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# Subtracting Fractions

Top
 Sub Topics Fractions can be a part of a object or whole, and are expressed as the ratio of two numbers used for comparison between parts and the whole. A fraction is a number expressed in the form of $\frac{p}{q}$, where the numerator 'p' signifies the number of parts it has, and the denominator 'q' is the number of parts the whole is divided into. We know that all the mathematical operators can be performed on the fractions. Subtraction is one among the mathematical operations. Subtraction with like denominators is easy, subtracting fractions takes two fractions and subtracts the second one from the first one. To learn how to subtract fractions, we need to first convert the mixed fractions to improper fractions. We say that the subtraction of the fractions with different denominators is possible. Also if we subtract zero from any fraction, the result remains the same.

## Steps for Subtracting Fractions

Steps for subtracting fractions are given below :
1. For the given problem, if there are like denominators then subtract the numerators keeping the denominator same else move to the next step.
2. Find the lowest common multiple of the denominators.
3. Rename the fractions to have the lowest common multiple and subtract the numerators of the fractions.
4. Numerator will be the difference and the denominator will be the lowest common multiple.
5. Simplify the fraction.

Adding fractions with the same denominator

Example: Add $\frac{5}{6}$ + $\frac{7}{6}$

Solution: Consider $\frac{5}{6}$ and $\frac{7}{6}$

$\frac{5}{6}$ + $\frac{7}{6}$

= $\frac{12}{6}$

= 2

Example: Add $\frac{2}{5}$ + $\frac{7}{2}$

Solution: Given $\frac{5}{6}$ and $\frac{7}{6}$

$\frac{2}{5}$ + $\frac{7}{2}$

For the denominator the lowest common multiple is 10.

= $\frac{39}{10}$

## Subtracting Fractions with Variables

Based on the steps explained above given below is an example for subtracting fractions with variables.

Example: $\frac{3}{p}$ + $\frac{7}{p - 2}$

Solution: $\frac{3}{p}$ + $\frac{7}{p - 2}$

= $\frac{3}{p}+\frac{7}{p-2}$

= $\frac{3(p-2)+7p}{p(p-2)}$

= $\frac{3p - 6 + 7p}{p(p-2)}$

= $\frac{10p-6}{p(p-2)}$

## Subtracting Fractions with Like Denominators

When we subtract fractions with like denominator, simply we subtract the numerators and place the result over the common denominator.

Below you could see some examples for subtracting fractions with like denominators.
Example 1: $\frac{3}{5}$ - 0

Solution: $\frac{3}{5}$ - 0 = $\frac{3}{5}$

Example 2: $\frac{7}{3}$ - $\frac{4}{3}$

Solution: $\frac{7}{3}$ - $\frac{4}{3}$ = $\frac{3}{3}$
= 1

Example 3: $\frac{12}{7}$ - 1

Solution : $\frac{12}{7}$ - 1 = $\frac{12}{7}$ - $\frac{7}{7}$

= $\frac{5}{7}$

## Subtracting Fractions with Unlike Denominators

When we want to subtract fractions with different denominators, mainly we use LCD method and proceed as with rational expressions with like denominators.

Given below are the examples of subtracting fractions with unlike denominators.
Example 1: $\frac{3}{5}$ - $\frac{2}{3}$

Solution:
Since the fractions have different denominators, they cannot be subtracted until they have the same denominator. The lowest common multiple is 15.

= $\frac{9}{15}$ - $\frac{10}{15}$

= $\frac{-1}{15}$

Example 2: $\frac{5}{7}$ - $\frac{8}{3}$

Solution: Since the fractions have different denominators, they cannot be subtracted until they have the same denominator. The lowest common multiple is 21.

= $\frac{15}{21}$ - $\frac{56}{21}$

= - $\frac{41}{21}$

## Subtracting Fractions with Exponents

Exponents are used to express roots and it tells us how many times a number can be used in multiplication.

Example 1: $\frac{5}{y^{2}}$ - $\frac{4}{y}$

Solution:

Given: $\frac{5}{y^{2}}$ - $\frac{4}{y}$

$\frac{5}{y^{2}}$ - $\frac{4}{y}*\frac{y}{y}$

= $\frac{5}{y^{2}}$ - $\frac{4y}{y^{2}}$

= $\frac{5-4y}{y^{2}}$

Example 2: $\frac{5}{yz}$ - $\frac{1}{x^{2}y^{2}}$ - $\frac{4}{x^{2}y^{2}z^{2}}$

Solution:

Given: $\frac{5}{yz}$ - $\frac{1}{x^{2}y^{2}}$ - $\frac{4}{x^{2}y^{2}z^{2}}$

= $\frac{5}{yz}$ * $\frac{x^{2}yz}{x^{2}yz}$ - $\frac{1}{x^{2}y^{2}}$ * $\frac{z^{2}}{z^{2}}$ - $\frac{4}{x^{2}y^{2}z^{2}}$

= $\frac{5x^{2}yz}{x^{2}y^{2}z^{2}}$ - $\frac{z^{2}}{x^{2}y^{2}z^{2 }}$ - $\frac{4}{x^{2}y^{2}z^{2}}$

= $\frac{5x^{2}y^{2}z^{2}-z^{2}-4}{x^{2}y^{2}z^{2}}$

## Subtracting Complex Fractions

Complex fraction contains a fraction in its numerator or denominator. To subtract complex fractions, first we need to simplify complex fractions as single fractions.

Then divide by multiplying by the reciprocal of the denominator.

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

Solution: Consider $\frac{\frac{1}{3}}{\frac{2}{5}}$

= $\frac{1}{3}$ * $\frac{5}{2}$

= $\frac{5}{6}$

Therefore, 5 + $\frac{3}{2}$ + $\frac{\frac{1}{3}}{\frac{2}{5}}$

= 5 + $\frac{3}{2}$ + $\frac{5}{6}$

Now Consider, $\frac{3}{2}$ + $\frac{5}{6}$

= $\frac{14}{6}$

= $\frac{7}{3}$

$\rightarrow$ 5 + $\frac{7}{3}$
= $\frac{22}{3}$