GCSE · Maths · AQA · Spec 8300

Multiplicative relationship as ratio or fraction

A recipe uses 3 times as much flour as butter. Is the butter 3/1 of the flour, or 1/3? It's one fact, and you can write it three ways.

Drag it and watch

Flour is always 3 times the butter

the amount of butter: 2/8. Butter 50 g. Flour 150 g. Flour ÷ butter 150 ÷ 50 = 3. Flour : butter 150 : 50 = 3 : 1. Butter : flour 50 : 150 = 1 : 3. Butter as a fraction of flour 50/150 = 1/3

Butter50 gFlour150 gFlour ÷ butter150 ÷ 50 = 3Flour : butter150 : 50 = 3 : 1Butter : flour50 : 150 = 1 : 3Butter as a fraction of flour50/150 = 1/3

The bar is the butter. The flour is worked out from it, and every readout updates as you drag. Watch the amounts change while the simplified ratios and the fraction stay put.

Watch out: Look at the two ratio readouts. Flour : butter is 3 : 1, but butter : flour is 1 : 3. The fraction flips as well: flour ÷ butter is 3, butter ÷ flour is 1/3. Same recipe, compared the other way round. So before you write anything, ask: compared with what?

Your turn to think

Cows and sheep: which answer would you give?

On a farm there are 4 times as many sheep as cows.

The number of cows is what fraction of the number of sheep? Pick the answer closest to yours.

4. The question says 4 times, so the fraction is 4/1.1/4. For every 4 sheep there's 1 cow, so cows ÷ sheep = 1/4.1/5. Picture 1 cow and 4 sheep. That's 5 animals, and 1 of them is a cow.

How sure are you?

I'm sureI think soI'm guessing

4. The question says 4 times, so the fraction is 4/1.

This is a common misconception

That 4 is real, but it belongs to the sheep. "4 times as many sheep as cows" compares the sheep with the cows. This question turns it round: cows compared with sheep. There are fewer cows than sheep, so the answer has to be less than 1. Cows ÷ sheep = 1/4.

1/4. For every 4 sheep there's 1 cow, so cows ÷ sheep = 1/4.

This matches the evidence

Yes. The cows are compared with the sheep: 1 cow for every 4 sheep, so the cows are 1/4 of the sheep. Check with real numbers: 20 cows means 80 sheep, and 20/80 = 1/4 ✓. As a ratio, cows : sheep = 1 : 4. It's the same fact.

1/5. Picture 1 cow and 4 sheep. That's 5 animals, and 1 of them is a cow.

This is a common misconception

Your picture is right, but you've compared the cow with all 5 animals. That answers a different question: what fraction of all the animals are cows? This question compares cows with sheep only. 1 cow against 4 sheep is 1/4. Leave the total out and compare one quantity with the other.

Whatever you're comparing WITH goes on the bottom of the fraction and second in the ratio. It's the other quantity, never the total.

When the multiplier isn't a whole number

Problem

A car park has 2.5 times as many cars as vans. Write cars : vans as a ratio in its simplest whole-number form. Then write the number of cars as a fraction of the number of vans, and the number of vans as a fraction of the number of cars.

Now you drive

Work backwards from a fraction

Mia's savings are 3/4 of Jake's savings. Write Jake : Mia as a ratio, then say how many times as much Jake has saved as Mia.

Compared with what? Mia is compared with Jake. So picture Jake's savings split into 4 equal parts. Mia has 3 parts of that same size.
missing step

Which line is step 2?

So Mia : Jake = 4 : 3.So Mia : Jake = 3 : 4.So Mia : Jake = 3 : 1.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

"Three times as much" is one fact. You can write it as a ratio or as a fraction, either way round, as long as you know what is being compared with what.

What you need to know

A multiplicative relationship says one quantity is a number of times another: "A is k times B".
As a ratio, A : B = k : 1. Ratios are normally written with whole numbers, so if k is a decimal or a fraction, multiply both parts by the same number.
As fractions, A ÷ B = k and B ÷ A = 1/k. So B is 1/k of A.
Order matters. Decide what is being compared with what before you write anything.

The big picture

When one quantity is a number of times another ("A is k times B"), that single fact can be written as a ratio, A : B = k : 1, or as a fraction either way round: A ÷ B = k, and B is 1/k of A. Scaling both quantities together never changes the ratio or the fraction. What does change the answer is the direction, so always decide what is being compared with what, and compare with the other quantity, never with the total.

Key points

1Scaling both quantities by the same number never changes the ratio or the fraction: 25 : 75 and 200 : 600 are both 1 : 3.

2Swap the order of a ratio and the fraction turns upside down: 3 : 1 becomes 1 : 3, and 3 becomes 1/3.

3The quantity you are comparing WITH goes second in the ratio and on the bottom of the fraction.

4Compare one quantity with the other one, never with the two added together.

5Working backwards, turn the fraction upside down: if B is 2/3 of A, then A is 3/2 = 1.5 times B, and A : B = 3 : 2.

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Worked example

Problem

A small jug holds 0.6 times as much as a large jug. Write small : large as a ratio of whole numbers. Then write each jug's capacity as a fraction of the other's.

⚠ Watch out

Writing the fraction upside down because of where the number sits in the sentence. If A is 5 times B, B is not 5/1 of A. B is the smaller quantity, so B is 1/5 of A. The 5 describes A compared with B.

🧠

Memory hook

One link, three ways to write it. Before you write any of them, ask: compared with what?

✓

Check yourself

A is 3.5 times B. Write A : B using whole numbers, then write B as a fraction of A. Check your answers by choosing a value for B and working out A.

Flashcards

(6)

"A is k times B." How do you write A : B?

A : B = k : 1. If k isn't a whole number, multiply both parts by the same number to get whole numbers.

"A is k times B." What fraction of A is B?

B is 1/k of A. Comparing the other way round turns k upside down.

What should you ask before you write a ratio or a fraction?

Compared with what? The quantity you compare with goes second in the ratio and on the bottom of the fraction.

A is 3 times B. Why is B not 1/4 of A?

1/4 compares B with A and B added together. B compared with A alone is 1/3.

If both quantities are doubled, what happens to their ratio?

Nothing. They grow at the same multiplier, so the simplified ratio and the fraction stay the same.

B is p/q of A. How many times B is A?

Turn the fraction upside down: A is q/p times B, and A : B = q : p.

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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Cows and sheep: which answer would you give?

Work backwards from a fraction

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