Fractions whose denominators are the same are called similar fractions. Fractions that are not similar are called dissimilar fractions. Hence, the fractions $\frac{1}{8}$, $\frac{3}{8}$, and $\frac{5}{8}$ are similar fractions, while the fractions $\frac{2}{3}$ and $\frac{1}{2}$ are dissimilar fractions. In this post, we are going to learn how to add fractions.

Adding similar fractions is very easy.  In adding similar fractions, you just add the numerator and copy the denominator.  Here are a few examples.

Example 1 $\displaystyle \frac{1}{5} + \frac{2}{5} = \frac{1 + 2}{5} = \frac{3}{5}$

Example 2 $\displaystyle \frac{1}{8} + \frac{2}{8} + \frac{4}{8} = \frac{1 + 2 + 4}{8} = \frac{7}{8}$

Example 3 $\displaystyle \frac{1}{9} + \frac{3}{9} + \frac{7}{9} = \frac{11}{9}$

In most cases, improper fractions or fractions whose denominator is less than its numerator such as the third example is converted to mixed form. The mixed form of $\frac{11}{9}$ is $1 \frac{2}{9}$. We will discuss how to make such conversion in the near future.

Addition of dissimilar fractions is a bit more complicated than adding similar fractions. In adding dissimilar fractions, you must determine the least common multiple (LCM)  of their denominator which is known as the least common denominator. Next, you have to convert all the addends to equivalent fractions whose denominator is the LCM. Having the same denominator means that the fractions are already similar.  Here are a few examples.

Example 1 $\displaystyle \frac{1}{2} + \frac{1}{3}$

Solution

a. Get the least common multiple (LCM) of 2 and 3.

Multiples of 2: 2, 4, 6, 8, 10, 12

Multiples of 3: 3, 6, 9, 12,  15

LCM of 2 and 3 is 6.

b. Convert the fractions into fractions whose denominator is the LCM which is 6.

First Addend: $\displaystyle \frac{1}{2} = \frac{m}{6}$ $m = (6 \div 2) \times 1 = 3$.

So, the equivalent of $\frac{1}{2}$ is $\frac{3}{6}$.

Second Addend: $\displaystyle \frac{1}{3} = \frac{n}{6}$ $n = (6 \div 3) \times 1 = 2$

So, the equivalent fraction of $\frac{1}{3}$ is $\frac{2}{6}$. $\displaystyle \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$.

So, $\displaystyle \frac{1}{2} + \frac{2}{3} = \frac{5}{6}$.

Example 2 $\displaystyle \frac{2}{3} + \frac{1}{5}$

Solution

a. Get the LCM of 3 and 5.

Multiples of 3: 3, 6, 9, 12, 15, 18

Multiples of 5: 5, 10, 15, 20

Therefore, the LCM of 3 and 5 is 15.

b. Convert the given fractions into equivalent fractions whose denominator is 15

First Addend: $\displaystyle \frac{2}{3} = \frac{p}{15}$ $p = (15 \div 3) \times 2 = 10$

So, the equivalent fraction of $\frac{2}{3}$ is $\frac{10}{15}$.

Second Addend: $\displaystyle \frac{1}{5} = \frac{q}{15}$ $q = 15 \div 5 \times 1 = 3$

So, the equivalent fraction of $\frac{1}{5}$ is $\frac{3}{15}$. $\displaystyle \frac{10}{15} + \frac{3}{15} = \frac{13}{15}$.

So, $\displaystyle \frac{2}{3} + \frac{1}{5} = \frac{13}{15}$

Example 3 $\displaystyle \frac{2}{3} + \frac{1}{6} + \frac{1}{8}$

Solution

a. Get the LCM of 3, 6 and 8.

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24

Multiples of 6: 6, 12, 18, 24, 30

Multiples of 8: 8, 16, 24, 32, 40

LCM of 3, 6 and 8 is 24.

b. Convert the given fractions into equivalent fractions whose denominator is 24.

First Addend: $\displaystyle \frac{2}{3} = \frac{x}{24}$ $x = (24 \div 3) \times 2 = 8 \times 2 = 16$.

Therefore, the equivalent fraction of $\frac{2}{3}$ is $\frac{16}{24}$

Second Addend: $\displaystyle \frac{1}{6} = \frac{y}{24}$ $y = (24 \div 6) \times 1 = 4$

Therefore, the equivalent fraction of $\frac{1}{6}$  is $\frac{4}{24}$

Third Addend: $\displaystyle \frac{1}{8} = \frac{z}{24}$ $z = (24 \div 8) \times 1 = 3$

Therefore, the equivalent fraction of $\frac{1}{8}$ is $\frac{3}{24}$. $\displaystyle \frac{16}{24} + \frac{4}{24} + \frac{3}{24} = \frac{23}{24}$