KS3 · Maths

Expressing one number as a fraction of another

Is 50 a lot? Next to 60, it's nearly all of it. Next to 25, it's double. It only becomes a fraction once you decide what the whole is.

Maths · Ratio and proportion

Same number, different whole

the amount: 2/6. Amount 18. As a fraction of 45 18/45 = 2/5. As a fraction of 18 18/18 = 1

Amount18As a fraction of 4518/45 = 2/5As a fraction of 1818/18 = 1

The bar counts up in 9s. Drag along it and read each amount two ways: as a fraction of 45, and as a fraction of 18.

One method, three quick uses

Problem

(a) What fraction of 250 is 150? (b) What fraction of 30 is 20? (c) A function machine turns 27 into 24. What fraction is it multiplying by?

Which number is the whole?

What fraction of 45 is 18?

Four learners answered the question 'What fraction of 45 is 18?'. Each one had a reason.

Which of these is closest to what you would have said?

45 is the whole, so it goes on the bottom. 18/45 simplifies to 2/5.The bigger number goes on top, so it's 45/18.45 comes first in the question, so it goes first: 45/18.18 is 2/5 here, so 18 will be 2/5 in any question.

How sure are you?

I'm sureI think soI'm guessing

45 is the whole, so it goes on the bottom. 18/45 simplifies to 2/5.

This matches the evidence

Spot on. The question asks what fraction OF 45, so 45 is the whole. Divide 18 and 45 by 9 and you get 2/5. Check: one fifth of 45 is 9, and two fifths is 18.

The bigger number goes on top, so it's 45/18.

This is a common misconception

Size doesn't decide it. The whole decides it. Here 45 is the whole, so 45 goes on the bottom. 45/18 answers a different question: what fraction of 18 is 45? That answer is 5/2, more than 1, because 45 is more than a whole 18.

45 comes first in the question, so it goes first: 45/18.

This is a common misconception

Reading order is a trap. Look for the word 'of'. The number straight after 'of' is the whole, and the whole goes on the bottom. 'What fraction of 45' makes 45 the whole, so the answer is 18/45 = 2/5.

18 is 2/5 here, so 18 will be 2/5 in any question.

This is a common misconception

A fraction is always an amount compared with ITS whole. Change the whole and the fraction changes too. £30 out of £100 is 3/10 of the whole. The same £30 out of £60 is 1/2 of the whole.

Before you write anything, find the whole. It goes on the bottom, every time.

Predict, then check

You know 24 is 3/8 of 64. Now swap the roles.

24 is 3/8 of 64. What fraction of 24 is 64?

Spot the slip

Close enough isn't good enough

3 metres of material cost £10.50. How much do 10 metres cost?

A student's answer — which line goes wrong?

3 × 2 = 6, so 6 metres cost £10.50 × 2 = £21.
For 10 metres I need the multiplier from 3 to 10: 3 × ? = 10, so ? = 10 ÷ 3.
10 ÷ 3 = 3.3, so the multiplier is 3.3.
£10.50 × 3.3 = £34.65
So 10 metres cost £34.65.

Your turn

Scale it with the exact multiplier

30 cookies need 195 g of chocolate chips. How much do 250 cookies need? Then: 30 packets cost £25.80. How much do 115 packets cost?

Cookies: we know 30 cookies need 195 g, and we want 250 cookies.
missing step

Which line is step 2?

Multiplier from 30 to 250: 250 ÷ 30 = 25/3Multiplier from 30 to 250: 30 ÷ 250 = 3/25Multiplier from 30 to 250: 250 ÷ 30 = 8.3

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Find the whole, put it on the bottom, simplify. Then flip it when the roles swap.

What you need to know

The whole goes on the bottom. In 'what fraction of A is B?', A is the whole, so the answer is B/A.
Simplify by dividing the top and bottom by the same number. That's dividing by 1, so the value doesn't change.
If the amount is bigger than the whole, the fraction is bigger than 1. That is allowed.
The multiplier between two numbers is found by dividing, and it is the same as the fraction one is of the other.
Keep multipliers exact. A recurring decimal must stay as a fraction; a decimal that stops is fine.
Swap the part and the whole and the fraction flips to its reciprocal. A number times its reciprocal is 1.

The big picture

To write one number as a fraction of another, put the whole on the bottom and simplify by dividing the top and bottom by the same number. The fraction can be more than 1 when the amount is bigger than the whole. The same fraction is the multiplier from the whole to the amount. Keep a multiplier exact: if dividing gives a recurring decimal, use the fraction. When the part and the whole swap, the fraction flips to its reciprocal.

Key points

1Whole on the bottom, amount on the top, then simplify.

2Fractions bigger than 1 happen whenever the amount is bigger than the whole.

3The fraction one number is of another is also the multiplier between them.

4Use the exact fraction for a multiplier like 10/3. Rounding it to 3.3 gives a wrong answer.

5Reciprocal pairs: if B is p/q of A, then A is q/p of B. Change mixed numbers to improper fractions before flipping.

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Worked example

Problem

What fraction of 84 is 35? Then: what fraction of 35 is 84?

⚠ Watch out

Putting the bigger number on the bottom out of habit. The bottom is for the whole, and sometimes the whole is the smaller number. Then the answer is bigger than 1.

🧠

Memory hook

'Of' points to the floor: the number after 'of' is the whole, and the whole goes on the bottom.

✓

Check yourself

Without looking back: what fraction of 105 is 75? And what fraction of 75 is 105? Say which number is the whole each time.

Flashcards

(13)

In 'what fraction of A is B?', which number is the whole?

A, the number straight after 'of'. It goes on the bottom: B/A.

Why doesn't simplifying change a fraction's value?

You divide the top and bottom by the same number, which is the same as dividing by 1.

Can the answer to 'what fraction of A is B?' be bigger than 1?

Yes. If the amount is bigger than the whole, the fraction is bigger than 1. For example, 205 is 5/3 of 123.

How do you find the multiplier that turns one number into another?

Divide, because division undoes multiplication. 3 × ? = 81, so ? = 81 ÷ 3 = 27.

What is the multiplier from 3 to 10?

10/3. Not 3.3, because 3.3 × 3 = 9.9.

What is a terminating decimal?

A decimal with a finite number of digits after the point, like 1.5 or 0.75. A recurring decimal goes on for ever.

When is it fine to use a decimal multiplier?

When it terminates. For example 3/2 = 1.5 is exact, so 1.5 is fine. 10/3 is not, so keep the fraction.

What does 3 × 16/3 equal, and why?

16. A fraction is a division, so 16/3 means 16 ÷ 3, and multiplying by 3 undoes it.

A function machine turns 30 into 42. What is its multiplier?

Output ÷ input = 42/30 = 7/5.

6,804 is 9/4 of 3,024. What fraction of 6,804 is 3,024?

4/9. Swapping the part and the whole flips the fraction.

What do you get when you multiply a number by its reciprocal?

1. For example, 9/4 × 4/9 = 1.

How do you find the reciprocal of a mixed number like 1 3/4?

Change it to an improper fraction first, 7/4, then flip it: 4/7.

What does 'proportion' mean when comparing an amount with a whole?

The amount with respect to its whole. The same amount is a different proportion of a different whole.

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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What fraction of 45 is 18?

Close enough isn't good enough

Scale it with the exact multiplier

More KS3 Maths topics