MathsThe maths behind the money30 min★★★ difficulty

Percentage change: interest, discounts and original prices

Sale signs, pay rises, interest rates and VAT are all one skill: percentage change. The hardest and most useful version is working backwards to the original amount, which is how you catch a fake discount.

Lesson overview

What you'll learn in this lesson

Solve problems involving percentage change, including percentage increase and decrease and original value problems

Key learning points

• Increase and decrease with one multiplier
• Simple interest on savings
• Finding the original price
• Spotting a misleading claim

This lesson at a glance

30 minutes
19 parts to scroll through
4 quick checks
Marked quiz at the end
Gentle pace: short sittings with pauses

Words to know

problemsinvolvingpercentagechangeincrease

Scroll down — the lesson carries on below

Part 1 of 19 · Discover5%

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Part 1 of 19

Watch first

Percentage change: interest, discounts and original prices

An English teaching video on this topic from mathantics. Watch it, then scroll down for the lesson.

In a nutshell

Solve problems involving percentage change, including percentage increase and decrease and original value problems

What you already know

Part 2 of 19

Before we start

What you already know

You can convert between fractions, decimals and percentages and use decimal multipliers.

Recap: Percentages of money you can do in your head

Key words

Part 3 of 19

Key words

Words you'll need today

Percentage increase: A rise expressed as a percentage of the starting value.
Percentage decrease: A fall expressed as a percentage of the starting value.
Multiplier: The decimal you multiply by to apply a change, such as 1.05 or 0.9.
Original value problem: Working back from a changed amount to the amount before the change.
Simple interest: Interest calculated only on the original amount each period.

Learning cycle

Part 4 of 19

Learning cycle 1 of 2

Applying a change

A short piece of teaching, then a check to make sure it has landed.

Explore the idea

Part 5 of 19

Learn

Increase and decrease with one multiplier

A 5% pay rise on £1,800 is £1,800 × 1.05 = £1,890. A 30% discount on £45 trainers is £45 × 0.7 = £31.50. Build the multiplier from 1: add the decimal for an increase, subtract it for a decrease. To express a change as a percentage, divide the change by the original and multiply by 100: rent rising from £650 to £689 is 39/650 = 0.06 = a 6% increase.

Explore the idea

Part 6 of 19

Learn

Simple interest on savings

Simple interest pays a fixed percentage of the original sum each year. £800 at 3% simple interest earns £24 a year, so after four years the interest is £96 and the balance is £896. Real savings accounts usually pay compound interest instead, which is why the next lesson matters, but simple interest is the right model for many fixed-term arrangements and a useful first estimate.

Quick check

Part 7 of 19

Quick check

£1,800 after a 5% rise is

Quick check

Part 8 of 19

Quick check

£45 with 30% off costs

Reset break

Part 9 of 19

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now

Learning cycle

Part 10 of 19

Learning cycle 2 of 2

Working backwards

A short piece of teaching, then a check to make sure it has landed.

Explore the idea

Part 11 of 19

Learn

Finding the original price

A jacket costs £42 in a 25% off sale. The £42 is 75% of the original, so divide: 42 ÷ 0.75 = £56. A restaurant bill of £54 includes 20% service, so the food was 54 ÷ 1.2 = £45. Reversing a change means dividing by the multiplier, never applying the same percentage to the new figure. Taking 25% of £42 gives £10.50 and a wrong answer of £52.50.

Explore the idea

Part 12 of 19

Learn

Spotting a misleading claim

A price rises 20% then falls 20%: £100 → £120 → £96. You are not back where you started, because the second percentage is taken from a bigger number. Advertisers use this. 'Was £80, now £60' is a 25% reduction, not 20%, because the change is measured against the original. Checking the arithmetic is genuinely how you avoid overpaying.

Quick check

Part 13 of 19

Quick check

A coat is £42 after 25% off. The original price was

Quick check

Part 14 of 19

Quick check

£100 rises 20% then falls 20%. The final price is

Common mix-ups

Part 15 of 19

Common mix-ups

Things people often get wrong

Reversing a discount by adding the same percentage back. Divide by the multiplier instead.
Thinking +20% then −20% returns the original value. It does not.
Measuring change against the new value rather than the original.
Using simple interest where interest compounds, and underestimating growth.
Applying two discounts by adding them: 10% then 10% is 19% off, not 20%.

Challenge round

Part 16 of 19

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

Challenge round

Part 17 of 19

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

Build the multiplier from 1: add the decimal for an ____, subtract it for a decrease.

To express a change as a ____, divide the change by the original and multiply by 100: rent rising from £650 to £689 is 39/650 = 0.06 = a 6% increase.

Reversing a ____ means dividing by the multiplier, never applying the same percentage to the new figure.

Mastery quiz

Part 18 of 19

Marked quiz

End of lesson quiz: Percentage change: interest, discounts and original prices

6 questions, marked with the reasoning shown. No timer.

1. Rent rises from £650 to £689. That is
2. £800 at 3% simple interest for four years earns
3. A bill of £54 includes 20% service. The food cost
4. Which multiplier applies a 12% increase?
5. 'Was £80, now £60' is a reduction of
6. Two successive 10% discounts give a total reduction of

Lesson round-up

Part 19 of 19

Lesson round-up

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Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

Work out the total cost of the trip if 50 pupils go.
Show why the price per pupil falls when more pupils go, using numbers.
At £20 each, how many pupils must go before the trip makes money? Show your working.

Working out
Working out
Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
Writing
Copy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working out
Write your own real-life problem that uses today's method, then solve it.
1 problem + solution

Write your answer, notes or working here

Saved as you type.

Photos of paper work, diagrams or working outHow many minutes did the written work take?

Part 1 of 19 · Watch & discover

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