SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS

We can use one of the following methods to subtract two fractions with unlike denominators. 

1. Cross-Multiplication method

2. LCM Method.

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 subtract of two fractions given below.

1/3 - 1/8

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

For 3 and 8, there is no common divisor other than 1.

So 3 and 8 are co-prime.

Here we have to apply cross-multiplication method to find the subtraction of two fractions 1/3 and 1/8 as shown below.

LCM Method (Least Common Multiple 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 subtract of two fractions given below.

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

2 and 4

So 12 and 20 are not co-prime.

In the next step, we have to find the LCM (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, the LCM of 12 and 20 is

=  2² x 3 x 5

=  4 x 3 x 5

=  60

Now, make the denominator of each fraction as 60 using multiplication and then subtract them as shown below. 

Practice Problems

Problem 1 : 

Find the value of : 

3/5 - 2/3

Solution : 

In the above two fractions, denominators are 5 and 3.

For 5 and 3, there is no common divisor other than 1.

So 5 and 3 are co-prime.

Here we have to apply cross-multiplication method to find the subtraction of two fractions 1/8 and 1/3 as shown below.

3/5 - 2/3  =  (9 - 10) / 15

3/5 - 2/3  =  - 1/15

Problem 2 : 

Find the value of : 

7/12 - 5/18

Solution : 

In the above two fractions, denominators are 12 and 18.

For 12 and 18, if there is at least one common divisor other than 1, then 12 and 18 are not co-prime.

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

2, 3, and 6

So 12 and 20 are not co-prime.

Here, we have to apply LCM method to find the subtraction of the two fractions. 

Find the LCM of 12 and 18.

LCM of (12, 18)  =  36

Now, make the denominator of each fraction as 36 using multiplication and subtract the fractions with like denominators. 

Then, we have  

7/12 - 5/18  =  21/36 - 10/36

7/12 - 5/18  =  (21 - 10) / 36

7/12 - 5/18  =  11/36

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