GCSE · Maths · AQA · Spec 8300

Percentages — definition, conversion, comparison

In one year group, 42 of 120 students walk to school. In another, 36 of 80 do. Which has the bigger share of walkers? The bigger number might fool you.

Drag it and watch

Your class's charity target is £40

bar sections filled: 3/6. Percentage of target 75%. Raised £30. Same share of £100 £75. Raised ÷ target × 100 30 ÷ 40 × 100 = 75

Percentage of target75%Raised£30Same share of £100£75Raised ÷ target × 10030 ÷ 40 × 100 = 75

The bar runs from £0 to £60. The target of £40 is the whole, so it sits at the 100% mark, two-thirds of the way along. Drag the handle and watch every readout keep pace.

Watch out: Here the whole is the £40 target, not the full length of the bar. Everything past the 100% mark is more than the whole, and that is exactly what a percentage over 100% means.

Same number, three outfits

Put each value where it really lives

Treat 1 as one whole, so 100% sits at 1. To place a percentage, divide it by 100 (30% → 0.3). To place a fraction, divide the top by the bottom (3/4 → 0.75). Now drag each value to its spot, then check.

Place 30%Place 3/10Place 5%Place 0.75Place 5/4Place 130%

Percentages are multipliers

How would you find 15% of £40?

A game costs £40 and you need to find 15% of the price. Be honest: which of these is closest to what you'd do?

Pick the idea you hold right now, then say how sure you are.

Multiply by 15: 40 × 15Multiply by 1.5: 40 × 1.5Divide by 15: 40 ÷ 15Multiply by 0.15: 40 × 0.15

How sure are you?

I'm sureI think soI'm guessing

Multiply by 15: 40 × 15

This is a common misconception

That gives £600, which is fifteen whole games! The 15 isn't the multiplier. 15% means 15 per hundred, so the multiplier is 15 ÷ 100 = 0.15, and 40 × 0.15 = £6.

Multiply by 1.5: 40 × 1.5

This is a common misconception

1.5 is 150%, one and a half games, which is £60. The slip is moving the decimal point one place instead of two. Dividing by 100 takes 15 to 0.15, not 1.5.

Divide by 15: 40 ÷ 15

This is a common misconception

It's tempting, because 10% of an amount is the same as dividing by 10. But that's a coincidence of 10, not a rule. 40 ÷ 15 ≈ 2.67, which is 1/15 of the price, about 6.7%. For 15%, multiply by 0.15.

Multiply by 0.15: 40 × 0.15

This matches the evidence

Yes. 15% = 15/100 = 0.15, so 15% of £40 = 40 × 0.15 = £6. The same idea reads a percentage change too. After a 15% rise you have 100% + 15% = 115% of the original, so you multiply by 1.15. After a 15% fall you have 85% left, so you multiply by 0.85.

A percentage is a multiplier in disguise: divide it by 100 to get the number to multiply by. 15% of → × 0.15. 115% of → × 1.15. Careful: a 5% rise is × 1.05, but a 30% rise is × 1.3, not × 1.03.

Predict, then check

Commit to an answer before you calculate anything.

In Year 10, 42 of the 120 students walk to school. In Year 11, 36 of the 80 students walk. Which year has the greater proportion of walkers?

Spot the slip

Where does this answer go wrong?

Write 45p as a percentage of £1.50.

A student's answer — which line goes wrong?

The part is 45p and the whole is £1.50.
Part over whole: 45 / 1.50
45 ÷ 1.50 = 30
30 × 100 = 3000%

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Turn any amount into 'out of 100' of its own whole, and converting, comparing and going past 100% all become the same idea.

What you need to know

What 'per cent' really means
How to swap between percentages, fractions and decimals, including values over 100%
How to use a percentage as a multiplier
How to write one amount as a percentage of another, and use that to compare fairly

The big picture

A percentage is a number of parts per hundred of a whole. That one idea lets you write the same value as a percentage, a fraction or a decimal, use a percentage as a multiplier, write one amount as a percentage of another, and compare amounts that come from different-sized wholes. And because 100% is the whole, a percentage over 100% simply means more than the whole.

Key points

1A percentage is a number of parts per hundred: 35% means 35 out of every 100.

2A percentage, a fraction and a decimal can name the same number: 35% = 35/100 = 7/20 = 0.35.

3Percentage → decimal: divide by 100. Decimal → percentage: multiply by 100. Fraction → decimal: divide the top by the bottom.

4100% is the whole. A percentage over 100% means more than the whole: 150% of something is 1.5 times it.

5A percentage works as a multiplier: 35% of an amount is amount × 0.35. After a 35% rise you have 135% of it, which is × 1.35.

6To write A as a percentage of B: put them in the same unit, then work out A ÷ B × 100.

7To compare parts of different-sized wholes, turn each into a percentage of its own whole, then compare.

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Worked example

Problem

A seedling was 8 cm tall. A month later it is 14 cm tall. Write its new height as a percentage of its old height.

⚠ Watch out

Dividing the wrong way round. 'A as a percentage of B' is A ÷ B × 100: the amount after 'of' goes underneath. So 12 as a percentage of 48 is 12 ÷ 48 × 100 = 25%, not 400%.

🧠

Memory hook

Per CENT = per hundred. Think of the 100 centimetres in a metre. Then remember the method: same units, part over whole, times 100.

✓

Check yourself

Without a calculator, write 35% as a decimal and as a fraction in its simplest form. Which is bigger, 3/8 or 35%? (Answers: 0.35 and 7/20. 3/8 is bigger: it is 37.5%.)

Flashcards

(10)

What does 'per cent' mean?

Per hundred. 23% means 23 out of every 100.

How do you turn a percentage into a decimal?

Divide by 100, which moves the digits two places: 7% = 0.07 and 130% = 1.3.

How do you turn a decimal into a percentage?

Multiply by 100: 0.4 = 40% and 2.4 = 240%.

How do you turn a fraction into a percentage?

Divide the top by the bottom, then multiply by 100: 3/5 = 0.6 = 60%. Or scale it to 'out of 100': 3/5 = 60/100 = 60%.

What is 100% of a quantity?

The whole quantity, all of it.

What does a percentage over 100% mean?

More than the whole. 250% of an amount is 2.5 times that amount.

Which number do you multiply by to find 6% of an amount?

0.06, because 6 ÷ 100 = 0.06. Not 6 and not 0.6.

Which multipliers give an amount after a 20% increase, and after a 20% decrease?

× 1.2 (you have 120%) and × 0.8 (you have 80%).

How do you write A as a percentage of B?

Put A and B in the same unit, then A ÷ B × 100. The amount after 'of' goes underneath.

Why use percentages to compare two groups of different sizes?

Each part gets scaled to 'out of 100' of its own whole, so the comparison is fair even when the wholes differ.

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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Put each value where it really lives

How would you find 15% of £40?

Where does this answer go wrong?

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