Counting Money and Making Change

The idea in one sentence

Coins are worth different amounts, so counting money means counting value, not counting coins.

Start with the biggest

Six coins can be worth less than two. To total a handful, sort by value and count down from the largest.

A £1, two 20p, and a 5p: start at 100, add 20 to get 120, add 20 more for 140, then 5 for

1. That’s £1.45.

Going largest-first keeps the running total simple, because you’re adding smaller and smaller adjustments to a number you already have. Starting with the small coins means recounting the big ones into an awkward total.

Two ways to write it

£1.45 and 145p are the same amount. The decimal point separates whole pounds from the pence left over, which is exactly place value doing its usual job — the columns after the point are tenths and hundredths of a pound, and a penny is one hundredth.

That makes money the most useful decimal practice there is, because the amounts are real and mistakes are obvious. Nobody needs convincing that £0.50 is more than £0.05.

One writing rule: two digits after the point, always. £3.5 should be written £3.50. Amounts under a pound are usually written in pence — 45p rather than £0.45 — though both are correct.

Making change by counting on

The school method is subtraction. The shopkeeper’s method is faster and less error-prone: start from the price and count up to what was handed over.

Buy something for £3.40 with a £5 note:

• 10p takes you to £3.50
• 50p takes you to £4.00
• £1 takes you to £5.00

Change is £1.60, and you’ve also worked out which coins to hand over. Counting on turns a borrowing problem into a series of easy steps to the next round number.

Which coins to use

Given a total, the fewest-coins answer comes from taking the largest coin that fits, then repeating. For 87p: 50p leaves 37p, 20p leaves 17p, 10p leaves 7p, 5p leaves 2p, then two 1p coins. Six coins.

This greedy approach works for British and American coins because of how the values are chosen. It isn’t guaranteed to work for every possible set of coin values, which is a genuinely interesting fact — with coins of 1, 3, and 4, making 6 greedily gives 4+1+1 when 3+3 is better.

Common mix-ups

• Counting coins instead of value. Five 1p coins lose to one 10p.
• Mixing units. Add pence to pence and pounds to pounds, or convert first.
• Writing £0.5 for fifty pence. Two decimal places.
• Forgetting the change from a round number. £10 minus £6.35 is £3.65, and the 65 comes from counting up to the next pound.

Check yourself

You buy a book for £6.75 and pay with a £10 note. What’s the change, counted on?

25p reaches £7, then £3 reaches £10. The change is £3.25.