Equivalent Fractions Explained: KS3 Maths

What does equivalent mean for fractions?

Two fractions are equivalent if they sit at exactly the same position on a number line. For example, ½, 2/4, 3/6 and 50/100 are all equivalent because each equals 0.5 as a decimal.

The underlying reason is that multiplying a fraction's top and bottom by the same number is really multiplying by a form of 1 (for example, 2/2 = 1), so the fraction's value is unchanged.

How do you make an equivalent fraction with a given denominator?

This skill is essential for adding and subtracting fractions with different denominators.

Steps:

1. Decide what number the original denominator must be multiplied by to reach the target denominator.
2. Multiply the numerator by the same number.
3. Write the result.

Worked example: Write 3/4 as an equivalent fraction with denominator 20.

1. 4 × 5 = 20, so multiply top and bottom by 5.
2. Numerator: 3 × 5 = 15.
3. Answer: 15/20.

Check: 15 ÷ 20 = 0.75 and 3 ÷ 4 = 0.75 ✓

Another example: Write 7/12 as an equivalent fraction with denominator 60.

1. 12 × 5 = 60.
2. 7 × 5 = 35.
3. Answer: 35/60.

How do you simplify a fraction using equivalence?

Simplifying (also called cancelling or reducing) produces a simpler equivalent fraction by dividing top and bottom by their highest common factor (HCF).

Worked example: Simplify 18/24.

1. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. HCF = 6.
2. Divide both by 6: 18 ÷ 6 = 3; 24 ÷ 6 = 4.
3. Simplified fraction: 3/4.

You can also simplify in stages — divide by 2 first, then 3, for example — as long as you keep dividing until no common factor remains.

How do you use equivalent fractions to compare two fractions?

When comparing fractions with different denominators, first convert both to a common denominator.

Worked example: Which is larger, 3/8 or 2/5?

1. Find the lowest common multiple (LCM) of 8 and 5: LCM = 40.
2. Convert: 3/8 = 15/40 and 2/5 = 16/40.
3. Compare numerators: 15 < 16, so 2/5 > 3/8.

The table confirms 2/5 is larger.

How do equivalent fractions link to adding and subtracting fractions?

You cannot add fractions directly unless their denominators are equal. Equivalent fractions let you rewrite both fractions using a common denominator before adding.

Example: 1/3 + 1/4.

LCM(3, 4) = 12. Convert: 1/3 = 4/12 and 1/4 = 3/12. Add: 4/12 + 3/12 = 7/12.

Without equivalent fractions, this calculation is impossible to perform correctly.

What mistakes do students commonly make?

Mistake 1 — Adding the same number to top and bottom. Doing 1/2 + 1/1 = 2/3 is wrong; this does not preserve the value. You must MULTIPLY (or divide) top and bottom by the same number.

Mistake 2 — Forgetting to apply the same operation to the numerator. If 4 × 5 = 20 for the denominator, the numerator must also be multiplied by 5.

Mistake 3 — Dividing by a common factor that is not the HCF. Dividing 18/24 by 2 gives 9/12, which is equivalent but not fully simplified. Always check whether a further common factor remains.

Mistake 4 — Confusing equal numerators with equal fractions. 3/5 and 3/7 have the same numerator but are not equivalent (0.6 ≠ 0.43).

Frequently asked questions

How can I tell quickly whether two fractions are equivalent?

Cross-multiply: for a/b and c/d, multiply a × d and b × c. If both products are equal, the fractions are equivalent. For example, 3/4 and 9/12: 3 × 12 = 36 and 4 × 9 = 36, so they are equivalent. This works because equivalent fractions satisfy ad = bc.

Is there a limit to how many equivalent fractions a given fraction has?

No — every fraction has infinitely many equivalents. Start from 1/2 and keep multiplying top and bottom by 2, 3, 4, … to get 2/4, 3/6, 4/8, and so on forever. The simplest form (lowest terms) is the one where top and bottom share no factor other than 1.

Do equivalent fractions always have the same decimal value?

Yes, always. This is the most reliable check: convert each fraction by dividing numerator by denominator. If the decimal values match, the fractions are equivalent. For example, 5/8 = 0.625 and 15/24 = 0.625 — equivalent.

Why does dividing top and bottom by the same number also give an equivalent fraction?

Because dividing by n/n is the same as multiplying by its reciprocal 1/n ÷ 1/n = 1. Multiplying or dividing by any form of 1 leaves the value unchanged. So 6/10 ÷ 2/2 = 3/5, and 3/5 represents the same proportion as 6/10.

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