We will learn how to solve addition of unlike fractions.

In order to add unlike fractions, first we convert them as like fractions with same denominator in each fraction with the help of method explained earlier and then we add the fractions.

Let us consider some of the examples of adding unlike fractions:

1. Add $$\frac{1}{2}$$, $$\frac{2}{3}$$ and $$\frac{4}{7}$$.

Solution:

Let us find the LCM of the denominators 2, 3 and 7.

The LCM of 2, 3 and 7 is 42.

$$\frac{1}{2}$$ = $$\frac{1 × 21}{2 × 21}$$ = $$\frac{21}{42}$$

$$\frac{2}{3}$$ = $$\frac{2 × 14}{3 × 14}$$ = $$\frac{28}{42}$$

$$\frac{4}{7}$$ = $$\frac{4 × 6}{7 × 6}$$ = $$\frac{24}{42}$$

Therefore, we get the like fractions $$\frac{1}{2}$$, $$\frac{2}{3}$$ and $$\frac{4}{7}$$.

Now, $$\frac{21}{42}$$ + $$\frac{28}{42}$$ + $$\frac{24}{42}$$

= $$\frac{21 + 28 + 24}{42}$$

= $$\frac{73}{42}$$

2. Add $$\frac{7}{8}$$ and $$\frac{9}{10}$$

Solution:

The L.C.M. of the denominators 8 and 10 is 40.

$$\frac{7}{8}$$ = $$\frac{7 × 5}{8 × 5}$$ =  $$\frac{35}{40}$$, (because 40 ÷ 8 = 5)

$$\frac{7}{8}$$ = $$\frac{9 × 4}{10 × 4}$$ = $$\frac{36}{40}$$, (because 40 ÷ 10 = 4)

Thus, $$\frac{7}{8}$$ + $$\frac{9}{10}$$

= $$\frac{35}{40}$$ + $$\frac{36}{40}$$

= $$\frac{35 + 36}{40}$$

= $$\frac{71}{40}$$

= 1$$\frac{31}{40}$$

3. Add $$\frac{1}{6}$$ and $$\frac{5}{12}$$

Solution:

Let L.C.M. of the denominators 6 and 12 is 12.

$$\frac{1}{6}$$ = $$\frac{1 × 2}{6 × 2}$$ = $$\frac{2}{12}$$, (because 12 ÷ 6 = 2)

$$\frac{5}{12}$$ = $$\frac{5 × 1}{12 × 1}$$ = $$\frac{5}{12}$$, (because 12 ÷ 12 = 1)

Thus, $$\frac{1}{6}$$ + $$\frac{5}{12}$$

= $$\frac{2}{12}$$ + $$\frac{5}{12}$$

= $$\frac{2 + 5}{12}$$

= $$\frac{7}{12}$$

4.
Add $$\frac{2}{3}$$, $$\frac{1}{15}$$ and $$\frac{5}{6}$$

Solution:

The L.C.M. of the denominators 3, 15 and 6 is 30.

$$\frac{2}{3}$$ = $$\frac{2 × 10}{3 × 10}$$ = $$\frac{20}{30}$$, (because 30 ÷ 3 = 10)

$$\frac{1}{15}$$ = $$\frac{1 × 2}{15 × 2}$$ = $$\frac{2}{30}$$, (because 30 ÷ 15 = 2)

$$\frac{5}{6}$$  = $$\frac{5 × 5}{6 × 5}$$ = $$\frac{25}{30}$$, (because 30 ÷ 6 = 5)

Thus, $$\frac{2}{3}$$ + $$\frac{1}{15}$$ + $$\frac{5}{6}$$

= $$\frac{20}{30}$$ + $$\frac{2}{30}$$ + $$\frac{25}{30}$$

= $$\frac{20 + 2 + 25}{30}$$

= $$\frac{47}{30}$$

= 1$$\frac{17}{30}$$

More examples on Addition of Unlike Fractions (Fractions having Different Denominators)

5. Add $$\frac{1}{6}$$ + $$\frac{3}{4}$$

 Solution:First Method:Step I: Find the L.C.M. of the denominators 6 and 4.L.C.M. of 6 and 4 = 2 × 3 × 2 =12

Step II:Write the equivalent fractions of $$\frac{1}{6}$$ and $$\frac{3}{4}$$ with denominator 12.

$$\frac{1 × 2}{6 × 2}$$ =  $$\frac{2}{12}$$

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

$$\frac{1}{6}$$ + $$\frac{3}{4}$$

=  $$\frac{2}{12}$$ + $$\frac{9}{12}$$

= $$\frac{2 + 9}{12}$$

= $$\frac{11}{12}$$

Add $$\frac{1}{6}$$ + $$\frac{3}{4}$$

Second Method:

 $$\frac{1}{6}$$ + $$\frac{3}{4}$$L.C.M. of 6 and 4 is 12=  $$\frac{(12 ÷ 6) × 1 + (12 ÷ 4) × 3}{12}$$= $$\frac{(2 × 1) + (3 × 3}{12}$$= $$\frac{2 + 9}{12}$$= $$\frac{11}{12}$$ Steps:Divide 12 by I denominator. Multiply the quotient with I numerator.Divide 12 by II denominator. Multiply the quotient with II numerator.
 6. Add $$\frac{3}{8}$$ + $$\frac{2}{4}$$ + $$\frac{6}{16}$$Solution:L.C.M. of 8, 4, 16 = 2 × 2 × 2 × 2 = 16$$\frac{3}{8}$$ + $$\frac{2}{4}$$ + $$\frac{6}{16}$$= $$\frac{(16 ÷ 8) × 3 + (16 ÷ 4) × 2 + (16 ÷ 16) × 6}{16}$$= $$\frac{(2 × 3) + (4 × 2) + (1 × 6)}{16}$$= $$\frac{6 + 8 + 6}{16}$$= $$\frac{20}{16}$$= $$\frac{5}{4}$$= 1$$\frac{1}{4}$$

7. Add 2$$\frac{2}{6}$$ + 5$$\frac{1}{3}$$ + 1$$\frac{4}{5}$$

Solution:

First Method:

Separate the whole numbers and proper fractions.

2$$\frac{2}{6}$$ + 5$$\frac{1}{3}$$ + 1$$\frac{4}{5}$$ = (2 + 5 + 1) + $$\frac{2}{6}$$ + $$\frac{1}{3}$$ + $$\frac{4}{5}$$

= 8 + $$\frac{2}{6}$$ + $$\frac{1}{3}$$ + $$\frac{4}{5}$$

L.C.M. of 6, 3 and 5 is 30.

= 8 + $$\frac{(30 ÷ 6) × 2 + (30 ÷ 3) × 1 + (30 ÷ 5) × 4}{30}$$

= 8 + $$\frac{(5 × 2) + (10 × 1) + (6 × 4)}{30}$$

= 8 + $$\frac{10 + 10 + 24}{30}$$

= 8 + $$\frac{44}{30}$$

= 8 + $$\frac{22}{15}$$

= 8 + 1$$\frac{7}{15}$$

Second Method:

Add 2$$\frac{2}{6}$$ + 5$$\frac{1}{3}$$ + 1$$\frac{4}{5}$$

Convert the mixed number into improper fractions and find the sum

2$$\frac{2}{6}$$ = $$\frac{(2 × 6) + 2}{6}$$ = $$\frac{14}{6}$$

5$$\frac{1}{3}$$ = $$\frac{(5 × 3) + 1}{3}$$ = $$\frac{16}{3}$$

1$$\frac{4}{5}$$ = $$\frac{(1 × 5) + 4}{5}$$ = $$\frac{9}{5}$$

Therefore, 2$$\frac{2}{6}$$ + 5$$\frac{1}{3}$$ + 1$$\frac{4}{5}$$ = $$\frac{14}{6}$$ + $$\frac{16}{3}$$ + $$\frac{9}{5}$$

= $$\frac{14 × 5}{6 × 5}$$ + $$\frac{16 × 10}{3 × 10}$$ + $$\frac{9 × 6}{5 × 6}$$

= $$\frac{70}{30}$$ + $$\frac{160}{30}$$ + $$\frac{54}{30}$$

= $$\frac{70 + 160 + 54}{30}$$

= $$\frac{284}{30}$$

= $$\frac{142}{15}$$

= 9$$\frac{7}{15}$$

8. Add $$\frac{2}{6}$$, 4 and $$\frac{7}{12}$$

 Solution:4 = $$\frac{4}{1}$$$$\frac{2}{6}$$ + 4 + $$\frac{7}{12}$$= $$\frac{2}{6}$$ + $$\frac{4}{1}$$ + $$\frac{7}{12}$$           L.C.M. of 6, 1, 12 is 12= $$\frac{(12 ÷ 6) × 2 + (12 ÷ 1) × 4 + (12 ÷ 12) × 7}{12}$$= $$\frac{(2 × 2) + (12 × 4) + (1 × 7)}{12}$$= $$\frac{4 + 48 + 7}{12}$$= $$\frac{59}{12}$$= 4$$\frac{11}{12}$$

To add unlike fractions, we first convert them into like fractions. In order to make a common denominator we find the LCM of all different denominators of the given fractions and then make them equivalent fractions with a common denominator.

Word Problems on Addition of Unlike Fractions:

1. On Monday Michael read $$\frac{5}{16}$$ of the book. On Wednesday he reads $$\frac{4}{8}$$ of the book. What fraction of the book has Michael read?

Solution:

On Monday Michael read $$\frac{5}{16}$$ of the book.

On Wednesday he reads $$\frac{4}{8}$$ of the book.

$$\frac{5}{16}$$ + $$\frac{4}{8}$$

Let us find the LCM of the denominators 16 and 8.

The LCM of 16 and 8 is 16.

$$\frac{5}{16}$$ = $$\frac{5 × 1}{16 × 1}$$ = $$\frac{5}{16}$$

$$\frac{4}{8}$$ = $$\frac{4 × 2}{8 × 2}$$ = $$\frac{8}{16}$$

Therefore, we get the like fractions $$\frac{5}{16}$$ and $$\frac{8}{16}$$.

Now, $$\frac{5}{16}$$ + $$\frac{8}{16}$$

= $$\frac{5 + 8}{16}$$

= $$\frac{13}{16}$$

Therefore, Michael read in two days $$\frac{13}{16}$$ of the book.

2. Sarah ate $$\frac{1}{3}$$ part of the pizza and her sister ate $$\frac{1}{2}$$ of the pizza. What fraction of the pizza was eaten by both sisters?

Solution:

Sarah ate $$\frac{1}{3}$$ part of the pizza.

Her sister ate $$\frac{1}{2}$$ of the pizza.

$$\frac{1}{3}$$ + $$\frac{1}{2}$$

Let us find the LCM of the denominators 3 and 2.

The LCM of 3 and 2 is 6.

$$\frac{1}{3}$$ = $$\frac{1 × 2}{3 × 2}$$ = $$\frac{2}{6}$$

$$\frac{1}{2}$$ = $$\frac{1 × 3}{2 × 3}$$ = $$\frac{3}{6}$$

Therefore, we get the like fractions $$\frac{2}{6}$$ and $$\frac{3}{6}$$.

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

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

= $$\frac{5}{6}$$

Therefore, $$\frac{5}{6}$$ of the pizza was eaten by both sisters.

3. Catherine is preparing for her final exam. She study $$\frac{9}{22}$$ hours on Wednesday and $$\frac{5}{11}$$ hours on Sunday. How many hours she studied in two days?

Solution:

Catherine study $$\frac{9}{22}$$ hours on Wednesday.

Again, she study $$\frac{5}{11}$$ hours on Sunday.

$$\frac{9}{22}$$ + $$\frac{5}{11}$$

Let us find the LCM of the denominators 22 and 11.

The LCM of 22 and 11 is 22.

$$\frac{9}{22}$$ = $$\frac{9 × 1}{22 × 1}$$ = $$\frac{9}{22}$$

$$\frac{5}{11}$$ = $$\frac{5 × 2}{11 × 2}$$ = $$\frac{10}{22}$$

Therefore, we get the like fractions $$\frac{9}{22}$$ and $$\frac{10}{22}$$.

Now, $$\frac{9}{22}$$ + $$\frac{10}{22}$$

= $$\frac{9 + 10}{22}$$

= $$\frac{19}{22}$$

Therefore, Catherine studied a total $$\frac{9}{22}$$ hours in two days.

1. Add the following Unlike Fractions:

(i) $$\frac{3}{4}$$ + $$\frac{5}{6}$$

(ii) $$\frac{1}{7}$$ + $$\frac{2}{3}$$ + $$\frac{6}{7}$$

(iii) $$\frac{7}{8}$$ + $$\frac{5}{6}$$ + $$\frac{4}{10}$$

(iv) $$\frac{3}{7}$$ + $$\frac{2}{5}$$ + $$\frac{6}{11}$$

(v) 3$$\frac{5}{8}$$ + 4$$\frac{1}{6}$$ + 4$$\frac{7}{12}$$

1. (i) 1$$\frac{7}{12}$$

(ii) 1$$\frac{2}{3}$$

(iii) 2$$\frac{13}{120}$$

(iv) 1$$\frac{144}{385}$$

(v) 12$$\frac{3}{8}$$

## You might like these

• ### Types of Fractions |Proper Fraction |Improper Fraction |Mixed Fraction

The three types of fractions are : Proper fraction, Improper fraction, Mixed fraction, Proper fraction: Fractions whose numerators are less than the denominators are called proper fractions. (Numerator < denominator). Two parts are shaded in the above diagram.

• ### Word Problems on Fraction | Math Fraction Word Problems |Fraction Math

In word problems on fraction we will solve different types of problems on multiplication of fractional numbers and division of fractional numbers.

• ### Equivalent Fractions | Fractions |Reduced to the Lowest Term |Examples

The fractions having the same value are called equivalent fractions. Their numerator and denominator can be different but, they represent the same part of a whole. We can see the shade portion with respect to the whole shape in the figures from (i) to (viii) In; (i) Shaded

• ### Subtraction of Fractions having the Same Denominator | Like Fractions

To find the difference between like fractions we subtract the smaller numerator from the greater numerator. In subtraction of fractions having the same denominator, we just need to subtract the numerators of the fractions.

• ### Comparison of Like Fractions | Comparing Fractions | Like Fractions

Any two like fractions can be compared by comparing their numerators. The fraction with larger numerator is greater than the fraction with smaller numerator, for example $$\frac{7}{13}$$ > $$\frac{2}{13}$$ because 7 > 2. In comparison of like fractions here are some

• ### Comparison of Fractions having the same Numerator|Ordering of Fraction

In comparison of fractions having the same numerator the following rectangular figures having the same lengths are divided in different parts to show different denominators. 3/10 < 3/5 < 3/4 or 3/4 > 3/5 > 3/10 In the fractions having the same numerator, that fraction is

• ### Like and Unlike Fractions | Like Fractions |Unlike Fractions |Examples

Like and unlike fractions are the two groups of fractions: (i) 1/5, 3/5, 2/5, 4/5, 6/5 (ii) 3/4, 5/6, 1/3, 4/7, 9/9 In group (i) the denominator of each fraction is 5, i.e., the denominators of the fractions are equal. The fractions with the same denominators are called

• ### Fraction of a Whole Numbers | Fractional Number |Examples with Picture

Fraction of a whole numbers are explained here with 4 following examples. There are three shapes: (a) circle-shape (b) rectangle-shape and (c) square-shape. Each one is divided into 4 equal parts. One part is shaded, i.e., one-fourth of the shape is shaded and three

• ### Worksheet on Equivalent Fractions | Questions on Equivalent Fractions

In worksheet on equivalent fractions, all grade students can practice the questions on equivalent fractions. This exercise sheet on equivalent fractions can be practiced by the students to get more ideas to change the fractions into equivalent fractions.

• ### Worksheet on Addition of Like Fractions | Addition of Fractions

In worksheet on addition of fractions having the same denominator, all grade students can practice the questions on adding fractions. This exercise sheet on fractions can be practiced by the students to get more ideas how to add fractions with the same denominators.

Related Concept