About Operations on Fractions
- Operations on Fractions
- Addition and Subtraction of Fractions
- Multiplication of Fractions
- Example of Multiplication of Two Fractions
- Example of Multiplication of Fraction with Mixed Number
- Division of Fractions
- Examples of Fraction Divided by Another Fraction
- Examples of a Whole Number Divided by a Fraction
- Real Life Modeling Questions
- Frequently Asked Questions
Operations on Fractions
There are four basic operations that can be performed on numbers and they are addition, subtraction, multiplication and division. We have learnt how to perform these operations on numbers since junior grades.
For example:
Addition: \( 3+2=5 \)
Subtraction: \( 7-4=3 \)
Multiplication: \( 5\times 3=15 \)
Division: \( \frac{25}{5}=5 \)
We can perform similar operations on fractions as well. Let us learn more about it.
Addition and Subtraction of Fractions
To add and subtract fractions we use equivalent fractions to write the fractions with a common denominator. To find a suitable common denominator we first find the LCM (least common multiple) of fractions with uncommon denominators. Let us understand this by solving a few examples:
Example 1: Find \( \frac{1}{8}+\frac{3}{5} \).
Solution:
Let’s use equivalent fractions to rewrite the fractions with a common denominator.
8 is not a multiple of 5, so let us rewrite each fraction with a denominator which is LCM of 8 and 5 which can be found as :
\( 8 \times 5=40 \)
\( \frac{1}{8} + \frac{3}{5} = \frac{5}{40} + \frac{24}{40}\) Rewrite \( \frac{1}{8} \) as \( \frac{1\times 5}{8\times 5} = \frac{5}{40} \) and \( \frac{3}{5} \) as \( \frac{3\times 8}{5\times 8} = \frac{24}{40} \)
\( =\frac{5+24}{40} \) Solve 5 + 24
\( =\frac{29}{40} \)
Example 2: Find \( \frac{6}{7}-\frac{3}{4} \).
Solution:
Let’s use equivalent fractions to rewrite the fractions with a common denominator.
7 is not a multiple of 4, so let us rewrite each fraction with a denominator which is LCM of 7 and 4 which can be found as :
\( 7\times 4=28 \)
\( \frac{6}{7}-\frac{3}{4}=\frac{24}{28}-\frac{21}{28} \) Rewrite \( \frac{6}{7} \) as \( \frac{6\times 4}{7\times 4}=\frac{24}{28} \) and \( \frac{3}{4} \) as \( \frac{3\times 7}{4\times 7}=\frac{21}{28} \)
\( =\frac{24-21}{28} \) Solve 24 – 21
\(~~~~~~~~~~ =\frac{3}{28} \)
Multiplication of Fractions
To perform multiplication of fractions is an easy task. All we need to do is multiply the numerator to the numerator and denominator to the denominator. Solving a few examples will help you understand the method of multiplying fractions.
Example of Multiplication of Two Fractions
Find: \( \frac{1}{7}\times \frac{3}{4} \)
Solution :
\( \frac{1}{7}\times \frac{3}{4}=\frac{1\times 3}{7\times 4} \) Multiply the numerator to the numerator and denominator to the denominator.
\( =\frac{3}{28} \)
Example of Multiplication of Fraction with Mixed Number
Find: \( \frac{2}{3}\times 5\frac{3}{4} \)
Solution:
\( \frac{2}{3}\times 5\frac{3}{4}=\frac{2}{3}\times \frac{23}{4} \) Write \( 5\frac{3}{4}\) as the improper fraction \( \frac{23}{4}\).
\( =\frac{2\times 23}{3\times 4}\) Multiply the numerator to the numerator and denominator to the denominator.
\( =\frac{46}{12}\) or \( \frac{23}{6}\) Simplify.
Division of Fractions
When we divide a fraction by another, we need to multiply the dividend with the reciprocal of the divisor. We can observe the method of writing the reciprocal with this explanation.
Reciprocal of \( \frac{a}{b}\) is \( \frac{b}{a}\)
For example:
Reciprocal of \( \frac{3}{7}\) is \( \frac{7}{3}\) and Reciprocal of \( \frac{9}{13}\) is \( \frac{13}{9}\)
If we notice closely the product of the fraction and its reciprocal is 1.
For example :
\( \frac{3}{7}\times \frac{7}{3}=1 \) and \( \frac{9}{13}\times \frac{13}{9}=1 \).
Note: Reciprocals are also called multiplicative inverses.
Now, let us see how reciprocals are used to perform division of fractions. Here are some examples.
Examples of Fraction Divided by Another Fraction
Find: \( \frac{1}{5}\div \frac{1}{9} \)
Solution:
\( \frac{1}{5}\div \frac{1}{9}=\frac{1}{5}\times \frac{9}{1} \) Multiply by the reciprocal of \( \frac{1}{9} \) which is \( \frac{9}{1} \)
\( =\frac{1\times 9}{5\times 1} \) Multiply the fractions.
\( =\frac{9}{5} \) Simplify.
Examples of a Whole Number Divided by a Fraction
Find: \( \frac{4}{7}\div 5 \)
Solution:
\( \frac{4}{7}\div 5=\frac{4}{7}\times \frac{1}{5} \) Multiply by the reciprocal of 5 which is \( \frac{1}{5} \)
\( =\frac{4\times 1}{7\times 5} \) Multiply the fractions.
\( =\frac{4}{35} \) Simplify.
Real Life Modeling Questions
You have \( \frac{4}{5} \) of a pizza. You divide the remaining pizza in 5 equal parts to distribute among your family members. What portion of the original pizza will each member get?
Solution:
We need to divide \( \frac{4}{5} \) into five equal parts.
\( \frac{4}{5}\div 5= \frac{4}{5}\times \frac{1}{5} \) Multiply by the reciprocal of 5 which is \( \frac{1}{5} \)
\( =\frac{4\times 1}{5\times 5} \) Multiply the fractions.
\( =\frac{4}{25} \) Simplify.
Hence, each family member gets \( \frac{4}{25} \) of the pizza.
As every whole number can be divided by 1 and it won’t change its value, so reciprocal of a whole number like 4 can be written as \( \frac{1}{4} \).
Reciprocal of zero is not defined as we cannot divide any number with zero.
0 can be written as \( \frac{0}{1} \) but, \( \frac{1}{0} \) is not defined.