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 

​$$\frac {28x^3y^2} {7 x^2y}$$​​

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

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

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

Example 2

Simplify

​$$\frac {5x^2-30x} {15xy}$$​​

Factorise the numerator before cancelling.

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

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

​$$\frac {10x^2}{15y^3} \times \frac {25y}{50x} \\$$​​

​$$\begin{aligned}\frac {10x^2}{15y^3} \times \frac {25y}{50x} &= \frac {^2\cancel {10}x^2}{^3\cancel{15}y^3} \times \frac {^1\cancel{25}y}{^2\cancel{50}x} \\\\ &=\frac {2x^2}{3y^3} \times \frac {y}{2x} \\\\ &=\frac {2x^2y}{6xy^3} \\\\ &=\frac {^1\cancel2 \times \cancel x \times x \times \cancel y}{^3\cancel 6 \times \cancel x \times \cancel y \times y \times y} \\\\ &=\underline{\frac {x}{3y^2}} \\\end{aligned}$$​​

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

​$$\frac {6x^3}{8} \div \frac {9x^4}{2}$$​​

​$$\begin{aligned}\frac {6x^3}{8} \div \frac {9x^4}{2}&=\frac {6x^3}{8} \times \frac {2}{9x^4} \\\\&=\frac {^2\cancel 6x^3}{^4\cancel8} \times \frac {^1\cancel2}{^3\cancel 9x^4} \\\\&= \frac {2x^3} {4}\times \frac {1}{3x^4}\\\\&= \frac {^1\cancel2x^3}{^6\cancel{12}x^4} \\\\&= \underline{\frac {1}{x}}\end{aligned}$$​​

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.

​$$\frac {1}{x+2}+ \frac {1}{x+3}$$​​

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