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/4
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/4
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/4
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
How Do You Add Two Rational Expressions with Different Denominators?
Note:
Adding rational expressions together? Don't have common denominators? No problem! Find the least common denominator (LCD) and change each rational expression into an equivalent expression with that LCD. Once you have common denominators, you're ready to add and simplify! Watch it all in this tutorial!
Keywords:
- problem
- rational expression
- fraction
- algebraic fraction
- polynomial
- add
- sum
- add fractions
- different denominator
- unequal denominator
- simplify
- simplest form
- add polynomials
- lcd
- lowest common denominator
- numerator
- denominator
Background Tutorials
Fraction Basics
What's a Numerator and What's A Denominator?
Numerators and denominators are the key ingredients that make fractions, so if you want to work with fractions, you have to know what numerators and denominators are. Lucky for you, this tutorial will teach you some great tricks for remembering what numerators and denominators are all about.
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Simplifying Expressions
What are Like Terms?
Combining like terms together is a key part of simplifying mathematical expressions, so check out this tutorial to see how you can easily pick out like terms from an expression
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Product of a Sum and a Difference
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Factors
What's a Factor?
Factors are a fundamental part of algebra, so it would be a great idea to know all about them. This tutorial can help! Take a look!
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Difference of Squares and Cubes
Simplifying Rational Expressions
What's a Rational Expression?
Got a fraction with a polynomial in the numerator and denominator? You have a rational expression! Learn about rational expressions in this tutorial.
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Adding and Subtracting with Unlike Denominators
What's the Least Common Denominator?
When you're working with fractions, you may need to find the least common denominator (LCD) in order to get the fractions to have a common denominator so that you can add or subtract them. The LCD is the smallest multiple that the denominators have in common. Learn about the LCD in this tutorial!
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Further Exploration
Adding and Subtracting with Like Denominators
Adding and Subtracting with Unlike Denominators
How Do You Subtract Two Rational Expressions with Different Denominators?
Subtracting rational expressions? Don't have common denominators? No problem! Find the least common denominator (LCD) and change each rational expression into an equivalent expression with that LCD. Once you have common denominators, you're ready to subtract and simplify! Watch it all in this tutorial!
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Solving Rational Equations
How Do You Solve a Word Problem with a Rational Equation?
This tutorial provides a great real world application of math! This tutorial shows you how to take the information given in a word problem and turn it into a rational equation. Then, you'll see how to solve that equation and get your answer!
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