Multiplication of Fractions: Properties, Methods & Solved Examples

Multiplication of Fractions: A fraction denotes a part of the whole. When we divide a whole into equal parts, then each part is called a fraction. A fraction has two parts, the numerator, and the denominator. We use the fundamental operations on fractions like addition, subtraction, multiplication, and division in our day-to-day life.

We know that multiplication is known as the repeating addition. We can multiply a fraction with a fraction or a whole number. There are some sets of rules for the multiplication of fractions. In this article, we will learn them in detail one by one.

What is Fraction?

A fraction is a number that represents a part of the whole. The whole may be a single object or multiple objects. A fraction is written as \(\frac{x}{y},\) where \(x\) and \(y\) are whole numbers and \(y≠0.\) Numbers such as \(\frac{1}{4},\frac{2}{5},\frac{4}{9},\frac{{11}}{7}\) are known as the fractions. 

For example: Draw a circle with any suitable radius. Then, divide the circle into four equal parts (sectors). Each equal part is considered as \(\frac{1}{4}.\)

Now, the number below the fraction bar is called the denominator. It tells us how many equal parts a whole is divided. The number above the line is called the numerator. It tells us how many equal parts are taken or considered. Have a look at the figure.

If four parts of the seven equal parts of the circle are shaded, we say four-seventh \(\frac{4}{7}\) of the circle is shaded three- seventh of the circle is not.

Similarly, If five parts of the seven equal parts of the circle are shaded, we say five-seventh \(\frac{5}{7}\) of the circle is shaded two- seventh of the circle is not.

Types of Fractions 

There are different types of fractions. Let us understand each type.

Proper Fractions

A fraction whose numerator is less than its denominator is called a proper fraction. For examples, \(\frac{3}{4},\frac{7}{{10}},\frac{1}{4},\frac{3}{7},\) etc, are all proper fractions. 

The value of the proper fraction is always less than \(1.\)

Improper Fractions

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction. For examples, \(\frac{7}{4},\frac{{13}}{5},\frac{{11}}{6},\frac{{23}}{7}\) etc, are all improper fractions. 

Mixed Fractions

A combination of a whole number and a proper fraction is called a mixed fraction. For examples, \(2\frac{2}{4},6\frac{5}{{10}},5\frac{1}{5},6\frac{2}{{13}}\) etc., are all mixed fractions. 

Multiplication of Fractions

Multiplication is known as repeating addition. Let us see the pictorial representation of it.

The fraction \(\frac{2}{3}\) is repeated four times, we can write this addition in a simpler way as \(4 \times \frac{2}{3}.\)
A fraction can be multiplied by a whole number or a fraction.

Multiplication of a Fraction by a Whole Number

Multiplying a fraction by a whole number means multiplying the whole number by the fraction’s numerator and keeping the denominator the same. After multiplication, simplify the fraction if required to get the product in the simplest form. 

For example: \(\frac{3}{7} \times 5 = \frac{{15}}{7}\)
Here, the fraction \(\frac{{15}}{7}\) is in its simplest form as the HCF of the numerator, and the denominator is \(1.\)

For example: \(\frac{2}{5} \times 10 = \frac{{20}}{5}\)
Here, \(\frac{{20}}{5}\) is not in its simplest form as the HCF of the numerator and the denominator is not \(1.\)
HCF\((20, 5)=5\)

Now, we can reduce the fraction by dividing the numerator and the denominator with this HCF.
So, \(\frac{{20 \div 5}}{{5 \div 5}} = \frac{4}{1} = 4\)
\(\therefore \frac{2}{5} \times 10 = 4\)

Multiplication of a Fraction by a Fraction

When two fractions are multiplied, the numerator and denominators are multiplied separately. The numerator of the first is multiplied with the numerator of the second, and the denominator of the first is multiplied by the denominator of the second. At last, we will reduce the fraction to its lowest term. A fraction is in its lowest form if \(1\) is the only common factor between its numerator and denominator.

Multiplication of a Proper Fraction by a Proper Fraction

Let us learn the multiplication of a proper fraction by a proper fraction.
For example, \(\frac{2}{3} \times \frac{6}{7} = \frac{{2 \times 6}}{{3 \times 7}} = \frac{{12}}{{21}}\)

Here, \(\frac{{12}}{{21}}\) is not in its simplest form as the HCF of the numerator and the denominator is not 1.
HCF\((12, 21)=3\)
Now, we can reduce the fraction by dividing the numerator and the denominator with this HCF.
Thus, \(\frac{{12 \div 3}}{{21 \div 3}} = \frac{4}{7}\)
\(\frac{2}{3} \times \frac{6}{7} = \frac{4}{7}\)

Multiplication of a Proper Fraction by an Improper Fraction

Let us learn the multiplication of a proper fraction by an improper fraction.
For example, \(\frac{4}{5} \times \frac{{15}}{8}\)
Here, \(\frac{4}{5}\) is a proper fraction and \(\frac{{15}}{8}\) is an improper fraction.
\( = \frac{{4 \times 15}}{{5 \times 8}} = \frac{{60}}{{40}}\)

Here, \(\frac{{60}}{{40}}\) is not in its simplest form as the HCF of the numerator and the denominator is not \(1.\)
HCF\((60, 40)=20\)
Now, we can reduce the fraction by dividing the numerator and the denominator with this HCF.
Thus, \(\frac{{60 \div 20}}{{40 \div 20}} = \frac{3}{2}\)
So, \(\frac{4}{5} \times \frac{{15}}{8} = \frac{3}{2}\)

Multiplication of an Improper Fraction by an Improper Fraction

For example, \(\frac{5}{3} \times \frac{{12}}{7}\)
Now, multiplying the numerators and the denominators we have,
\(\frac{{5 \times 12}}{{3 \times 7}} = \frac{{60}}{{21}}\) Here, \(\frac{{60}}{{21}}\) is not in its simplest form as the HCF of the numerator and the denominator is not 1.
HCF\((60, 21)=3\)
Thus, \(\frac{{60 \div 3}}{{21 \div 3}} = \frac{{20}}{7}\)
So, \(\frac{5}{3} \times \frac{{12}}{7} = \frac{{20}}{7}\)

Multiplication of a Mixed Fraction by an Improper Fraction

If a mixed fraction and an improper fraction are being multiplied by each other, we need to convert the mixed fraction into an improper fraction.
For example, \(2\frac{3}{7} \times \frac{{14}}{3}\)
Let us convert the mixed fraction into an improper fraction.
\(2\frac{3}{7} = \frac{{2 \times 7 + 3}}{7} = \frac{{14 + 3}}{7} = \frac{{17}}{7}\)

Now, multiplying the numerators and the denominators of both the fractions we have,
\(\frac{{17}}{7} \times \frac{{14}}{3} = \frac{{17 \times 14}}{{7 \times 3}} = \frac{{238}}{{21}}\)

Here, \(\frac{{238}}{{21}}\) is not in its simplest form.

HCF\((238, 21)=7\)
Thus, \(\frac{{238 \div 7}}{{21 \div 7}} = \frac{{34}}{3}\)
\(2\frac{3}{7} \times \frac{{14}}{3} = \frac{{34}}{3}\)

Multiplication of a Mixed Fraction by a Proper Fraction

Let us take an example and see how we multiply a mixed fraction and a proper fraction.
\(3\frac{2}{5} \times \frac{5}{8}\)
Let us convert the mixed fraction into an improper fraction.
\(3\frac{2}{5} = \frac{{17}}{5}\)
Now, multiplying the numerators and the denominators of both the fractions we have,
\(\frac{{17 \times 5}}{{5 \times 8}} = \frac{{85}}{{40}}\)
Here, \(\frac{{85}}{{40}}\) is not in its simplest form.
HCF\((85, 40)=5\)
Thus, \(\frac{{85 \div 5}}{{40 \div 5}} = \frac{{17}}{8}\)
So, \(3\frac{2}{5} \times \frac{5}{8} = \frac{{17}}{8}\)

Multiplication of a Mixed Fraction by a Mixed Fraction

Before doing the multiplication of two mixed fractions, we need to convert them into an improper fraction.
Example: \(1\frac{2}{5} \times 2\frac{2}{3}\)
Let us convert the mixed fraction into an improper fraction.
\(1\frac{2}{5} = \frac{{1 \times 5 + 2}}{5} = \frac{7}{5}\)
\(2\frac{2}{3} = \frac{{2 \times 3 + 2}}{3} = \frac{8}{3}\) Now, multiplying the numerators and the denominators of both the fractions we have,
\(\frac{{7 \times 8}}{{5 \times 3}} = \frac{{56}}{{15}}\)
Here, the fraction \(\frac{{56}}{{15}}\) is in its simplest form as the HCF of the numerator and the denominator is \(1.\)

Multiplication of a Fraction Using the Operator ‘of’

The word ‘of’ denotes multiplication. 

For example, \(\frac{2}{6}\) of \(12\) cakes mean \(4\) cakes i.e. \(\frac{2}{6} \times 12 = 4.\)

Properties of Multiplication of Fractions

1. If a non-zero fraction is multiplied by \(1\) the result is the fractional number itself.

Example: \(\frac{3}{8} \times 1 = \frac{3}{8}\)

2. If zero is multiplied by a non-zero fraction, then the result is zero.

Example: \(\frac{3}{8} \times 1 = \frac{3}{8}\)

3. The product of a fraction and its reciprocal is always \(1.\)

Example: \(\frac{2}{5} \times \frac{5}{2} = \frac{{2 \times 5}}{{5 \times 2}} = \frac{{10}}{{10}} = 1\) (reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\))

Solved Examples – Multiplication of Fractions

Q.1. Solve \(\frac{2}{3} \times \frac{9}{{10}}.\)
Ans: Given \(\frac{2}{3} \times \frac{9}{{10}}\)
\( = \frac{{2 \times 9}}{{3 \times 10}} = \frac{{18}}{{30}}\)
HCF\((18, 30)=6\)
So, \(\frac{{18 \div 6}}{{30 \div 6}} = \frac{3}{5}\)
\(\therefore \frac{2}{3} \times \frac{9}{{10}} = \frac{3}{5}\)

Q.2. Simplify \(5\left( {\frac{8}{{11}} \times \frac{{22}}{5}} \right).\)
Ans: Given, \(5\left( {\frac{8}{{11}} \times \frac{{22}}{5}} \right)\)
\( = 5 \times \left( {\frac{8}{{11}} \times \frac{{22}}{5}} \right)\)
\( = 5 \times \left( {\frac{8}{1} \times \frac{2}{5}} \right)\)
\( = 5 \times \frac{{16}}{5}\)
\( = 16\)
\(\therefore 5\left( {\frac{8}{{11}} \times \frac{{22}}{5}} \right) = 16\)

Q.3. To bake a cake, \(1\frac{1}{2}\) cups of flour are needed. How many cups of flour are needed for baking \(6\) cakes?
Ans: Given, to bake a cake, \(1\frac{1}{2}\) cups of flour are needed.
The numbers of cups of flour required to bake \(1\) cake\( = 1\frac{1}{2} = \frac{3}{2}\)
So, the numbers of cups of flour required to bake \(6\) cakes\( = \frac{3}{2} \times 6 = 9\) cups

Q.4. Find the area of a square whose side length is \(5\frac{1}{2}\;{\rm{cm}}\).
Ans: We know that the area of a square\( = {\mathop{\rm side}\nolimits} \, \times {\rm{side}}\)
\(5\frac{1}{2} = \frac{{5 \times 2 + 1}}{2} = \frac{{11}}{2}\)
Hence, the area of the square whose side length is \(5\frac{1}{2}\;{\rm{cm}} = 5\frac{1}{2} \times 5\frac{1}{2} = \frac{{11}}{2} \times \frac{{11}}{2} = \frac{{121}}{4}\;{\rm{c}}{{\rm{m}}^2}\)

Q.5. There are \(20\) students in a class, and \(\frac{1}{4}\) of them are boys. Find out the number of boys in the class?
Ans: Given that there are \(20\) students in a class, and \(\frac{1}{4}\) of them are boys.
Here, we need to find \(\frac{1}{4}\) of \(20\) that means \(\frac{1}{4} \times 20 = 5\)
Hence, the number of boys in the class is 5.

Summary

In this article, we have covered the definition and types of fractions, multiplication of a fraction with a whole number, multiplication of different types of fractions, etc.

FAQs on Multiplication of Fractions

Q.1. How do you multiply mixed fractions?
Ans: Before doing the multiplication of two mixed fractions, we need to convert them into an improper fraction.
Example: \(1\frac{1}{5} \times 5\frac{2}{3}\)
Let us convert the mixed fraction into an improper fraction.
\(1\frac{1}{5} = \frac{6}{5},5\frac{2}{3} = \frac{{17}}{3}\) Now, multiplying the numerators and the denominators of both the fractions we have,
\(\frac{{6 \times 17}}{{5 \times 3}} = \frac{{102}}{{15}}\)
Here, the fraction \(\frac{{102}}{{15}}\) is not in its simplest form as the HCF of the numerator, and the denominator is \(3.\)
Now, we can reduce the fraction by dividing the numerator and the denominator with this HCF.
So, \(\frac{{102 \div 3}}{{15 \div 3}} = \frac{{34}}{5}\).
\(\therefore 1\frac{1}{5} \times 5\frac{2}{3} = \frac{{34}}{5}\)

Q.2. How do you multiply a fraction with a whole number?
Ans: Multiplying a fraction by a whole number means multiplying the whole number by the fraction’s numerator and keeping the denominator the same. After multiplication, simplify the fraction if required to get the product in the simplest form.
For example: \(\frac{3}{7} \times 6 = \frac{{18}}{7}\)
Here, the fraction \(\frac{{18}}{7}\) is in its simplest form as the HCF of the numerator, and the denominator is \(1.\)

Q.3. How do you multiply a fraction by a fraction?
Ans: When two fractions are multiplied, the numerator and denominators are multiplied separately. The numerator of the first is multiplied with the numerator of the second. Then, the denominator of the first is multiplied by the denominator of the second. At last, we will reduce the fraction in its lowest term.
For example, \(\frac{2}{3} \times \frac{1}{7} = \frac{{2 \times 1}}{{3 \times 7}} = \frac{2}{{21}}\)

Q.4. What do we get as a result when a fraction is multiplied by its reciprocal?
Ans: If we multiply a fraction with its reciprocal, it gives \(1\) as result.
Let us take a fraction \(\frac{3}{4}.\)
Reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}.\)
Now the product of \(\frac{3}{4}\) and \(\frac{4}{3} = \frac{3}{4} \times \frac{4}{3} = 1\)

Q.5 What do we get as a result if we multiply a fraction by \(1\)?
Ans: If a non-zero fraction is multiplied by \(1\) the result is the fractional number itself.
Example: \(\frac{5}{8} \times 1 = \frac{5}{8}\)

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