Good Learning Co.
SavannahYear 8
← Maths
MathsThe maths behind the money30 min★★★ difficulty

Compound interest with a formula

Compound interest is interest on interest. Substituting into one formula tells you what savings will grow to — and what a debt will grow to if you leave it alone, which is the same maths pointed the other way.

Lesson overview

What you'll learn in this lesson

Substitute numerical values into formulae and expressions, including scientific formulae

Key learning points

  • • The formula and how to substitute
  • • Small differences, large effects
  • • The same formula for debt
  • • Real returns after inflation

This lesson at a glance

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

Words to know

substitutenumericalvaluesformulaeexpressions

Scroll down — the lesson carries on below

Part 1 of 18 · Discover6%
1

Watch & discover

Part 1 of 18

Watch first

Compound interest with a formula

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

In a nutshell

Substitute numerical values into formulae and expressions, including scientific formulae

2

What you already know

Part 2 of 18

Before we start

What you already know

You can use decimal multipliers for percentage change and evaluate powers.

Recap: Percentage change: interest, discounts and original prices
3

Key words

Part 3 of 18

Key words

Words you'll need today

Principal
The starting amount saved or borrowed, written P.
Compound interest
Interest added to the balance so that later interest is calculated on it too.
AER
Annual Equivalent Rate, the standardised yearly rate for comparing savings accounts.
Formula
A rule written with letters, into which values are substituted.
Real return
Growth after inflation is taken off.
4

Learning cycle

Part 4 of 18

Learning cycle 1 of 2

Substituting into the formula

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

5

Explore the idea

Part 5 of 18

Learn

The formula and how to substitute

For a principal P, an annual rate r as a percentage, and n years, the balance is A = P(1 + r/100)^n. Substituting carefully means doing the bracket first. For P = 500, r = 4, n = 3: 1 + 4/100 = 1.04, then 1.04³ = 1.124864, then A = 500 × 1.124864 = £562.43 to the nearest penny. Interest earned is £62.43, whereas simple interest would have given only £60.

6

Explore the idea

Part 6 of 18

Learn

Small differences, large effects

Substitution lets you test scenarios instead of guessing. £2,000 at 5% for ten years gives 2,000 × 1.05¹⁰ = £3,257.79. At 3% it is £2,687.83 — a gap of about £570 from two percentage points. Doubling the time roughly squares the multiplier, which is why starting early matters far more than most people expect.

7

Quick check

Part 7 of 18

Quick check

£500 at 4% compound for 3 years gives

8

Reset break

Part 8 of 18

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
9

Quick check

Part 9 of 18

Quick check

In A = P(1 + r/100)^n, what do you calculate first?

10

Learning cycle

Part 10 of 18

Learning cycle 2 of 2

Growth, debt and inflation

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

11

Explore the idea

Part 11 of 18

Learn

The same formula for debt

Unpaid debt compounds too. £1,500 on a card at 24% a year, untouched, becomes 1,500 × 1.24³ = £2,858.98 after three years. Card interest is usually applied monthly, which compounds faster still: use r as the monthly rate and n as the number of months. Compounding is neutral — it rewards savers and punishes borrowers with identical arithmetic.

12

Explore the idea

Part 12 of 18

Learn

Real returns after inflation

A savings account paying 4% while inflation runs at 3% gives a real return of roughly 1%. Your balance grows but your buying power barely does. A useful way to see it is 1.04 ÷ 1.03 ≈ 1.0097. When comparing accounts, use the AER, which standardises how often interest is paid, and check that the rate is not a short-term bonus that drops after a year.

13

Quick check

Part 13 of 18

Quick check

£2,000 at 5% compound for 10 years is closest to

14

Quick check

Part 14 of 18

Quick check

Interest 4% with inflation 3% gives a real return of about

15

Common mix-ups

Part 15 of 18

Common mix-ups

Things people often get wrong

  • Multiplying P by r × n. That is simple interest and undercounts compound growth.
  • Forgetting to evaluate the bracket before the power.
  • Writing 4% as 4 in the multiplier. It becomes 1.04.
  • Using an annual rate with a monthly period, or the reverse.
  • Ignoring inflation and treating nominal growth as a real gain.
16

Challenge round

Part 16 of 18

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

17

Mastery quiz

Part 17 of 18

Marked quiz

End of lesson quiz: Compound interest with a formula

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

  1. 1. The multiplier for 6% is

  2. 2. £1,500 at 24% compound for 3 years is about

  3. 3. Compound interest differs from simple interest because

  4. 4. AER is used to

  5. 5. For monthly compounding you should use

  6. 6. Starting to save earlier helps because

18

Lesson round-up

Part 18 of 18

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

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

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. 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

Saved as you type.

Part 1 of 18 · Watch & discover

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