Myriora Maths › 11+ maths › Chapter 2: Classic word problems

Coins and making amounts

11+Chapter 2 · Lesson 13

In this lesson

For the fewest coins, take the biggest coin that fits, again and again.
For the number of ways, start with the most big coins, and work down.
And count which coins you use, not the order.

Your turn

How many ways can you make forty pence, using twenty pence, ten pence and five pence coins?

Show the answer

With two twenties, there's one way. With one twenty, make the other twenty pence from tens and fives: three ways. With no twenties, make forty pence from tens and fives: five ways. One plus three plus five is nine ways!

The common mistake

A common mistake is counting the same coins twice, in a different order. Ten, five, five, and five, ten, five are the same coins! The order doesn't make a new way: only count which coins you use.

Try it at home

Empty a purse: what's the fewest coins you need for 87p? How many ways can you make 20p?

Read the whole lesson

Hello, and welcome to Myriora Maths! Coin puzzles come up a lot in the 11 plus. There are two kinds. What's the fewest coins you need? And how many different ways can you make an amount?

Make eighty-seven pence with the fewest coins. Always take the biggest coin that fits. Fifty pence leaves thirty-seven. Twenty pence leaves seventeen. Ten pence leaves seven. Five pence leaves two. And then a two pence coin. That's five coins.

Why biggest first? A big coin does the work of lots of small ones. So taking the biggest coin each time leaves the least to make. With British coins, this always gives the fewest.

Now the second kind. How many ways can you make twenty pence, using ten pence, five pence and two pence coins? The trick is to be systematic: start with as many ten pences as you can, then fewer and fewer.

With two ten pences, that's twenty already: one way. With one ten pence, we need ten more: two fives, or five twos. That's two ways. With no ten pences, we need twenty: four fives, two fives and five twos, or ten twos. That's three ways. One plus two plus three: six ways altogether.

Why work in order? Going from the most big coins down to none, you can't miss a way, and you can't count one twice.

A common mistake is counting the same coins twice, in a different order. Ten, five, five, and five, ten, five are the same coins! The order doesn't make a new way: only count which coins you use.

Your turn! How many ways can you make forty pence, using twenty pence, ten pence and five pence coins? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! With two twenties, there's one way. With one twenty, make the other twenty pence from tens and fives: three ways. With no twenties, make forty pence from tens and fives: five ways. One plus three plus five is nine ways!

Quick review. For the fewest coins, take the biggest coin that fits, again and again. For the number of ways, start with the most big coins, and work down. And count which coins you use, not the order. See you next time!