Addition of Unlike Fractions

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

LCM of 6 and 4

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


\(\frac{1 × 2}{6 × 2}\) =  \(\frac{2}{12}\)

\(\frac{3 × 3}{4 × 3}\) =  \(\frac{9}{12}\)

Step III:Add the equivalent fractions

\(\frac{1}{6}\) + \(\frac{3}{4}\)

=  \(\frac{2}{12}\) + \(\frac{9}{12}\)

= \(\frac{2 + 9}{12}\)

= \(\frac{11}{12}\)

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\(\frac{1}{4}\)

L.C.M. of 8, 4 and 16

Addition of Mixed Numbers

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\(\frac{11}{12}\)

L.C.M. of 6, 1 and 12
Addition of Unlike Fractions

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.

Now add the two fractions

\(\frac{5}{16}\) + \(\frac{4}{8}\)

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.

Now add the two fractions

\(\frac{1}{3}\) + \(\frac{1}{2}\)

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.

Now add the two fractions

\(\frac{9}{22}\) + \(\frac{5}{11}\)

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.


Questions and Answers Addition of Unlike Fractions:

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\(\frac{7}{12}\)


Answer:

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

(ii) 1\(\frac{2}{3}\) 

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

(iv) 1\(\frac{144}{385}\) 

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

Related Concept





4th Grade Math Activities

From Addition of Unlike Fractions to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Multiplication of a Number by a 3-Digit Number |3-Digit Multiplication

    Mar 28, 24 06:33 PM

    Multiplying by 3-Digit Number
    In multiplication of a number by a 3-digit number are explained here step by step. Consider the following examples on multiplication of a number by a 3-digit number: 1. Find the product of 36 × 137

    Read More

  2. Multiply a Number by a 2-Digit Number | Multiplying 2-Digit by 2-Digit

    Mar 27, 24 05:21 PM

    Multiply 2-Digit Numbers by a 2-Digit Numbers
    How to multiply a number by a 2-digit number? We shall revise here to multiply 2-digit and 3-digit numbers by a 2-digit number (multiplier) as well as learn another procedure for the multiplication of…

    Read More

  3. Multiplication by 1-digit Number | Multiplying 1-Digit by 4-Digit

    Mar 26, 24 04:14 PM

    Multiplication by 1-digit Number
    How to Multiply by a 1-Digit Number We will learn how to multiply any number by a one-digit number. Multiply 2154 and 4. Solution: Step I: Arrange the numbers vertically. Step II: First multiply the d…

    Read More

  4. Multiplying 3-Digit Number by 1-Digit Number | Three-Digit Multiplicat

    Mar 25, 24 05:36 PM

    Multiplying 3-Digit Number by 1-Digit Number
    Here we will learn multiplying 3-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. 1. Multiply 201 by 3 Step I: Arrange the numb…

    Read More

  5. Multiplying 2-Digit Number by 1-Digit Number | Multiply Two-Digit Numb

    Mar 25, 24 04:18 PM

    Multiplying 2-Digit Number by 1-Digit Number
    Here we will learn multiplying 2-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. Examples of multiplying 2-digit number by

    Read More