KS3 · Maths

Percentage increase and decrease

If a price rises by 13% and then falls by 13%, is it back where it started? Hold that thought. One idea answers it: the original is always 100%.

Maths · Percentage change

Pin the original at 100%, then slide

Cells shaded (each cell is 5%): 20/30. Percentage of the original 100%. What happened no change. Multiplier 100 ÷ 100 = 1. £80 becomes £80. 450 becomes 450

Percentage of the original100%What happenedno changeMultiplier100 ÷ 100 = 1£80 becomes£80450 becomes450

Two different amounts, £80 and 450, both start at 100%. Drag the marker along the bar: both amounts change in step, and the multiplier is the same for both.

Watch out: Whether you can go past 100% depends on the situation. You can't score 110% on a test or eat 110% of a cake, but the price of a house can increase by 110%.

Same question, different words

Which multiplier does the job?

Each card describes a change to an amount. Put it with the single multiplier that makes that change in one step.

Still to sort

× 1.35 (0)

1 + 0.35: the new amount is 135% of the original.

× 0.65 (0)

1 − 0.35: 65% of the original is left.

Where the line is: Only 0.01 away from × 0.66. Read the percentage carefully: 35% off leaves 65%, 34% off leaves 66%.

× 0.66 (0)

1 − 0.34: 66% of the original is left.

× 3 (0)

1 + 2: the new amount is 300% of the original.

7 of 7 still to sort.

Watch out: 'Increase by 35%' and 'decrease by 35%' land in different piles. The same percentage gives a different multiplier for an increase and a decrease.

No calculator? No problem

Fill in the missing steps

Without a calculator, work out 48 increased by 32%, and then 48 decreased by 68%.

Increasing by 32% means finding 100% + 32% = 132% of 48, and 132 ÷ 100 gives the multiplier.
missing step

Which line is step 2?

The multiplier is 1.32The multiplier is 1.032The multiplier is 0.68

What do you think?

The 5% trap

You want to increase £40 by 5% in one step.

Which multiplier would you use? Pick the one closest to what you really think.

× 1.5, because the 5 goes after the 1.× 0.05, because 5% as a decimal is 0.05.× 1.05, because the new amount is 105% of the original.

How sure are you?

I'm sureI think soI'm guessing

× 1.5, because the 5 goes after the 1.

This is a common misconception

Tempting, but 1.5 is 150% — that's a 50% increase: £40 × 1.5 = £60. A percentage only becomes a decimal when you divide by 100, so 5% is 0.05, not 0.5. The multiplier is 1 + 0.05 = 1.05.

× 0.05, because 5% as a decimal is 0.05.

This is a common misconception

Your conversion is spot on: 5% is 0.05. But £40 × 0.05 = £2, and £2 is only the increase — the extra bit. The question wants the new amount, which is 100% + 5% = 105% of £40, so the multiplier is 1.05.

× 1.05, because the new amount is 105% of the original.

This matches the evidence

Yes. The original is 100%; add 5% and you want 105%, and 105 ÷ 100 = 1.05. £40 × 1.05 = £42: the £40 you started with plus the £2 increase, in one step.

Divide the percentage by 100, then add it to 1. +5% is × 1.05, and × 1.5 would be a 50% increase.

Predict, then check

Commit to an answer before you reveal. Imagine a jacket that starts at £100.

A price goes up by 13%. Then the new price goes down by 13%. Where does it end up?

Spot the slip

Where does this sale price go wrong?

A T-shirt costs £25. In a sale, its price is reduced by 24%. Find the sale price.

A student's answer — which line goes wrong?

The original price, £25, is 100%.
24% as a decimal is 24 ÷ 100 = 0.24.
The multiplier is 1 + 0.24 = 1.24.
25 × 1.24 = 31
The sale price is £31.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The original is always 100%. Every increase or decrease is just a new percentage of it, and one multiplier finds it.

What you need to know

The original amount is always 100%, which is 1 as a decimal.
Increase by x% means find (100 + x)% of the original. Decrease by x% means find (100 − x)%. Decreasing 600 by 10% is the same as finding 90% of 600: 540.
On a bar model, find 10% first. 5% is half of 10%, and 1% is a tenth of it. Increasing 240 by 35%: 240 + 3 × 24 + 12 = 324.
The multiplier is the new percentage divided by 100. Increase by 20%: × 1.2. Decrease by 20%: × 0.8.
After a change, the new amount becomes the new 100%.

The big picture

Percentage change always starts from the same place: the original amount is 100%. Increasing by a percentage means finding more than 100% of the original, and decreasing means finding less. You can build the answer from 10%, 5% and 1% on a bar model, or do it in one step with a multiplier: the new percentage divided by 100.

Key points

1For an increase, add the percentage as a decimal to 1: +35% is × 1.35, +4% is × 1.04, +200% is × 3.

2For a decrease, take it away from 1: −35% is × 0.65, −4% is × 0.96, −2.3% is × 0.977.

3Make the decimal by dividing by 100: 5% is 0.05, so +5% is × 1.05, never × 1.5.

4Increasing and decreasing by the same percentage use different multipliers: +23% is × 1.23 but −23% is × 0.77.

5Without a calculator, split the multiplier: × 1.32 is the same as × 132 then × 0.01.

6Whether a change can go past 100% depends on the context: a house price can rise by 110%, but nobody scores 110% on a test.

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Worked example

Problem

A bike costs £640. Its price goes up by 41%. What is the new price?

⚠ Watch out

Writing 5% as 1.5 in a multiplier. A percentage becomes a decimal by dividing by 100, so 5% is 0.05 and an increase of 5% is × 1.05. Multiplying by 1.5 is a 50% increase.

🧠

Memory hook

Start at 100. Step up or down by the percentage. Divide by 100. Multiply once. (+20% → 120 → × 1.2; −20% → 80 → × 0.8.)

✓

Check yourself

Without a calculator: what single number increases an amount by 7%? What single number decreases it by 7%? Why does doing one straight after the other not take you back to the start?

Flashcards

(13)

What percentage is the original amount?

Always 100%, which is 1 as a decimal.

'Increase by x%' is the same as finding what?

(100 + x)% of the original. Increase 70 by 12% = find 112% of 70.

'Decrease by x%' is the same as finding what?

(100 − x)% of the original. Decrease 780 by 21% = find 79% of 780.

How do you turn a percentage into a decimal?

Divide by 100: 35% → 0.35 and 4% → 0.04.

Multiplier to increase by 35%?

1.35: add 0.35 to 1.

Multiplier to decrease by 34%?

0.66: take 0.34 away from 1.

Multiplier to increase by 200%?

3, because the new amount is 300% of the original.

Why is the multiplier for a 5% increase not 1.5?

5% is 0.05, so the multiplier is 1.05. × 1.5 is a 50% increase.

How do you get 5% and 1% quickly?

Find 10% by dividing by 10. 5% is half of 10%, and 1% is 10% divided by 10.

Using 100%, 50% and 5%, what is 155% of 120?

120 + 60 + 6 = 186.

How can you work out 700 × 1.38 without a calculator?

700 × 138 = 96 600, then × 0.01 gives 966.

Up 13%, then down 13%: back to the start?

No. 1.13 × 0.87 = 0.9831, so you end at 98.31% of the original.

Can a percentage increase be more than 100%?

It depends on the context: a house price can rise by 110%, but you cannot score 110% on a test.

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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Which multiplier does the job?

Fill in the missing steps

The 5% trap

Where does this sale price go wrong?

More KS3 Maths topics