Equivalent Fractions and Simplifying

What you will learn

• Explain why $\frac{1}{2}$, $\frac{2}{4}$ and $\frac{4}{8}$ are the same amount.
• Find equivalent fractions by multiplying or dividing.
• Simplify a fraction to its lowest terms.
• Check whether two fractions are equal.

Before you start: you should know what a fraction is and be able to find a common factor. See Fractions Explained and Factors and Multiples.

The idea

Take a strip and shade half of it. Cut every part in half again, and you have 4 parts with 2 shaded. Cut again, and you have 8 parts with 4 shaded. The shaded amount never changed, only the number of cuts. So $\frac{1}{2} = \frac{2}{4} = \frac{4}{8}$. These are called equivalent fractions.

Making equivalent fractions

Multiply the top and the bottom by the same number. Both parts change, so the size of each piece changes in step with the number of pieces.

$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

You can only multiply by the same number on both sides. $\frac{2}{3}$ and $\frac{2 + 4}{3 + 4}$ are not equal, because adding changes the amount.

Simplifying

Simplifying goes the other way. Divide the top and the bottom by a number that goes into both. A fraction is in its simplest form when no number except $1$ divides both. The quickest way is to divide by the highest common factor. See HCF and LCM.

Worked examples

Example 1: Find a missing number

Fill the gap: $\frac{3}{4} = \frac{\square}{20}$.

1. Look at the bottoms: $4 \to 20$ is $\times 5$.
2. Do the same to the top: $3 \times 5 = 15$.
3. Check: $\frac{15}{20}$ divided by $5$ on both sides gives $\frac{3}{4}$.

Answer: $\frac{15}{20}$

Example 2: Simplify fully

Simplify $\frac{24}{36}$.

1. Factors of $24$: $1, 2, 3, 4, 6, 8, 12, 24$. Factors of $36$: $1, 2, 3, 4, 6, 9, 12, 18, 36$.
2. The highest common factor is $12$.
3. $24 \div 12 = 2$ and $36 \div 12 = 3$.
4. $\frac{2}{3}$ cannot be simplified further, since $2$ and $3$ share only the factor $1$.

Answer: $\frac{2}{3}$

Example 3: Are they equal?

Are $\frac{5}{7}$ and $\frac{15}{21}$ equivalent?

1. Check the top: $5 \to 15$ is $\times 3$.
2. Check the bottom: $7 \to 21$ is $\times 3$.
3. Both are multiplied by $3$, so they are equivalent.

Answer: Yes.

Tip: if you divide and a number is still left that goes into both, keep going. Dividing by $2$ then by $3$ is the same as dividing by $6$.

Why it matters

Adding, subtracting and comparing fractions all start with writing them over the same bottom number, and that means equivalent fractions. Simplifying makes answers easier to read: a recipe that needs $\frac{4}{8}$ of a cup is easier to follow as half a cup.

Ratios, percentages and probabilities are fractions too, so the same skill returns again and again.