $\frac{4950}{720}$ [ multiplying the numbers in the numerator and denominator ]

                                = $\frac{4950\div 90}{720\div 90}$ [ dividing the numerator and the denominator by the HCF ]

                               = $\frac{55}{8}$ [ simplest form of the fraction which is in improper form ]

                               = $ 6\frac{7}{8}$ Final answer in the form of mixed fraction.

  Working:
To find the Highest Common Factor of 4950 and 720, let us find the prime factors of each.

fraction22
4950 = 2 x 3 x3 x 5 x 5 x 11
720 = 2 x 2 x 2 x 2 x 3 x 3 x 5
The common factors are , 2,  3, 3, 5
Highest Common Factor HCF = 2 x 5 x 3  x 3 = 90

Practice Questions: 

Multiply the following fractions and express your result in simplest form.
1. $2\frac{4}{5}\times \frac{5}{7}$

2. $3\frac{3}{4}\times 2\frac{1}{2}\times \frac{16}{25}$

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