Everything to learn better...

Home

Maths

Algebra

Algebraic fractions - Higher

Algebraic fractions - Higher

Select Lesson

Exam Board

Select an option

Explainer Video

Loading...
Tutor: Bilal

Summary

Algebraic fractions

​​In a nutshell

Algebraic fractions can be simplified, added, subtracted, multiplied and divided just like fractions with numbers. The same principles behind performing operations on fractions can be extended to algebraic fractions, like cancelling fractions by finding a common factor or finding a common denominator with addition or subtraction. In addition to these principles, algebraic fractions require the expressions on the numerator or denominator to be factorised.



Cancelling algebraic fractions

Algebraic fractions can be cancelled if they have a common factor, this can be either a number or variable that divides into the numerator and denominator.


Example 1

Simplify 

28x3y27x2y\frac {28x^3y^2} {7 x^2y}​​


Write out the expressions on the numerator and denominator in full, then cancel down.

28x3y27x2y=28×x×x×x×y×y7×x×x×y=428×x×x×x×y×y17×x×x×y=4xy\begin{aligned}\frac {28x^3y^2} {7x^2y} &=\frac {28 \times x \times x \times x \times y \times y} {7 \times x \times x \times y} \\ \\&=\frac {^4\cancel{28} \times \cancel x \times \cancel x \times x \times \cancel y \times y} {^1 \cancel7 \times \cancel x \times \cancel x \times \cancel y} \\ \\&= \underline{4xy}\end{aligned}​​


Sometimes you need to factorise the numerator or denominator before cancelling.


Example 2

Simplify

5x230x15xy\frac {5x^2-30x} {15xy}​​


Factorise the numerator before cancelling.

5x230x15xy=5x(x6)15xy=15x(x6)315xy=x63y\begin{aligned}\frac {5x^2-30x} {15xy} &= \frac {5x(x-6)}{15xy} \\\\&= \frac {^1\cancel 5 \cancel x(x-6)}{^3\cancel{15}\cancel xy} \\\\&= \underline{\frac {x-6}{3y}}\end{aligned}​​



Multiplying algebraic fractions

Algebraic fractions can be multiplied in a similar way to multiplying numerical fractions. Multiply the numerators together and multiply the denominators together and divide the two results. You can cancel down the algebraic fraction before or after the multiplication.


Example 3

Multiply and simplify

10x215y3×25y50x\frac {10x^2}{15y^3} \times \frac {25y}{50x} \\​​


10x215y3×25y50x=210x2315y3×125y250x=2x23y3×y2x=2x2y6xy3=12×x×x×y36×x×y×y×y=x3y2\begin{aligned}\frac {10x^2}{15y^3} \times \frac {25y}{50x} &= \frac {^2\cancel {10}x^2}{^3\cancel{15}y^3} \times \frac {^1\cancel{25}y}{^2\cancel{50}x} \\\\ &=\frac {2x^2}{3y^3} \times \frac {y}{2x} \\\\ &=\frac {2x^2y}{6xy^3} \\\\ &=\frac {^1\cancel2 \times \cancel x \times x \times \cancel y}{^3\cancel 6 \times \cancel x \times \cancel y \times y \times y} \\\\ &=\underline{\frac {x}{3y^2}} \\\end{aligned}​​



Dividing algebraic fractions

To divide algebraic fractions, change the divide sign into a muliplication sign and turn the second fraction upside down. Then perform the multiplication.


Example 4

Divide and simplify

6x38÷9x42\frac {6x^3}{8} \div \frac {9x^4}{2}​​


6x38÷9x42=6x38×29x4=26x348×1239x4=2x34×13x4=12x3612x4=1x\begin{aligned}\frac {6x^3}{8} \div \frac {9x^4}{2}&=\frac {6x^3}{8} \times \frac {2}{9x^4} \\\\&=\frac {^2\cancel 6x^3}{^4\cancel8} \times \frac {^1\cancel2}{^3\cancel 9x^4} \\\\&= \frac {2x^3} {4}\times \frac {1}{3x^4}\\\\&= \frac {^1\cancel2x^3}{^6\cancel{12}x^4} \\\\&= \underline{\frac {1}{x}}\end{aligned}​​



Adding or subtracting algebraic fractions

To add or subtract algebraic fractions, multiply up the fractions so that they have the same denominator. Then, add the numerators before simplifying the result.


Example 5

Add the fractions and simplify the answer.

1x+2+1x+3\frac {1}{x+2}+ \frac {1}{x+3}​​


1x+2+1x+3=1×(x+3)(x+2)(x+3)+1×(x+2)(x+2)(x+3)=x+3+x+2(x+2)(x+3)=2x+5(x+2)(x+3)\begin{aligned}\frac {1}{x+2} + \frac {1}{x+3} &= \frac {1 \times (x+3)}{(x+2)(x+3)} + \frac {1 \times (x+2)}{(x+2)(x+3)} \\\\&= \frac {x+3 + x+2}{(x+2)(x+3)} \\\\&= \underline{\frac {2x+5}{(x+2)(x+3)}} \\\end{aligned}​​



Create an account to read the summary

Exercises

Create an account to complete the exercises

FAQs - Frequently Asked Questions

How do you multiply or divide algebraic fractions?

How do you cancel algebraic fractions?

What are algebraic fractions?

Beta

I'm Vulpy, your AI study buddy! Let's study together.