GCSE · Maths · AQA · Spec 8300

Express one quantity as a fraction of another

3 cm and 4 cm: is that 3/4 or 4/3? Both! 3 cm is 3/4 of 4 cm, and 4 cm is 4/3 of 3 cm. 'Of' decides which.

Part of the whole — or more than it

How much of B does A make?

A on the 24 cm bar: 9/24. A as a fraction of B 3/4. Before simplifying 9/12. Compared with 1 whole Less than 1. Length of A 9 cm

A as a fraction of B3/4Before simplifying9/12Compared with 1 wholeLess than 1Length of A9 cm

B is 12 cm long and counts as one whole: it is the stretch from 0 to the 1 mark. Each small cell is 1 cm. Drag the end of A shorter than B, level with it, then past it, and watch the fraction.

What would you write?

Sam's run as a fraction of Priya's

Sam ran 1.5 km. Priya ran 900 m. Write the distance Sam ran as a fraction of the distance Priya ran.

Which answer is closest to what you would write?

900/1500 = 3/5. The smaller number always goes on top.1.5/900 — just use the numbers in the question.It comes out as 1500/900, but a fraction can't be bigger than 1, so I'd flip it to 3/5.1.5 km = 1500 m. Priya's 900 m is the whole, so it's 1500/900 = 5/3.

How sure are you?

I'm sureI think soI'm guessing

900/1500 = 3/5. The smaller number always goes on top.

This is a common misconception

That answers a different question: Priya's distance as a fraction of Sam's. Size doesn't decide which number goes where — the word 'of' does. Here the amount after 'of' is Priya's 900 m, so that's the whole and it goes on the bottom. Sam ran further than Priya, so his fraction should come out bigger than 1.

1.5/900 — just use the numbers in the question.

This is a common misconception

Those numbers are in different units, so 1.5/900 compares kilometres with metres. It would mean Sam ran a tiny sliver of Priya's distance, but he actually ran further. Get both into metres first: 1.5 km = 1500 m.

It comes out as 1500/900, but a fraction can't be bigger than 1, so I'd flip it to 3/5.

This is a common misconception

You did the hard part right, then undid it. A fraction can be bigger than 1: 5/3 means one whole and two thirds more. Flipping it gives 3/5, which is Priya's distance as a fraction of Sam's — the reverse question.

1.5 km = 1500 m. Priya's 900 m is the whole, so it's 1500/900 = 5/3.

This matches the evidence

Exactly right. Same units first, the amount after 'of' on the bottom, then simplify by dividing top and bottom by 300. The answer is more than 1 because Sam ran further than Priya — his distance is more than one whole of hers.

Neither size nor order decides which number goes on the bottom — the word 'of' does. And an answer bigger than 1 is not a mistake to fix.

Your turn to make the decisions

2 hours as a fraction of 90 minutes

Express 2 hours as a fraction of 90 minutes. Give your answer in its simplest form.

There are two amounts: 2 hours and 90 minutes. Before calculating, decide which one is the whole.
missing step

Which line is step 2?

The whole is 90 minutes, because it comes after 'of'. So 90 minutes goes on the bottom.The whole is 2 hours, because it is the bigger amount.The whole is 2 hours, because it is mentioned first.

Put it on the line

Where does each answer land?

Work out each fraction, then slide it to its place. On this line, 1 means one whole of the amount after 'of'.

Q1 — 30 cm as a fraction of 40 cmQ2 — 1.2 kg as a fraction of 800 gQ3 — 15 seconds as a fraction of 1 minuteQ4 — 1 m as a fraction of 100 cmQ5 — £3 as a fraction of £2.40

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

That amount goes on the bottom. And when the first amount is bigger than the whole, the fraction is bigger than 1 — which is completely fine.

What you need to know

'A as a fraction of B' asks: how much of B does A make? B — the amount after 'of' — is the whole, so the fraction is A/B with B on the bottom.
Both amounts must be in the same unit before you write the fraction. Useful swaps: 1 m = 100 cm, 1 km = 1000 m, 1 kg = 1000 g, 1 hour = 60 minutes, £1 = 100p. Changing to the smaller unit usually avoids decimals.
Simplify the fraction by dividing the top and bottom by their highest common factor. For example, 45/60 = 3/4 (divide both by 15).
If A is smaller than B the fraction is less than 1. If A and B are equal it is exactly 1. If A is bigger than B the fraction is greater than 1, such as 5/3. You can leave it as an improper fraction or write it as a mixed number (5/3 = 1 2/3), but never flip it to make it smaller than 1.

The big picture

To write A as a fraction of B, ask: how much of B does A make? B, the amount after 'of', is the whole, so it goes on the bottom. Put both amounts in the same unit first, write the fraction A/B, then simplify it. If A is smaller than B the fraction is less than 1; if A is bigger than B it is more than 1, and that top-heavy answer is correct.

Key points

1The amount after 'of' is the whole: A as a fraction of B is A/B.

2Same units first, then write the fraction, then simplify.

3A smaller than B → less than 1. A bigger than B → more than 1. Both are correct answers.

4Swap the wording and the fraction turns over: if A is 3/4 of B, then B is 4/3 of A.

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Worked example

Problem

Express 600 g as a fraction of 2 kg. Give your answer in its simplest form.

⚠ Watch out

Turning the fraction over to make it less than 1. For 20 cm as a fraction of 15 cm, writing 15/20 = 3/4 answers the reverse question. The whole is 15 cm, so it's 20/15 = 4/3 — more than 1, because 20 cm is longer than 15 cm.

🧠

Memory hook

'Of' goes on the floor: the amount after 'of' is the whole, so it sits at the bottom of the fraction.

✓

Check yourself

Write 50p as a fraction of £2. Then write £2 as a fraction of 50p. (Answer: 50/200 = 1/4, and 200/50 = 4, which is bigger than 1.)

Flashcards

(6)

In 'A as a fraction of B', which amount is the whole?

B — the amount after 'of'. It goes on the bottom, so the fraction is A/B.

What must you check before you write the fraction?

That both amounts are in the same unit. Convert first, e.g. 1 m = 100 cm or 1 hour = 60 minutes.

A is bigger than B. What do you know about A as a fraction of B?

It is greater than 1 — an improper fraction such as 7/4. That is a correct answer, so don't turn it over.

What is the last step after writing the fraction?

Simplify: divide the top and bottom by their highest common factor, e.g. 36/48 = 3/4 (divide by 12).

If A is 2/5 of B, what fraction of A is B?

5/2. Swapping which amount comes after 'of' turns the fraction over.

Write 3 hours as a fraction of a day.

A day is 24 hours, so it's 3/24 = 1/8.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Sam's run as a fraction of Priya's

2 hours as a fraction of 90 minutes

Where does each answer land?

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