Simplifying fractions

When we have a fraction, such as \frac{15}{30}, we often want to write it as simply as we can.

Fraction equivalence

To simplify fractions, we need to understand one simple rule: if we multiply or divide both the top and bottom of a fraction by the same number, we get an equivalent fraction. In other words, the value of the fraction does not change.

For example, if we have the fraction \frac{4}{6}, we can divide both the top and bottom by 2 to get:

• \frac{4\div2}{6\div2}=\frac{2}{3}

This fraction is equivalent to \frac{4}{6}, but it is in a simpler form.

Simplifying fractions

To simplify a fraction, we need to find the highest number that goes into both the numerator (top) and denominator (bottom) of the fraction.

This is called the highest common factor (HCF) - see here for more details on how to find it.

Example: Simplify \frac{18}{24}

• The HCF of 18 and 24 is 6.
• We can divide both the top and bottom of the fraction by 6:
  • \frac{18\div6}{24\div6}=\frac{3}{4}
• Answer: \frac{3}{4}

Example: Simplify \frac{45}{60}

• The HCF of 45 and 60 is 15.
• We can divide both the top and bottom of the fraction by 15:
  • \frac{45\div15}{60\div15}=\frac{3}{4}
• Answer: \frac{3}{4}

Example: Simplify \frac{56}{98}

• The HCF of 56 and 98 is 14.
• We can divide both the top and bottom of the fraction by 14:
  • \frac{56\div14}{98\div14}=\frac{4}{7}
• Answer: \frac{4}{7}

Example: Simplify \frac{81x}{108x}

• The HCF of 81x and 108x is 27x.
• We can divide both the top and bottom of the fraction by 27x:
  • \frac{81x\div27x}{108x\div27x}=\frac{3}{4}
• Answer: \frac{3}{4}

Example: Simplify \frac{50y^2}{75y}

• The HCF of 50y^2 and 75y is 25y.
• We can divide both the top and bottom of the fraction by 25y:
  • \frac{50y^2\div25y}{75y\div25y}=\frac{2y}{3}
• Answer: \frac{2y}{3}