Add & Subtract Fractions

Let's now learn how to add and subtract fractions.

Fractions with the same denominator

Let's first consider fractions who have the same denominator.

When fractions have the same denominator, you only have to add (or subtract) the numerators and maintain the denominator.

Example 1

Solve \(\frac{2}{9} + \frac{4}{9}\)

Solution

\(\frac{2}{9} + \frac{4}{9}\)

\(\Rightarrow \frac{2 \ + \ 4}{9}\)

\(\Rightarrow \frac{6}{9}\)

\(\Rightarrow \frac{2 \times 3}{3 \times 3}\)

\(\Rightarrow \frac{2}{3}\)

Example 2

Simplify the expression: \(\frac{2}{3} + \frac{5}{3}\)

Solution

\(\frac{2}{3} + \frac{5}{3}\)

\(\Rightarrow \frac{2 \ + \ 5}{3}\)

\(\Rightarrow \frac{7}{3}\)

\(\Rightarrow 2\frac{1}{3}\)

Example 3

Simplify: \(\frac{1}{7} + \frac{2}{7} + \frac{4}{7}\)

Solution

\(\frac{1}{7} + \frac{2}{7} + \frac{4}{7}\)

\(\Rightarrow \frac{1 \ + \ 2 \ + \ 4}{7}\)

\(\Rightarrow \frac{7}{7}\)

\(\Rightarrow 1\)

Example 4

Simplify: \(\frac{1}{5} - \frac{2}{5} + \frac{4}{5}\)

Solution

\(\frac{1}{5} - \frac{2}{5} + \frac{4}{5}\)

\(\Rightarrow \frac{1 \ - \ 2 \ + \ 4}{5}\)

\(\Rightarrow \frac{3}{5}\)

Example 5

Simplify: \(1 - \frac{4}{9}\)

Solution

\(1 - \frac{4}{9}\)

\(\Rightarrow \frac{9}{9} - \frac{4}{9}\)

\(\Rightarrow \frac{9 \ - \ 4}{9}\)

\(\Rightarrow \frac{5}{9}\)

Remember:

When adding and/or subtracting fractions with the same denominator, maintain the denominator, then add and/or subtract the numerators.

Fractions with different denominators

If the fractions have different denominators, you need to find the Least Common Multiple (LCM) of the various denominators before you apply the operations on the fractions.

Consider the examples below:

Example 6

Simplify \(\frac{4}{5} - \frac{1}{3} + \frac{2}{9}\)

Solution

\(\frac{4}{5} - \frac{1}{3} + \frac{2}{9}\)

The L.C.M of \(5, 3\) and \(9\) is \(45\)

\(\Rightarrow \frac{9(4) \ - \ 15(1) \ + \ 5(2)}{45}\)

\(\Rightarrow \frac{36 \ - \ 15 \ + \ 10}{45}\)

\(\Rightarrow \frac{31}{45}\)

Example 7

Simplify \(\frac{1}{3} - \frac{1}{2} + \frac{2}{5}\)

Solution

\(\frac{1}{3} - \frac{1}{2} + \frac{2}{5}\)

The L.C.M of \(3, 2\) and \(5\) is \(30\)

\(\Rightarrow \frac{10(1) \ - \ 15(1) \ + 6(2)}{30}\)

\(\Rightarrow \frac{10 \ - \ 15 \ + 12}{30}\)

\(\Rightarrow \frac{7}{30}\)

Example 8

Simplify \(\frac{1}{2} - \frac{2}{3} + \frac{3}{4}\)

Solution

\(\frac{1}{2} - \frac{2}{3} + \frac{3}{4}\)

The LCM of \(2, 3\) and \(4\) is \(12\)

\(\Rightarrow \frac{6(1) \ - \ 4(2) \ + \ 3(3)}{12}\)

\(\Rightarrow \frac{6 \ - \ 8 \ + \ 9}{12}\)

\(\Rightarrow \frac{7}{12}\)

Example 9

Simplify \(\frac{1}{3} + \frac{1}{9} + \frac{1}{27}\)

Solution

\(\frac{1}{3} + \frac{1}{9} + \frac{1}{27}\)

The LCM of \(3, 9\) and \(27\) is \(27\)

\(\Rightarrow \frac{9(1) \ + \ 3(1) \ + \ 1(1)}{27}\)

\(\Rightarrow \frac{9 \ + \ 3 \ + \ 1}{27}\)

\(\Rightarrow \frac{13}{27}\)

Example 10

Simplify \(\frac{4}{7} - \frac{2}{3} + \frac{6}{21}\)

Solution

\(\frac{4}{7} - \frac{2}{3} + \frac{6}{21}\)

The LCM of \(7, 3\) and \(21\) is \(21\)

\(\Rightarrow \frac{3(4) \ - \ 7(2) \ + \ 1(6)}{21}\)

\(\Rightarrow \frac{12 \ - \ 14 \ + \ 6}{21}\)

\(\Rightarrow \frac{4}{21}\)

Remember:

When the fractions have different denominators, find the Least Common Multiple (L.C.M) of their denominators.

Mixed Fractions

When the fractions given are mixed fractions there are also two ways that you can simplify it.

\(*\) You can change the mixed fractions into improper fractions first before you add or subtract as done in example 1 and 2 below, or

\(*\) You can add and/or subtract the whole numbers first before you deal with the proper fractions.

Let's consider the examples below:

Example 11

Simplify \(2\frac{2}{3} \ + \ 1\frac{2}{5}\)

Solution

\(2\frac{2}{3} \ + \ 1\frac{2}{5}\)

Changing it to improper fractions:

\(\Rightarrow\) \(\frac{8}{3} \ + \ \frac{7}{5}\)

The LCM of \(3\) and \(5\) is \(15\)

\(\Rightarrow\) \(\frac{5(8) \ + \ 3(7)}{15}\)

\(\Rightarrow\) \(\frac{40 \ + \ 21}{15}\)

\(\Rightarrow\) \(\frac{61}{15}\)

\(\Rightarrow\) \(4\frac{1}{15}\)

Example 12

Simplify \(2\frac{4}{5} \ + \ 1\frac{2}{3}\)

Solution

\(2\frac{4}{5} \ + \ 1\frac{2}{3}\)

Changing it to improper fractions:

\(\Rightarrow\) \(\frac{14}{5} \ + \ \frac{5}{3}\)

The LCM of \(3\) and \(5\) is \(15\)

\(\Rightarrow\) \(\frac{3(14) \ + \ 5(5)}{15}\)

\(\Rightarrow\) \(\frac{42 \ + \ 25}{15}\)

\(\Rightarrow\) \(\frac{67}{15}\)

\(\Rightarrow\) \(4\frac{7}{15}\)

Alternatively, we can add or subtract the whole part of the fraction separately before adding or subtracting the fraction part. Consider the example below:

Example 13

Simplify \(2\frac{4}{5} \ + \ 1\frac{2}{3}\)

Solution

\(2\frac{4}{5} \ + \ 1\frac{2}{3}\)

\(\Rightarrow 2 + \ 1 \ + \frac{4}{5} \ + \ \frac{2}{3}\)

\(\Rightarrow 3 \ + \frac{3(4) + 5(2)}{15}\)

\(\Rightarrow 3 \ + \frac{12 + 10}{15}\)

\(\Rightarrow 3 \ + \frac{22}{15}\)

\(\Rightarrow 3 \ + 1\frac{7}{15}\)

\(\Rightarrow 3 \ + \ 1 + \ \frac{7}{15}\)

\(\Rightarrow 4 + \frac{7}{15}\)

\(\Rightarrow 4\frac{7}{15}\)

Example 14

Simplify \(2\frac{4}{15} - 1\frac{2}{3}\)

Solution

\(2\frac{4}{15} - 1\frac{2}{3}\)

\(\Rightarrow 2 - 1 + \frac{4}{15} - \frac{2}{3}\)

\(\Rightarrow 1 + \frac{1(4) - 5(2)}{15}\)

\(\Rightarrow 1 + \frac{4 - 10}{15}\)

\(\Rightarrow 1 + \frac{- 6}{15}\)

\(\Rightarrow \frac{15(1) - 1(6)}{15}\)

\(\Rightarrow \frac{15 - 6}{15}\)

\(\Rightarrow \frac{9}{15}\)

\(\Rightarrow \frac{3}{5}\)

Example 15

Simplify \(1\frac{1}{2} + 2\frac{1}{4} - 3\frac{5}{8}\)

Solution

\(1\frac{1}{2} + 2\frac{1}{4} - 3\frac{5}{8}\)

\(\Rightarrow 1 + 2 - 3 + \frac{1}{2} + \frac{1}{4} - \frac{5}{8}\)

\( \Rightarrow 3 - 3 + \frac{4(1) + 2(1) - 1(5)}{8} \)

\( \Rightarrow \frac{4 + 2 - 5}{8} \)

\( \Rightarrow \frac{1}{8} \)

Exercise

Simplify the fractions below:

\(3\frac{3}{8} - 1\frac{1}{12}\)
\(3\frac{3}{8} - 1\frac{1}{2}\)
\(2\frac{2}{3} - \frac{5}{6}\)
\((\frac{2}{3} - \frac{1}{2}) \div \frac{1}{6}\)
\(\frac{1}{2} - \frac{2}{3} + \frac{3}{4}\)

Solution

a. \(2\frac{7}{24}\)

b. \(1\frac{7}{8}\)

c. \(3\frac{1}{2}\)

d. 1

e. \(\frac{7}{12}\)

Quick Test

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Basic 7