# Subtracting Fractions

Subtracting fractions involves three basic steps:

• Identifying whether the denominators are the same. If they are the same, we move to step 2. If they are different, we find their common denominators.
• Subtracting the numerators
• Simplify the result if needed.

## With Like Denominators

To subtract fractions with the same denominators, we subtract only the numerators and write the difference over the common denominator.

Let us subtract the fractions ${\dfrac{4}{7}-\dfrac{2}{7}}$

Identifying the Denominators

Here, the denominators are 7

Subtracting the Numerators

${\dfrac{4-2}{7}}$

= ${\dfrac{2}{7}}$, which can not be simplified or reduced further.

Thus, the difference is ${\dfrac{2}{7}}$

Problem – Subtracting from a NEGATIVE FRACTION

E.g.1.

Subtract: ${-\dfrac{6}{11}-\dfrac{3}{11}}$

Solution:

Given, ${-\dfrac{6}{11}-\dfrac{3}{11}}$
= ${\dfrac{6-3}{11}}$
= ${\dfrac{3}{11}}$
Thus, the difference is ${\dfrac{3}{11}}$

## With Unlike Denominators

Let us subtract the unlike fractions ${\dfrac{4}{15}-\dfrac{1}{6}}$

Identifying the Denominators

Here, the denominators are 15 and 6, that are different.

Finding the LCM of the Denominators

The LCM of 15 and 6 is 30

Making the Denominators the Same

Converting ${\dfrac{4}{15}}$ and ${\dfrac{1}{6}}$ to their equivalent fractions with 30 as the denominator,

${\dfrac{4\times 2}{15\times 2}}$ = ${\dfrac{8}{30}}$

${\dfrac{1\times 5}{6\times 5}}$ = ${\dfrac{5}{30}}$

Now, we have ${\dfrac{8}{30}-\dfrac{5}{30}}$

Subtracting the Numerators

${\dfrac{8-5}{30}}$

= ${\dfrac{3}{30}}$

Simplifying

${\dfrac{3\div 3}{30\div 3}}$

= ${\dfrac{1}{10}}$

Thus, the difference is ${\dfrac{1}{10}}$

Subtract: ${\dfrac{11}{3}}$ from ${\dfrac{9}{2}}$

Solution:

The denominators are 2 and 3
The LCM of 2 and 3 is 6
Here, ${\dfrac{9}{2}-\dfrac{11}{3}}$
= ${\dfrac{9\times 3}{2\times 3}-\dfrac{11\times 2}{3\times 2}}$
= ${\dfrac{27}{6}-\dfrac{22}{6}}$
= ${\dfrac{27-22}{6}}$
= ${\dfrac{5}{6}}$
Thus, the difference is ${\dfrac{5}{6}}$

Problem – Subtracting NEGATIVE FRACTIONS

Subtract: ${\dfrac{2}{5}-\left( -\dfrac{1}{2}\right)}$

Solution:

The denominators are 5 and 2
The LCM of 5 and 2 is 10
Given, ${\dfrac{2}{5}-\left( -\dfrac{1}{2}\right)}$
= ${\dfrac{2\times 2}{5\times 2}-\dfrac{1\times 5}{2\times 5}}$
= ${\dfrac{4}{10}-\dfrac{5}{10}}$
= ${\dfrac{4-5}{10}}$
= ${\dfrac{-1}{10}}$
Thus, the difference is ${\dfrac{-1}{10}}$

## With Whole Numbers

To subtract a fraction from a whole number, we convert the whole number into the fraction form and then follow the steps of subtracting fractions with unlike denominators.

Given ${3-\dfrac{1}{2}}$

Converting the Whole Number to Fraction

${\dfrac{3}{1}-\dfrac{1}{2}}$

Identifying the Denominators

Here, the denominators are 1 and 2

Finding the LCM of the Denominators

The LCM of 1 and 2 is 2

Subtracting the Numerators

${\dfrac{3}{1}-\dfrac{1}{2}}$

= ${\dfrac{3\times 2}{1\times 2}-\dfrac{1\times 1}{2\times 1}}$

= ${\dfrac{6}{2}-\dfrac{1}{2}}$

= ${\dfrac{6-1}{2}}$

= ${\dfrac{5}{2}}$

= ${2\dfrac{1}{2}}$

Thus, the difference is ${2\dfrac{1}{2}}$

## With Mixed Numbers

To subtract mixed numbers, we first convert them into improper fractions and then subtract.

Given ${\dfrac{7}{3}-1\dfrac{1}{4}}$

Converting to Improper Fraction

= ${\dfrac{7}{3}-\dfrac{\left( 1\times 4\right) +1}{4}}$

= ${\dfrac{7}{3}-\dfrac{5}{4}}$

Identifying the Denominators

Here, the denominators are 3 and 4

Finding the LCM of the Denominators

The LCM of 3 and 4 is 12

Making the Denominators the Same and Subtracting the Numerators

${\dfrac{7\times 4}{3\times 4}-\dfrac{5\times 3}{4\times 3}}$

= ${\dfrac{28}{12}-\dfrac{15}{12}}$

= ${\dfrac{28-15}{12}}$

= ${\dfrac{13}{12}}$

= ${1\dfrac{1}{12}}$

Thus, the difference is ${1\dfrac{1}{12}}$

Subtract the following fractions: ${7\dfrac{1}{3}-2\dfrac{3}{4}}$

Solution:

Here, ${7\dfrac{1}{3}-2\dfrac{3}{4}}$
= ${\dfrac{\left( 7\times 3\right) +1}{3}-\dfrac{\left( 2\times 4\right) +3}{4}}$
= ${\dfrac{22}{3}-\dfrac{11}{4}}$
The denominators are 3 and 4
The LCM of 3 and 4 is 12
Now, we have
${\dfrac{22}{3}-\dfrac{11}{4}}$
= ${\dfrac{22\times 4}{3\times 4}-\dfrac{11\times 3}{4\times 3}}$
= ${\dfrac{88}{12}-\dfrac{33}{12}}$
= ${\dfrac{88-33}{12}}$
= ${\dfrac{55}{12}}$
= ${4\dfrac{7}{12}}$
Thus, the difference is ${4\dfrac{7}{12}}$