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How to Add and Subtract Rational Expressions with Unlike Denominators and Variables?

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  • Last Updated : 07 Feb, 2022

Numbers represented in the form of m/n are called Fractions. Here, ‘m’ or the upper part of the fraction is the numerator, and ‘n’ or the lower part of the fraction is called the denominator. Fractions with numerator lesser than Denominator, are Proper Fraction. Fractions with numerator greater than Denominator, fall in the category of Improper Fraction. Improper fractions are often denoted by Mixed Fractions, where there is a whole part and fractional part. 

Rational Numbers

Fractions in the form of m/n where n!=0 fall in the category of Rational Numbers. So, any fractions 4/5, 2/4, 1/8 fall in the category of rational numbers.  Rational numbers can be positive or negative. The only pre-requisite for any fractional number to be called rational number is that the denominator of the fraction must not be zero.

For same Denominators, Addition and Subtraction of Rational Expressions is relatively easier as the mathematical operation are performed straight  on  the numerators. When denominators are different or the rational numbers have different variables then simply addition of terms doesn’t work. For different denominators, to perform addition and subtraction we need to find LCM of the given terms first and then perform Mathematical operations. For different variables, we can’t perform addition and subtraction on them, only like variable can combine to perform Mathematical operations.

Add and Subtract Rational Expressions with Unlike Denominators and Variables

Below steps must be followed when we add or subtract rational expressions containing variables with different denominators:

Step 1: Since the denominators are different, mathematical operations such as addition/ subtraction can’t be performed directly. For this, we need to first find the LCM of different fractional terms and equalise the denominators.

Step 2: Calculations can be performed easily on the numerator as the denominators are equal. Perform addition and subtraction operations on the numerator part of rational numbers as desired.

Step 3: Simplify the result and reduce the expression to the lowest possible form. 

Sample Problems

Question 1: Add 11b/6 and 19b/6.

Solution: 

Since the denominator is same, we will directly add the numerators.

= 11b/6 + 19b/6

= 30b/6

= 5b

Question 2: Subtract 11b/6 from 19b/6.

Solution: 

Since the denominator is same, we will directly subtract the numerators.

= 19b/6 – 11b/6

= 8b/6

= 4b/3

Question 3: Add 10s/4 and 10s/3.

Solution: 

Since, the denominator is not the same, we will  take the LCM of denominators.

= 10s/4 + 10s/3

The LCM of 4 and 3 is 12.

So, (10s × 3)/(4 × 3) + (10s × 4)/(3 × 4)

= 30s/12 + 40s/12

= 70s/12

Question 4: Subtract 10s/4 from 10s/3

Solution: 

Since, the denominator is not the same, we will  take the LCM of denominators.

= 10s/3 – 10s/4pm

The LCM of 4 and 3 is 12.

So, (10s × 4)/(3 × 4) – (10s × 3)/(4 × 3)

= 40s/12 – 30s/12

= 10s/12

= 5s/6

Question 5: Add 3z/4 + 10y/3 + 4z/3.

Solution: 

Since, the denominator is not the same, take the LCM of denominators.

= 3z/4 +10y/3 +4z/3

The LCM of 4 and 3 is 12.

So, (3z × 3)/(4 × 3) + (10y × 4)/(3 × 4) + (4z × 4)/(3 × 4)

= 9z/12 + 40y/12 + 16z/12

Combine terms with Like Variables i.e add terms with Like Variables

= 25z/12 + 40y/12

Question 6: Subtract 7z/4 from 10z/3.

Solution: 

Since the denominator is not the same, take the LCM of denominators.

= 10z/3 – 7z/4pm

The LCM of 4 and 3 is 12.

So, (10z × 4)/(3 × 4) – (7z × 3)/(4 × 3)

= 40z/12 – 21z/12

= 19z/12

Question 7: Subtract 10i/4 from 10i/3

Solution:

Since, the denominator is not the same, take the LCM of denominators.

= 10i/3 – 10i/4pm

The LCM of 4 and 3 is 12.

So, (10i × 4)/(3 × 4) – (10i × 3)/(4 × 3)

= 40i/12 – 30i/12

= 10i/12

= 5i/6

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