Fractions are categorized under rational numbers. Students generally find fractions difficult to operate on. Fractions are considered as part of a whole, so while adding two or more fractions, they are all first related to the same whole and then added.

Let us look into the steps and methods used to adding fractions.

Let us say that we want to add $\frac{2}{3}$ and $\frac{3}{4}$.

Students are tempted to add the numerators together and denominators together and write the answer as $\frac{2+3}{3+4}$ = $\frac{5}{7}$. This is a wrong way of adding fractions. Each of the fraction here is related to the whole in a different way. The fraction $\frac{2}{3}$ is formed by dividing the whole into three equal parts and combining two of them together. This means we add two unit fractions $\frac{1}{3}$ together to get $\frac{2}{3}$. In the case of $\frac{3}{4}$ the whole is divided into four equal parts and three unit fractions $\frac{1}{4}$ is combined together to get $\frac{3}{4}$. Indeed we can add only same unit fractions together.

So before adding we need to ensure that each fraction is rewritten into a form, so that both of them are formed by combining the same unit fractions.

Let us now look at the steps involved in adding fractions.

Step 1:
Check whether the denominators of the fractions to be added are same. If yes, it means the fractions are formed by combining the same unit fractions. In that case, the numerators can be added together keeping the same denominator for the sum. Reduce the resulting sum if possible.

Step 2:
If the denominators of the fractions to be added are different, we need to find a common denominator so that each of the fractions can be written in an equivalent form with this common denominator. These equivalent fractions can then be added using the rule stated in step 1.

The following flow chart depicts the steps for be followed while adding the fractions.

Fractions with same denominators are called like denominators. Like denominators can be added by adding the numerators together keeping the same denominator.

Example:

$\frac{2}{9}$ + $\frac{4}{9}$ = $\frac{2+4}{9}$ = $\frac{6}{9}$ = $\frac{2}{3}$

Here the answer is given as a reduced fraction.
The following diagram depicts how each fraction is formed by combining equal unit fractions and how each fraction can be broken into unit fractions and combined together for adding.

When the sum so obtained is greater than 1, that is if the sum happens to be an improper fraction, it is written as a mixed number.

$\frac{3}{5}$ + $\frac{4}{5}$ = $\frac{7}{5}$ = 1$\frac{2}{5}$

Fractions with different denominators are known as unlike denominators. Such fractions are added in two steps.
Step 1:
Each of the fractions is written as an equivalent fraction with a common denominator. The LCM of the original denominators are generally used as the common denominator and hence called LCD, the lowest common denominator.

Example:
Suppose we add $\frac{3}{4}$ and $\frac{5}{6}$. The LCM of the two denominators 4 and 6 s 12 and hence to be used as the LCD.
The equivalent fractions for each fraction with denominator 12 is found as follows
$\frac{3}{4}$ = $\frac{9}{12}$. (As the denominator is multiplied by 3, the numerator is also multiplied by 3)

$\frac{5}{6}$ = $\frac{10}{12}$ (Both the denominator and numerator are multiplied by 2)

Step 2:
The equivalent fractions with same denominator are added by adding the numerators together and keeping the denominator same.
$\frac{3}{4}$ + $\frac{5}{6}$ = $\frac{9}{12}$ + $\frac{10}{12}$ = $\frac{19}{12}$ = 1$\frac{7}{12}$

Here the improper fraction resulting from addition is written as a mixed number.

### Solved Examples

Question 1: A Pizza is cut into 8 equal pieces and shared by three children Jerome, Jane and Robby. Jerome and Jane each got 3 pieces while little Robby got 2 pieces. Find the fraction of Pizza shared by
(a)  Jerome and Jane.
(b)  Jerome and Robby.
Solution:

While Jerome and Jane got $\frac{3}{8}$ of the Pizza, Robby got $\frac{2}{8}$ of it.

Fraction of Pizza shared by Jerome and Jane    = $\frac{3}{8}$ + $\frac{3}{8}$ = $\frac{6}{8}$ = $\frac{3}{4}$

Fraction of Pizza shared by Jerome and Robby = $\frac{3}{8}$ + $\frac{2}{8}$ = $\frac{5}{8}$

Question 2: More than two fractions can also be added applying the same methods discussed above.
Add $\frac{2}{5}$, $\frac{5}{6}$ and $\frac{13}{15}$.
Solution:

We have unlike denominators here. The LCM of the denominators 5, 6 and 15 = 30.

The equivalent fractions to be used for addition with LCD = 30  are

$\frac{2}{5}$ = $\frac{12}{30}$,      $\frac{5}{6}$ = $\frac{25}{30}$     and     $\frac{13}{15}$ = $\frac{26}{30}$

$\frac{2}{5}$ + $\frac{5}{6}$ + $\frac{13}{15}$ = $\frac{12}{30}$ + $\frac{25}{30}$ + $\frac{26}{30}$ = $\frac{63}{30}$ = $\frac{21}{10}$ = 2$\frac{1}{10}$

The resulting sum is reduced and expressed as a mixed number.

Question 3: Mixed numbers can be added as fractions, by writing them as improper fractions.
Add $3\frac{3}{4}$ and 2$\frac{2}{5}$
Solution:

Writing the mixed numbers as improper fractions we get 3$\frac{3}{4}$ = $\frac{15}{4}$     and   2$\frac{2}{5}$ = $\frac{12}{5}$
3$\frac{3}{4}$ + 2$\frac{2}{5}$ = $\frac{15}{4}$ + $\frac{12}{5}$
The LCD for the addition = 20. Hence,
3$\frac{3}{4}$ + 2$\frac{2}{5}$ = $\frac{15}{4}$ + $\frac{12}{5}$  = $\frac{75}{20}$ + $\frac{48}{20}$ = $\frac{123}{20}$ = 6$\frac{3}{20}$
Note here the sum is again given as a mixed number.

Question 4: Here are few examples in algebra.
$\frac{a}{c}$ + $\frac{b}{c}$ = $\frac{a+b}{c}$
Solution:

Both the algebraic fractions have like denominators. The addition is performed similar to the manner fractional numbers with like denominators are added.

Algebraic fractions with unlike denominators are also added using common denominator as in the case of numerical fractions.
$\frac{1}{x}$ + $\frac{1}{y}$ = $\frac{y}{xy}$ + $\frac{x}{xy}$ = $\frac{x+y}{xy}$