Five More Coins
This problem follows on from Five Coins so we would recommend looking at that one first.
How much money might he have? If possible, talk to someone else about your ideas.
Can you name an amount of money that Ben's five coins couldn't add up to? Why couldn't he have that amount of money?
Dayna is trying to guess how much money Ben has. She knows that altogether he has less than £1.
Dayna starts writing down all the amounts of money less than £1 that Ben's five coins could add up to.
Could Ben have any amount of money between 5p and £1 in his pocket, or do you think there are some amounts that would be impossible for him to have?
Have a go at writing out Dayna's list of numbers, showing how Ben could make each total with five coins. If there are any numbers missing from the list, can you explain why these amounts are impossible for Ben to have?
Well done to everybody who had a go at this activity. Xi from George Heriot's School in the UK sent in this solution:
First I thought all five coins were the same:
1p 1p 1p 1p 1p
10p 10p 10p 10p 10p
Second I thought there were two kinds of coins:
2p 1p 1p 1p 1p
50p 5p 5p 5p 5p
Third I thought there were three kinds of coins:
1p 1p 1p 2p 5p
5p 5p 5p 10p 20p
Fourth I thought there were four kinds of coins:
1p 1p 20p 50p 10p
Finally all the coins are different:
10p 1p 50p 20p 5p
£1 coins and £2 coins are missing from the list because the adding up amount would be more than £1.
This is a good explanation of one way to approach this task systematically, Xi! I wonder if there are any other possibilities where all the coins are different?
We received a solution to this task from a child in Monkfield, England, saying:
You can't make 98p with 5 coins.
Why is 98p not possible with 5 coins? Are there any other amounts under £1 that are not possible to make with 5 coins?
We also received this account from a teacher describing how they approached this task:
Our school, Barford St Peters Primary in Warwickshire, has an after-school Maths Club for juniors where we often look at NRICH problems. I explained this problem to them, and most of them decided to look at the different possible total amounts, from 1p up to 100p, and worked out if this was possible with five coins, and if so, which five coins would do it. Dean, Liam and Andrew worked up to about 40p in the time available, and decided that only 1p, 2p, 3p and 4p were impossible with five coins.
Then, in his own time, Andrew decided that he wanted to consider, for a given number of coins, what was the smallest total which required that number of coins. His first answer was that 38p was the smallest amount which required five coins, 88p the smallest to need six coins. He worked out that £5.88 was the smallest to need nine coins (£2, £2, £1, 50p, 20p, 10p, 5p, 2p, 1p), but that £7.88 needed ten coins (another £2 plus the ones before). He reckoned £17.88 needed fifteen coins, £27.88 needed twenty coins, £37.88 needed twenty five coins, £187.88 needed one hundred coins, and then went off into the realms of trillions of coins and googols of coins!
You've thought really hard about this, Andrew! I wonder what would happen with your larger amounts of money if we changed the activity slightly and said that Ben had 'five coins or notes' in his pocket instead?
So far, we haven't received any solutions for what Dayna's full list of numbers might be, or explanations for which amounts of money would be impossible for Ben to have in his pocket. Please do email us to let us know if you have any ideas.
Five More Coins
This problem follows on from Five Coins so we would recommend looking at that one first.
How much money might he have? If possible, talk to someone else about your ideas.
Can you name an amount of money that Ben's five coins couldn't add up to? Why couldn't he have that amount of money?
Dayna is trying to guess how much money Ben has. She knows that altogether he has less than £1.
Dayna starts writing down all the amounts of money less than £1 that Ben's five coins could add up to.
Could Ben have any amount of money between 5p and £1 in his pocket, or do you think there are some amounts that would be impossible for him to have?
Have a go at writing out Dayna's list of numbers, showing how Ben could make each total with five coins. If there are any numbers missing from the list, can you explain why these amounts are impossible for Ben to have?
Why do this problem?
This activity is an interesting context in which to practise addition and subtraction, and it helps learners to become more familiar with coin denominations. It requires a systematic approach and recording is key.
Possible approach
This task follows on from Five Coins, and so it would be worth having a go at that problem before tackling this one.
Introduce the first part of the problem, with Ben having five coins in his pocket. Give them a few minutes to talk to a partner about how much money Ben might have, and then ask some children to share their thoughts. Start to write up their suggestions on the board, for example by listing the five coins and the total. (You may wish to list all the different coins for reference.)
Challenge learners to suggest an amount of money that they think would be impossible to make from five coins, and importantly, why that total is impossible. For example, some pairs might offer a total that is less than 5p, or more than £10; some might suggest an amount which includes a fraction of a penny, such as 20.5p. (You may wish to explain that there used to be a halfpenny coin, but it was removed from circulation at the end of 1984.)
Key questions
If you can't find a way of making a particular total, are you certain you have tried all possibilities? Or is this total impossible? How do you know?
Possible support
Five Coins is a good problem to try before this one. Having coins at the ready will be helpful for many pupils.
Possible extension
Encouarge learners to tweak the task themselves by asking "I wonder what will happen if we...?"