Algebraic Fractions Simplifying Worksheet and Examples

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Algebraic Fractions

  • Algebraic fractions are expressions that contain one or more variables in the numerator or denominator, or both.
  • Algebraic fractions are similar to numerical fractions, but instead of numbers, they have algebraic expressions.

In this article, we will discuss: 

  1. What is an Algebraic Fraction?
  2. Steps for Adding, Subtracting, Multiplying and Dividing Algebraic Fractions

Here is one more link to practice a few extra questions: Maths Genie Algebraic Fractions Questions

What is an Algebraic Fraction?

  • An algebraic fraction is a fraction whose numerator or denominator or both are algebraic expressions.

Algebraic Fractions Denominator

  • For example, (3x2 + 2x)/(x + 1) is an algebraic fraction.

Steps to Simplify Algebraic Fractions

  • To simplify an algebraic fraction, follow the following steps:

Step #1: Factorise the numerator and denominator

Step #2: Cancel out any common factors in the numerator and denominator

Step #3: Write the simplified fraction

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Solved Example: 

Question 1: Simplify (3x2 + 9x)/ (x2 + 4x)

Solution: 

  • Step #1: Factorise the numerator and denominator

3x (x + 3)/x(x + 4)

  • Step #2: Cancel out the common factor ‘x’:

3(x + 3)/(x + 4)

  • Step #3: Write the simplified fraction

3(x + 3)/(x + 4)

Practice Questions

Question 1: Simplify (2y2 + 6y) / (y2 + 3y)

Answer :


Question 2: Simplify (4a2 - 16) / (a2 - 4)

Answer :

Steps for Adding and Subtracting Algebraic Fractions

  • To add or subtract algebraic fractions, follow the following steps:

Step #1: Find a common denominator

Step #2: Rewrite each fraction with the common denominator

Step #3: Add or subtract the numerators

Step #4: Simplify the resulting fraction, if necessary

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Solved Example: 

Question 1: Add (2x + 1)/(x2 – 1) and (x – 1)/(x + 1)

Solution:

  • Step #1: Find a common denominator

(x2 – 1)(x + 1)

  • Step #2: Rewrite each fraction with the common denominator

Algebraic Fractions Solved Example Question 1

  • Step #3: Add or subtract the numerators

Solved Example 1

Solved Example 2

  • Step #4: Simplify the resulting fraction

how to simplify algebraic fractions

simplifying algebraic fractions

Steps for Multiplying Algebraic Fractions

  • To multiply algebraic fractions, follow the following steps:

For multiplication:

Step #1: Multiply the numerators

Step #2: Multiply the denominators

Step #3: Simplify the resulting fraction, if necessary

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Solved Example

Question: Multiply (2x + 1)/ (x2 – 1) and (x – 1)/(x + 1)

Solution: 

  • Step #1: Multiply the numerators

(2x + 1)(x – 1)

  • Step #2: Multiply the denominators

(x2 – 1)(x + 1)

  • Step #3: Simplify the resulting fraction

(2x2 – x – 1)/(x3 – x)

Steps for Dividing Algebraic Fractions

  • To multiply algebraic fractions, follow the following steps:

For division:

Step #1: Flip the second fraction (divisor) and turn it into its reciprocal

Step #2: Multiply the first fraction (dividend) with the reciprocal of the second fraction (divisor)

Step #3: Simplify the resulting fraction, if necessary

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Solved Example

Question: Divide (2x + 1)/(x2 – 4x + 3) by (x – 1)/(x – 3)

Solution: 

  • Step #1: Flip the second fraction to get

(x – 3)/(x – 1)

  • Step #2: Multiply the first fraction with the reciprocal of the second fraction to get

Steps for Dividing Algebraic Fractions

  • Step #3: Simplify the resulting fraction by factoring the quadratic in the denominator and cancelling any common factors. The numerator can also be simplified using the distributive property.

Simplified algebraic fraction example showing the cancellation of common factors in the numerator and denominator.

Algebraic Fractions Dividing Final Answers

The simplified expression is (2x + 1)/(x – 1)2

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Practice Questions

Question 1: Simplify the algebraic fraction: (3x2 - 12x) / (x2 - 4)

Answer :


Question 2: Simplify the algebraic fraction: (2x2 + 5x - 3) / (x2 - 4x + 4)

Answer :

Solving Equations with Algebraic Fractions

When solving equations with algebraic fractions,

  • The first step is to get rid of the fractions by finding a common denominator.
  • Then, the equation can be solved using standard techniques like simplifying and factoring.

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Solved Example

Question: Solve for x: (2x + 1)/(x – 2) + 1 = 3/(x – 2)

Solution: 

  • Step #1: Find the common denominator by multiplying both sides by (x – 2), to get

(2x + 1) + (x – 2) = 3

  • Step #2: Simplify the resulting equation by combining like terms, to get

3x – 1 = 3

  • Step #3: Solve for x by adding 1 to both sides and then dividing by 3, to get

x = 4/3

Practice Questions

Question 1: Simplify the algebraic fraction: (5a2 + 7ab) / (3a2 - 4ab)

Answer :


Question 2: Simplify the algebraic fraction: (2x3 - 6x2 + 4x) / (4x2 - 8x + 4)

Answer :

Conclusion

  • Algebraic fractions is a keystone used for solving several problems at the basic algebra level.
  • What make algebraic fractions work out is that, you should be well aware of the rules for simplifying, adding, subtracting, multiplying, dividing, solving equations, and everything else within the scope of the subject. By having the knack and moving on the fractions can be simply conquered.

Worksheet on Algebraic Fraction

Question 1: Simplify the algebraic fraction:

(3a3 - 9a2b + 6ab2) / (6a2b - 12ab2 + 6b3)



Question 2: Simplify the algebraic fraction:

(x4 - 4x2 + 4) / (x2 - 2x + 1)



Question 3: Simplify the algebraic fraction:

(3xy2 - 6x2y) / (2x2y - 4xy2)



Question 4: Simplify the algebraic fraction:

(4x3 - 8x2 + 4x) / (2x2 - 4x + 2)