Adding and subtracting fractions are relatively difficult operations when compared to performing multiplication or division on fractions. The rules for subtracting fractions are same as the one's applied to adding fractions. Our objective here is to learn how to perform subtraction in fractions.

## Rules for Subtracting Fractions

Suppose we have a subtraction problem $\frac{3}{7}$ - $\frac{1}{4}$ to solve. The common error committed by students is find the difference for numerators and denominators separate and write the answer as $\frac{3-1}{7-4}$ = $\frac{2}{3}$(wrong).

Fractions are composites made up of unit fractions. Unit fractions are fractions with one in the numerator like $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$ etc.. The fraction $\frac{1}{3}$ refers to one part of the whole when the whole is divided into three equal parts. Other fractions are viewed to be formed by combining equal unit fractions together. This idea is revealed pictorially as follows:

Subtraction of fractions is viewed as the removal of a number of equal unit fractions. The rule or the method for subtraction applied for fractions depend on whether the two numbers involved in the operation, have the same or different denominators.

## How to solve Subtracting Fractions

Fractions can be subtracted by using these two steps:

Step 1:
If the subtraction is performed on fractions with like denominators, the subtraction is done only on numbers in the numerators, retaining the same denominator.

Step 2:
If the subtraction is performed on fractions with unlike denominators, the fractions are first written as equivalent fractions with a common denominator. Then the subtraction is done as it is done in the case of fractions with like denominators.

## Subtracting Fractions with Like Denominators

Fractions with like denominators are formed by combining the same unit fraction.

For example $\frac{2}{7}$ is formed by combining two unit fractions $\frac{1}{7}$ and $\frac{3}{7}$ is formed combining three of the same unit fraction $\frac{1}{7}$.

Finding the difference between two like fractions can be viewed as removing the unit fractions contained in the smaller from the greater of the two numbers.  So to perform subtraction, it is sufficient to find the difference of the numerators, retaining the same denominator.

Example:
$\frac{5}{9}$ - $\frac{3}{9}$ = $\frac{5-3}{9}$ = $\frac{2}{9}$.

The above operation can be pictorially explained as follows:

The resulting difference is reduced when ever it is possible. While performing subtraction on mixed numbers, the operation is performed after writing the mixed numbers as improper fractions.

## Subtracting Fractions with Unlike Denominators

Unlike fractions are formed by combining different unit fractions. For example, $\frac{2}{3}$ is formed combining two of $\frac{1}{3}$ while $\frac{2}{7}$ is formed by combining of two of $\frac{1}{7}$. Hence the subtraction can be performed on unlike fractions, only after expressing them as equivalent fractions sharing a common denominator.

The lowest common multiple of the two denominators is used in this process as the common denominator and hence called the lowest common denominator (LCD). Then the difference between the equivalent fractions are found in the same manner as with like fractions.

### Solved Example

Question: Find the difference of $\frac{1}{2}$ - $\frac{3}{8}$
Solution:

The first task is to find the common denominator to be used in the equivalent fractions $\frac{1}{2}$ and $\frac{3}{8}$.
The LCM of 2 and 8 is 8 and 8 is used as the common denominator.

Since the second fraction $\frac{3}{8}$ already has 8 as its denominator, it is only required to write $\frac{1}{2}$ in its equivalent form with denominator 8.

$\frac{1}{2}$ = $\frac{1\times 4}{2\times 4}$ = $\frac{4}{8}$.

Hence $\frac{1}{2}$ - $\frac{3}{8}$ = $\frac{4}{8}$ - $\frac{3}{8}$ = $\frac{1}{8}$.

## Examples of  Subtracting Fractions

Example 1:
When their father died Andrew and Abby each correspondingly inherited $\frac{1}{2}$ and $\frac{3}{8}$ of their father's wealth while their cousin Ann received the remaining part. How much of the father's wealth is gifted to Ann?
Solution:
We need to find the total amount inherited by the brothers and subtract that from 1 to find Ann/s share in the wealth.
The part of wealth inherited by the brothers = $\frac{1}{2}$ + $\frac{3}{8}$ (We add fractions with unlike denominators here)

= $\frac{4}{8}$ + $\frac{3}{8}$ (Equivalent fraction for $\frac{1}{2}$ is used)

= $\frac{7}{8}$

The part of wealth gifted to Ann = 1 - $\frac{7}{8}$

= $\frac{1}{1}$ - $\frac{7}{8}$ (1 is written as a rational number.)

= $\frac{8}{8}$ - $\frac{7}{8}$ (Equivalent fraction)

= $\frac{1}{8}$
Ann received $\frac{1}{8}$ of her uncle's wealth.

Example 2:
Subtraction can be performed in a group of fractions along with addition.
$\frac{2}{3}$ + $\frac{3}{4}$ - $\frac{5}{6}$

The LCD to be used is 12 ( LCM of the denominators 3, 4 and 6)

Rewriting the fractions in equivalent forms

$\frac{2}{3}$ = $\frac{2\times 4}{3\times 4}$ = $\frac{8}{12}$

$\frac{3}{4}$ = $\frac{3\times 3}{4\times 3}$ = $\frac{9}{12}$

$\frac{5}{6}$ = $\frac{5\times 2}{6\times 2}$ = $\frac{10}{12}$

$\frac{2}{3}$ + $\frac{3}{4}$ - $\frac{5}{6}$ = $\frac{8}{12}$ + $\frac{9}{12}$ - $\frac{10}{12}$ = $\frac{8+9-10}{12}$ = $\frac{7}{12}$

Example 3:
Subtraction on mixed numbers
4$\frac{2}{3}$ - 2$\frac{1}{2}$ = $\frac{14}{3}$ - $\frac{5}{2}$ (The mixed numbers are written as improper fractions)
= $\frac{28}{6}$ - $\frac{15}{6}$ (Equivalent fractions)

= $\frac{13}{6}$ (Subtraction performed with common denominator)

= $2\frac{1}{6}$ (The difference is written as a mixed number)

Example 4:
Subtraction on fractional algebraic expressions
$\frac{2x}{(x^{2}-1)} - \frac{2}{x+1}$

As x2 - 1 = (x + 1)(x - 1)

The LCD to be used is = x2 - 1

$\frac{2x}{(x^{2}-1)} - \frac{2}{x+1}$ = $\frac{2x}{(x^{2}-1)}$ - $\frac{2(x-1)}{(x+1)(x-1)}$

= $\frac{2x-2x +2}{x^{2}-1}$

= $\frac{2}{x^{2}-1}$