GCSE · Maths · AQA · Spec 8300

Fractions and percentages as operators

One shop says '25% off'. Another says 'Pay just 3/4 of the price'. Which is cheaper? Spot the multiplier hiding in each sentence and it takes one line.

The multiplier machine

Feed it any amount. What changes, and what never does?

the amount going in: 4/8. Goes in £40. Comes out £30.00. Out is always this much of in 3/4 = 75% = 0.75

Goes in£40Comes out£30.00Out is always this much of in3/4 = 75% = 0.75

This machine works out 3/4 of whatever you put in. Drag the bar to change the amount going in. Two readouts move as you drag. One of them never moves at all.

Sort by multiplier

Different words, same multiplication

Every statement below is secretly one multiplication. Put each one in the bin with its multiplier. For 'increase by' or 'decrease by', start from the whole amount (100%) and ask: what percentage do I end up with?

Still to sort

× 0.25 (0)

You end up with a quarter (25%) of the amount.

Where the line is: 'Decrease by 25%' does not go here. Taking 25% off leaves 75%, not 25%.

× 0.75 (0)

You end up with three quarters (75%) of the amount.

Where the line is: 'Decrease by 75%' does not go here. Taking 75% off leaves only 25%.

× 1.25 (0)

You keep the whole 100% and get 25% more on top: 125%.

Where the line is: Bigger than 1, so the amount grows. An increase of 25% is × 1.25, not × 0.25.

× 1.75 (0)

You keep the whole 100% and get 75% more on top: 175%.

Where the line is: An increase of 75% adds 75% to the 100% you started with, so the multiplier is above 1.

10 of 10 still to sort.

Finished? Look at which sentences ended up sharing a bin. Wordings that look nothing alike turn out to be the same machine, and that settles the shop question from the start: neither shop is cheaper.

Which multiplier?

The 15%-off trap

A jacket costs £60. In a sale, the price is decreased by 15%. You want the sale price in one step.

Which is closest to what you would do?

Multiply by 0.15. It's a 15% decrease, so the multiplier is 0.15.Multiply by 0.85. After 15% comes off, 85% of the price is left.Work out 60 − 15 = 45. It's 15% off, so take 15 away.Multiply by 1.15. The percentage is 15, and a multiplier has a 1 in front.

How sure are you?

I'm sureI think soI'm guessing

Multiply by 0.15. It's a 15% decrease, so the multiplier is 0.15.

This is a common misconception

£60 × 0.15 = £9. That's a real number, but it's the amount taken OFF, not the price you pay. The multiplier has to be what you end up with: 100% − 15% = 85%, so £60 × 0.85 = £51.

Multiply by 0.85. After 15% comes off, 85% of the price is left.

This matches the evidence

Spot on. 100% − 15% = 85%, and £60 × 0.85 = £51. One multiplication takes you straight to the sale price, with no need to find the 15% first.

Work out 60 − 15 = 45. It's 15% off, so take 15 away.

This is a common misconception

This treats 15% as £15. But 15% of £60 is only £9. A percentage is a share of the original, so you can't subtract it as if it were pounds. Use the multiplier instead: £60 × 0.85 = £51.

Multiply by 1.15. The percentage is 15, and a multiplier has a 1 in front.

This is a common misconception

£60 × 1.15 = £69, so the price went UP. 1.15 is the multiplier for a 15% increase. A decrease has to shrink the amount, so its multiplier is less than 1: 100% − 15% = 85%, so × 0.85.

For a decrease, ask: what percentage is LEFT? That's your multiplier. 15% off leaves 85%, so × 0.85.

Your turn

A percentage increase in one line

A gym has 1 250 members. Over a year, membership increases by 12%. How many members does it have now?

An increase of 12% means the gym ends up with 100% + 12% = 112% of its old membership.
missing step

Which line is step 2?

So the multiplier is 0.12.So the multiplier is 1.12.So the multiplier is 0.88.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

'Of', 'increase by' and 'decrease by' all come down to one multiplication.

What you need to know

An operator is a number that acts on an amount by multiplying it. Fractions and percentages can both be operators.
'3/4 of' and '75% of' are the same operator written two ways: both multiply by 0.75.
Increases and decreases are operators too. Start from the whole original, which is 100%.
The multiplier is what you END UP WITH compared with the original, never the change on its own.

The big picture

A fraction or a percentage can act on an amount as a multiplier, called an operator. 'A fraction of', 'a percentage of', 'increase by' and 'decrease by' each turn into one multiplication: find the multiplier, then multiply once. The multiplier is what you end up with, as a fraction or percentage of the original.

Key points

1Every wording becomes one move: find the multiplier, then multiply once.

2Percentage to multiplier: divide by 100. 35% becomes 0.35, and 140% becomes 1.4.

3Increase by p%: multiply by (100 + p)%. Decrease by p%: multiply by (100 − p)%.

4A multiplier above 1 makes the amount grow, below 1 makes it shrink, and exactly 1 (100%) leaves it unchanged.

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Worked example

Problem

Which is more: 3/5 of £45, or 45% of £60?

⚠ Watch out

Multiplying by the percentage change itself. For a 10% decrease, × 0.1 only finds the amount taken off. You keep 90%, so the multiplier is 0.9.

🧠

Memory hook

Find what you END UP with. That's the multiplier. Then multiply once.

✓

Check yourself

Write the multiplier for: (a) 2/5 of an amount, (b) a 40% increase, (c) a 40% decrease. Answers: 0.4, 1.4, 0.6. Notice (b) and (c) sit either side of 1.

Flashcards

(8)

What is an operator?

A multiplier that acts on an amount: whatever goes in comes out multiplied by the same number.

What single multiplier is '3/8 of'?

× 3/8, which is × 0.375. With 'of', the fraction itself is the multiplier.

How do you turn a percentage into a multiplier?

Divide it by 100. For example, 64% becomes × 0.64.

What is the multiplier for an increase of 20%?

× 1.2. You keep 100% and add 20%, so you end up with 120%.

What is the multiplier for a decrease of 20%?

× 0.8. Take 20% off 100% and 80% is left.

Why is × 0.15 wrong for a 15% decrease?

It only finds the 15% that is taken off. You end up with 85%, so the multiplier is 0.85.

'B is 3/4 of A.' Say it as a percentage.

B is 75% of A. The fraction and the percentage describe the same multiplier, 0.75.

An amount is multiplied by 1.6. Is it bigger, smaller or the same?

Bigger. 1.6 is 160%, an increase of 60%. Any multiplier above 1 makes the amount grow.

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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Different words, same multiplication

The 15%-off trap

A percentage increase in one line

More AQA GCSE Maths topics