Expressing One Quantity as a Fraction of Another: KS3 Maths

What does "express as a fraction of" mean?

The phrase "express A as a fraction of B" means write the fraction A/B and simplify it. You are asking: "what fraction of B is A?" The answer shows how A compares to B in proportional terms.

Examples in words:

• "Express 20 as a fraction of 80" → 20/80 = 1/4
• "Express 15 as a fraction of 25" → 15/25 = 3/5
• "Express 36 as a fraction of 48" → 36/48 = 3/4

How do you express one quantity as a fraction of another?

Method — three steps:

1. Write the first quantity (the part) as the numerator.
2. Write the second quantity (the whole) as the denominator.
3. Simplify the fraction by dividing top and bottom by their highest common factor (HCF).

Worked example: In a class of 30 students, 18 are girls. Express the number of girls as a fraction of the total class.

1. Numerator = 18 (the part you are describing).
2. Denominator = 30 (the total).
3. HCF(18, 30) = 6. Simplify: 18/30 = 3/5.

Worked example: A bookshelf holds 24 books. 9 are maths books. Express the number of maths books as a fraction of the total.

1. Fraction = 9/24.
2. HCF(9, 24) = 3.
3. Simplified: 3/8.

What happens when the quantities have different units?

Before writing the fraction, both quantities must be in the same unit.

Example: Express 45 minutes as a fraction of 3 hours.

• Convert 3 hours to minutes: 3 × 60 = 180 minutes.
• Fraction = 45/180.
• HCF(45, 180) = 45.
• Simplified: 1/4.

Example: Express 800 g as a fraction of 2 kg.

• Convert 2 kg to grams: 2000 g.
• Fraction = 800/2000.
• HCF(800, 2000) = 400.
• Simplified: 2/5.

Never write a fraction with mixed units (e.g., 45 minutes over 3 hours) — the answer would be numerically meaningless.

Can the fraction be greater than 1?

Yes, when the first quantity is larger than the second. For example, express 40 as a fraction of 25:

• Fraction = 40/25.
• HCF(40, 25) = 5.
• Simplified: 8/5 (an improper fraction, greater than 1).

This means 40 is more than a whole of 25 — it is 8/5 times as large.

How is this linked to finding a percentage?

Expressing as a fraction and expressing as a percentage are closely related. Once you have the fraction, multiply by 100 to convert to a percentage.

Example: 18 out of 30 students are girls.

• As a fraction: 18/30 = 3/5.
• As a percentage: 3/5 × 100 = 60%.

The skills are often tested together in the same question.

What mistakes should you avoid?

Mistake 1 — Reversing numerator and denominator. "18 girls as a fraction of 30 students" is 18/30, not 30/18. The quantity being described goes on top; the reference total goes on the bottom.

Mistake 2 — Forgetting to simplify. Leaving the answer as 18/30 is not wrong, but exam mark schemes usually expect the fraction in its simplest form. Always divide by the HCF.

Mistake 3 — Ignoring unit conversion. If the quantities use different units, converting is not optional — the fraction is meaningless without it.

Mistake 4 — Using a fraction > 1 when the question expects a proportion of the whole. If the answer comes out as an improper fraction, re-read the question to check whether the first quantity should indeed be larger than the second.

Frequently asked questions

Does it matter which number goes on top?

Yes, definitely. The order changes the meaning entirely. "Express A as a fraction of B" always puts A on top and B on the bottom. "A as a fraction of B" asks how many Bs fit inside A (as a fraction), not the other way round. Re-read the question carefully before writing the fraction.

How do I know whether to convert to the smaller or larger unit?

It does not matter which you choose, as long as both quantities end up in the same unit. Converting to the smaller unit (e.g., minutes instead of hours, grams instead of kilograms) usually gives whole numbers, which are easier to work with.

Can I leave the answer as a decimal instead of a fraction?

Usually no, because the question asks for a fraction. However, checking your answer as a decimal is a great way to verify it. For example, 3/5 = 0.6, and 18 ÷ 30 = 0.6 — they match, so the fraction is correct.

How does this skill appear on KS3 papers?

It often appears within a multi-step problem: for example, "there are 24 counters in a bag — 10 are red. Write the probability that a randomly chosen counter is red, as a fraction in its simplest form." Here, expressing 10 as a fraction of 24 is the first step, and simplification completes the answer.

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