Unlike Fractions with Examples and FAQ

What do you mean by Unlike Fractions?

Let us revise a bit about fractions before we know more about unlike fractions

What is a Fraction?

A fraction is a number representation by which a part of a whole object or a group of objects is described. The numerical representation of fraction is as follows:

\frac{a}{b} where ‘a’ is termed as the numerator and ‘b’ as the denominator.

Some examples of fractions are \frac{2}{5}, \frac{2}{4}, \frac{1}{7}, \frac{4}{9}, \frac{3}{8}, \frac{7}{12} .

The two types of fractions are Like and Unlike Fractions.

Like Fractions

A group of Fractions is like or similar fractions if all fractions have the same denominators. For Example, \frac{2}{5}, \frac{3}{5}, \frac{1}{5}, \frac{4}{5} is a group of like fractions.

Unlike Fractions

A group of fractions is Unlike fractions or dissimilar fractions if fractions have different denominators. For example,

\frac{2}{5}, \frac{2}{4}, \frac{1}{7}, \frac{4}{9}, \frac{3}{8}, \frac{7}{12} .

As the denominators are different so addition and subtraction cannot be done simply. First, the fractions have to be changed to like fractions and then the required operation can be performed.

How to Convert Unlike Fractions to Like Fractions?

Unlike fractions can be converted to like fractions. First, the LCM of the denominators is found out and the numerator and denominator of each fraction are multiplied by a number which changes the denominator value, such that the fractions become alike. 

For example, unlike fractions \frac{2}{3} \text { and } \frac{3}{5} can be converted to like fractions.

First, the LCM of the denominators 3, 5 needs to be calculated. The LCM is 15.

Now converting the fractions to like fractions.

\frac{2}{3} \times \frac{5}{5}=\frac{10}{15}

\frac{3}{5} \times \frac{3}{3}=\frac{9}{15}

Now the fractions are \frac{10}{15} \text { and } \frac{9}{15} are like fractions.

Different Operations on unlike fractions:

• Addition and Subtraction

The addition and subtraction of two, unlike fractions, can be performed only when the denominators are made equal this can be done in two ways. 

(1) Cross Multiplication 

(2) LCM

Let us understand both the methods with the help of examples:

Example: Add \frac{5}{6} \text { and } \frac{3}{7} \text {. }

Solution:

By Cross Multiplication

The numerator of the first fraction \frac{5}{6} i.e., 5 is multiplied by the denominator of the second fraction \frac{3}{7} i.e., 7.

Similarly, the numerator of the second fraction is multiplied by the denominator of the first fraction, i.e., 3 is multiplied by 6.

Now the denominator of both the fractions is multiplied.

\therefore \frac{5}{6}+\frac{3}{7}=\frac{(5 \times 7)+(3 \times 6)}{7 \times 6}

=\frac{35+18}{42}

=\frac{53}{42}

By LCM method

The LCM of the denominators, i.e., 6 and 7, is 42.

Now multiply the first fraction \frac{5}{6} with the fraction \frac{7}{7} and the second fraction \frac{3}{7} with the fraction \frac{6}{6} and then add the two.

\therefore \frac{5}{6}+\frac{3}{7}=\left(\frac{5}{6} \times \frac{7}{7}\right)+\left(\frac{3}{7} \times \frac{6}{6}\right)

=\frac{35}{42}+\frac{18}{42}

=\frac{35+18}{42}=\frac{53}{42}

Hence using both the methods the addition can be done of fractions which are unlike.

Example: Subtract \frac{3}{6} from \frac{3}{4}.

Solution:

By Cross Multiplication

The numerator of the first fraction \frac{3}{6} i.e., 3 is multiplied by the denominator of the second fraction \frac{3}{4} i.e., 4.

Similarly, the numerator of the second fraction is multiplied by the denominator of the first fraction, i.e., 3 is multiplied by 6.

Now the denominator of both the fractions is multiplied.

\frac{3}{4}-\frac{3}{6}=\frac{(3 \times 6)-(3 \times 4)}{4 \times 6}

=\frac{18-12}{24}

=\frac{6}{24}

The fraction can be simplified to \frac{1}{4} \text {. }       [since, 24 ÷ 6 = 4]

By LCM method

The LCM of the denominators, i.e., 4 and 6, is 12.

Now multiply the first fraction \frac{3}{6} with the fraction \frac{2}{2} and the second fraction  \frac{3}{4} with the fraction \frac{3}{3}.

\therefore \frac{3}{4}-\frac{3}{6}=\frac{3}{4} \times \frac{3}{3}-\left(\frac{3}{6} \times \frac{2}{2}\right)

=\frac{9}{12}-\frac{6}{12}

=\frac{3}{12}

which can simply further as 12 \div 3=4

\therefore \frac{3}{4}-\frac{3}{6}=\frac{1}{4}

• Multiplication of Unlike Fractions

To multiply unlike fractions, multiply the numerators then multiply the denominators separately, and then simplify the result.

Example: Multiply \frac{2}{5}\text{ and }\frac{1}{3}.

Solution:

\frac{2}{5} \times \frac{1}{3}=\frac{2 \times 1}{5 \times 3}=\frac{2}{15}

• Division of Unlike Fractions

To divide unlike fractions, replace the division sign with multiplication and take the reciprocal of the second fraction.

Example: \frac{2}{3} \div \frac{4}{5}.

Solution: 

\frac{2}{3} \div \frac{4}{5}=\frac{2}{3} \times \frac{5}{4}

=\frac{2 \times 5}{3 \times 4}

=\frac{10}{12}

Examples

Example 1: Which group of fractions are like and which are unlike among the given options:

\text { A. } \frac{1}{2}, \frac{2}{2}, \frac{4}{2}, \frac{7}{2}.

\text { D. } \frac{1}{2}, \frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{9}{10}

Solution:

In options (A) and (B) the denominator is the same for the full group of fractions hence they are like fractions.

In options (C) and (D) the denominators are different for each fraction and hence they are unlike fractions.

Example 2: Find the sum of the unlike fractions \frac{4}{7} and \frac{3}{2}.

Solution:

The given fractions are unlike as the denominators are 7 and 2.

The LCM of the denominators 7 and 2 is 14.

Multiply the numerator  and denominator of  \frac{4}{7} \text { by } 2

\frac{4}{7} \times \frac{2}{2}=\frac{8}{14}

multiply the numerator  and denominator of  \frac{3}{2} \text { by } 7

\frac{3}{2} \times \frac{7}{7}=\frac{21}{14}

\therefore \frac{4}{7}+\frac{3}{2}=\frac{8}{14}+\frac{21}{14}=\frac{8+21}{14}=\frac{29}{14}

Therefore, the sum of the two fractions is \frac{29}{14}.