# Simplify algebraic fractions

Lesson

Simplifying algebraic fractions usually involves the cancelling of common factors between the numerator and denominator. This could include both numeric and algebraic factors.

 $\frac{3xy}{6xz}$3xy6xz​ $=$= $\frac{xy}{2xz}$xy2xz​     Cancel the highest common numerical factor $=$= $\frac{y}{2z}$y2z​       Cancel the highest common powers of each variable

When trying to simplify expressions that involve algebraic fractions, it is best to initially cancel out any immediately recognisable common factors between the numerator and denominator. This should make the simplification process of a bigger expression a lot easier, and we can see this in the example below.

 $\frac{x^2}{x}+3x$x2x​+3x $=$= $\frac{x^2}{x}+\frac{3x^2}{x}$x2x​+3x2x​ | $\frac{x^2}{x}+3x$x2x​+3x $=$= $x+3x$x+3x $=$= $\frac{x^2+3x^2}{x}$x2+3x2x​ | $=$= $4x$4x $=$= $\frac{4x^2}{x}$4x2x​ | $=$= $4x$4x |

#### Worked Examples

##### Question 1

Express $\frac{72s^2t^2}{81s^3t}$72s2t281s3t in simplest form.

##### Question 2

Yuri wanted to know whether he is faster than his brother Jimmy at building a wooden shelf. Their woodworking instructor told Yuri that his speed was given by the expression $18xy$18xy and that Jimmy's speed was given by $6xy$6xy, where $x$x relates to the quality of the tools they use, and $y$y depends on the quality of the materials used.

1. If both brothers use the same tools and materials, write a fraction which represents the ratio of Yuri and Jimmy's construction speeds respectively. Do not simplify your answer.

$\frac{\editable{}}{\editable{}}$

2. By simplifying the fraction from the previous part, determine how many times faster Yuri is than Jimmy.

3. If both brothers always use the same tools and materials, does the quality of either one impact how many times faster Yuri is than Jimmy?

No

A

Yes

B

No

A

Yes

B

##### Question 3

Simplify the following expression:

$\frac{12y^3}{6y^2}+\frac{27y^2}{9y}+\frac{4y}{4}$12y36y2+27y29y+4y4