GCSE · Maths · AQA · Spec 8300

Fractions in ratio problems

A ratio looks like two numbers and a colon. Hidden inside it are fractions — and the whole trick is one question: a fraction of WHAT?

Ratio · Fractions

One ratio, two fractions

Ali's blocks: 3/8. Ali's share of £40 £15. Ali's share of £400 £150. Ali's fraction of the whole 3/8. Ali's share as a fraction of Ben's 3/5

Ali : Ben3 : 5Ali's share of £40£15Ali's share of £400£150Ali's fraction of the whole3/8Ali's share as a fraction of Ben's3/5

Ali and Ben share some money. The bar is the whole amount, cut into 8 equal blocks: the shaded blocks are Ali's, the rest are Ben's. At the start Ali has 3 of the 8 blocks — that's £15 of £40, and £150 of £400: the same 3/8 either way. Drag to change the split and watch both fractions. Whatever split you choose, the two money boxes always show the same fraction of each total, because the fraction belongs to the ratio, not to the amount.

Which fraction?

Be honest: what would you write?

A tin of paint is mixed from red and white in the ratio 2 : 3.

What fraction of the paint is red? Pick the idea closest to what you think right now.

2/3 — the ratio is 2 to 3, so red is 2 over 3.2/5 — there are 2 + 3 = 5 equal parts, and 2 of them are red.You can't say until you know how much paint there is.

How sure are you?

I'm sureI think soI'm guessing

2/3 — the ratio is 2 to 3, so red is 2 over 3.

This is a common misconception

This is the slip almost everyone makes at first, so don't worry. 2/3 is a real fraction here — but it's red compared with WHITE: for every 3 tins' worth of white there are 2 of red. The question asks about all of the paint. Draw it as blocks: 2 red + 3 white = 5 blocks altogether. Red is 2 of those 5, so red is 2/5 of the paint.

2/5 — there are 2 + 3 = 5 equal parts, and 2 of them are red.

This matches the evidence

Exactly right. Adding the parts gives you the whole: 5 blocks, 2 of them red, so 2/5 of the paint is red. White is the other 3/5 — and the two fractions of the whole add up to 1, which is a handy check.

You can't say until you know how much paint there is.

This is a common misconception

It feels as if you need an amount, but the ratio already fixes the fraction. Whether the tin holds 5 litres or 500 litres, the paint is still 5 equal parts with 2 of them red — just as the £40 and £400 boxes above always gave Ali the same fraction of each total. You need the amount to work out HOW MUCH red there is, not what fraction is red.

Fraction of the whole = one share's parts ÷ (all the parts added up). Fraction of the other share = one share's parts ÷ the other share's parts. Read the question to see which one it wants.

Worked example: from a fraction back to a ratio

Problem

In a choir, the number of boys is 3/4 of the number of girls. There are 84 children in the choir altogether. How many of them are boys?

Your turn

Fill in the missing steps

Sam's savings are 5/8 of Priya's savings. Together they have £390. How much does Priya have?

The fraction compares Sam with Priya: Sam has 5/8 of what Priya has.
missing step

Which line is step 2?

Sam : Priya = 5 : 3Sam : Priya = 5 : 8Sam : Priya = 8 : 5

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Of the whole: parts over the total number of parts. Of the other share: one part over the other.

What you need to know

Every ratio a : b is really a + b equal blocks — a blocks for the first share, b blocks for the second.
Before you write any fraction, ask: a fraction of the whole, or a fraction of the other share?
A fraction sentence can be turned back into a ratio, and then solved by finding the size of one part.

The big picture

A ratio a : b splits a whole into a + b equal parts. That gives you two different kinds of fraction. A share as a fraction of the WHOLE is its parts over the total number of parts: a/(a + b). A share as a fraction of the OTHER share is just a/b. These fractions belong to the ratio, so they stay the same however big the total is. You can also go the other way: a sentence like 'A is 3/4 of B' turns straight into the ratio A : B = 3 : 4, and from there you solve the problem by finding one part.

Key points

1A ratio a : b cuts the whole into a + b equal parts.

2The first share as a fraction of the whole is a/(a + b); the second share is b/(a + b). These two add up to 1.

3The first share as a fraction of the second share is a/b — a comparison, not a fraction of the whole.

4'A is m/n of B' means A : B = m : n. If m/n of a whole is one thing and the rest is another, the ratio is m : (n − m).

5The fractions come from the ratio, so they stay the same whatever the total amount is. The total only tells you the size of one part.

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Worked example

Problem

A bag contains only red and blue counters. 3/8 of the counters are red. There are 45 blue counters. How many red counters are there?

⚠ Watch out

Reading a ratio a : b as 'the first share is a/b of the whole'. a/b only compares the first share with the second. For a fraction of the whole, the denominator is ALL the parts added together: a + b.

🧠

Memory hook

Blocks out of ALL the blocks. 'Of the whole' → add the parts for the bottom number. 'Of the other one' → just put one number over the other.

✓

Check yourself

Flour and sugar are mixed 5 : 3. What fraction of the mixture is sugar? And sugar is what fraction of the flour? (Answers: 3/8 of the mixture; 3/5 of the flour.)

Flashcards

(7)

A ratio a : b splits the whole into how many equal parts?

a + b equal parts — a for the first share, b for the second.

In the ratio a : b, what fraction of the WHOLE is the first share?

a/(a + b): its parts over the total number of parts.

In the ratio a : b, the first share is what fraction of the SECOND share?

a/b — this compares the two shares, not a share with the whole.

'A is m/n of B.' What is the ratio A : B?

A : B = m : n. The fraction compares A with B, so the numerator goes with A and the denominator with B.

m/n of a whole is one thing and the rest is another. What is the ratio?

m : (n − m). The whole is n parts, so the rest gets the leftover n − m parts.

If the total amount changes but the ratio stays the same, does the fraction change?

No. The fraction belongs to the ratio. The total only changes how big each part is.

What's a quick check on the two fractions of the whole from a two-part ratio?

They add up to 1 — for example, 2/5 and 3/5.

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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Be honest: what would you write?

Fill in the missing steps

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