# ADDING TWO FRACTIONS WITH DIFFERENT DENOMINATORS

"Adding two fractions with different denominators"  is sometimes difficult job for some students who study math in school level.

Actually it is not a difficult one, once we have understood the concept.

Here, we explain two methods to add two fraction with different denominators.

Here students may have a question.

That is, "When do we have to use the appropriate method to add two fractions with different denominators? "

Answer for the above question is given with detailed explanation.

## Cross-multiplication method

If the denominators of the fractions are co-prime or relatively prime, we have to apply this method.

Fro example, let us consider the two fractions 1/8,  1/3.

In the above two fractions, denominators are 8 and 3.

For 8 and 3, there is no common divisor other than 1. So 8 and 3 are co-prime.

Here we have to apply cross-multiplication method to add the two fractions 1/8 and 1/3 as given below.

## L.C.M method

If the denominators of the fractions are not co-prime (there is a common divisor other than 1), we have to apply this method.

Fro example, let us consider the two fractions 5/12,  1/20.

In the above two fractions, denominators are 12 and 20.

For 12 and 20, if there is at least one common divisor other than 1, then 12 and 20 are not co-prime.

For 12 & 20, we have the following common divisors other than 1.

2 & 4

So 12 and 20 are not co-prime.

In the next step, we have to find the L.C.M (Least common multiple) of 12 and 20.

12 =  2² x 3

20 = 2² x 5

When we decompose 12 and 20 in to prime numbers, we find 2, 3 and 5 as prime factors for 12 and 20.

To get L.C.M of 12 and 20, we have to take 2, 3 and 5 with maximum powers found above.

So, L.C.M of 12 and 20 = 2² x 3 x 5

= 4 x 3 x 5

= 60

Now we have to make the denominators of both the fractions to be 60 and add the two fractions 5/12 and 1/20 as given below.

## Adding two fractions with different denominators - Practice problems

Problem 1 :

Add :  1 / 12  +  3 / 18

Solution :

The given two fractions are unlike fractions. Because, they have different denominators.

For 12 and 18, we have the following common divisors other than 1.

2, 3 and 6

So 12 and 18 are not co-prime.

In the next step, we have to find the L.C.M (Least common multiple) of 12 and 18.

L.C.M of (12 and 18)  =  36

Now we have to make the denominators of both the fractions to be 36.

To make the denominator to be 36, we have to multiply the numerator and denominator of the first fraction by 3 and and for the second fraction by 2.

Then, we have

1 / 12  +  3 / 18  =  3 / 36  +  2 / 36

=  (3 + 2) / 36

=  5 / 36

Hence, the sum of the two fractions is 5/36.

Let us look at the next problem on "Adding two fractions with different denominators"

Problem 2 :

Add :  3 / 20  +  7 / 30

Solution :

The given two fractions are unlike fractions. Because, they have different denominators.

For 20 and 30, we have the following common divisors other than 1.

2, 5 and 10

So 20 and 30 are not co-prime.

In the next step, we have to find the L.C.M (Least common multiple) of 20 and 30.

L.C.M of (20 and 30)  =  60

Now we have to make the denominators of both the fractions to be 60.

To make the denominator to be 60, we have to multiply the numerator and denominator of the first fraction by 3 and and for the second fraction by 2.

Then, we have

3 / 20  +  7 / 30  =  9 / 60  +  14 / 60

=  (9 + 14) / 60

=  23 / 60

Hence, the sum of the two fractions is 23/60.

Let us look at the next problem on "Adding two fractions with different denominators"

Problem 3 :

Add :  3 / 7  +  2 / 9

Solution :

The given two fractions are unlike fractions. Because, they have different denominators.

For 7 and 9, there is no common divisor other than 1.

So 7 and 9 are co-orime

Here, we have to apply cross multiplication method to add the two fractions.

To add the two fractions, we have to do the following three steps.

Step 1 :

Multiply the numerator of the first fraction by denominator of the second fraction.

Step 2 :

Multiply the numerator of the second fraction by denominator of the first fraction.

Step 3 :

Multiply the denominators of the two fractions.

When we do the above three steps, we will have

3 / 7  +  2 / 9  =  (27 + 14) / 63

=  41 / 63

Hence, the sum of the two fractions is 41 / 63.

Let us look at the next problem on "Adding two fractions with different denominators"

Problem 4 :

Add :  1 / 2  +  7 / 3  +  4 / 5

Solution :

The denominators of all the fractions are not same.

In this problem, we have more than two fractions.

If we have more than two fractions, we can use only L.C.M method.

In the next step, we have to find the L.C.M (Least common multiple) of 2, 3 and 5.

L.C.M of (2, 3 and 5)  =  30

Now we have to make the denominators of all the three fractions to be 30.

To make the denominator to be 30, we have to multiply the numerator and denominator of the first fraction by 15, for the second fraction by 10 and for the third fraction by 6.

Then, we have

1 / 2  +  7 / 3  +  4 / 5  =  15/30 + 70/30 + 24/30

=  (15+70+24) / 30

=  (109) / 30

=  109 / 30

Hence, the sum of the two fractions is 109/30.

Let us look at the next problem on "Adding two fractions with different denominators"

Problem 5 :

Add :  1 / 9  +  7 / 10   +  4 / 9

Solution :

The denominators of all the fractions are not same.

In this problem, we have more than two fractions.

If we have more than two fractions, we can use only L.C.M method.

In the next step, we have to find the L.C.M (Least common multiple) of 9, 10 and 9.

L.C.M of (9, 10 and 9)  =  90

Now we have to make the denominators of all the three fractions to be 90.

To make the denominator to be 90, we have to multiply the numerator and denominator of the first fraction by 10, for the second fraction by 9 and for the third fraction by 10.

Then, we have

1 / 9  +  7 / 10   +  4 / 9  =  10/90 + 63/90 + 40/90

=  (10+63+40) / 90

=  (113) / 90

Hence, the sum of the two fractions is 113/90

## Adding two fractions with different denominators - Worksheets

Worksheet - 1

Worksheet - 2

Worksheet - 3

Worksheet - 4

Worksheet - 5

After having gone through the stuff and examples, we hope that the students would have understood "Adding two fractions with different denominators".