Strange bank account (part 2)
Investigate different ways of making £5 at Charlie's bank.
Age
11 to 14
Challenge level
Problem
This problem follows on from Strange Bank Account.
Image
In Charlie's Bank you are only allowed to deposit £2 at a time and withdraw £3 at a time. You can also cancel transactions.
Alison found a way of increasing her account balance by £5:
Seven deposits and three withdrawals:
(+ £2) + (+ £2) + (+ £2) + (+ £2) + (+ £2) + (+ £2) + (+ £2) + (- £3) + (- £3) + (- £3)
which Alison wrote as $7\times (+ £2) + 3 \times (- £3)$
She then found another way:
One deposit and cancelling one withdrawal, which Alison wrote as $(+ £2) - (- £3)$
Are there other ways in which Alison can increase the amount of money in her account by £5? How many ways?
Can Alison change the balance in her account by other amounts in many different ways?
With thanks to Don Steward, whose ideas formed the basis of this problem.
Getting Started
Here is a table showing some ways of obtaining a £5 increase:
Can you continue the table in each direction?
| Deposits $(+ £2)$ | Withdrawals $(- £3)$ | Calculation | Outcome |
| ... | ... | ... | ... |
| $4$ | $1$ | $4 \times (+ £2) + 1 \times (- £3)$ | $+ £5$ |
| $7$ | $3$ | $7 \times (+ £2) + 3 \times (- £3)$ | $+ £5$ |
| $10$ | $5$ | $10 \times (+ £2) + 5 \times (- £3)$ | $+ £5$ |
| ... | ... | ... | ... |
Can you continue the table in each direction?
Student Solutions
We had lots of correct solutions submitted for this problem, so well done to everyone who had a go!
Christopher from High Rock Middle School explains why Alison can make $ £5$ in infinitely many ways:
There are infinitely many ways that Alison can increase the amount of money
in her account by £$5$. That's because she can initially increase her account to 5$5$
by depositing four lots of £$2$ four times and then withdrawing £$3$.
Then, you can keep on depositing £$2$ three times and withdrawing £$3$ two times on top of this, because it cancels out to an overall change of £$0$. So you can keep on adding (+ £$2$+ £$2$+ £$2$- £$3$- £$3$), and the amount of ways keeps on growing.
Sam from Bridgewater High School found a formula to explain this:
She can raise her balance by £$5$ by depositing (or cancelling if the number
is negative) the following number of transactions:
For any given integer $n$:
$3n+1$ lots of £$2$
$1-2n$ lots of £$3$
$2(3n+1)+3(1-2n)=(6n+2)+(3-6n)=5$.
Chinat from Harrow International School Hong Kong explains how Alison can make any amount of money in her account:
For other ways to change, Alison can increase the account balance by £$1$ by
either:
- Doing $2$ deposits and doing $1$ withdrawal: $2 \times ( £2) + 1 \times (- £3) = £1$
- Cancelling $4$ deposits and cancelling $3$ withdrawals: $4 \times (- £2) + 3 \times ( £3) = £1$
By repeating these sequences of transactions enough times, we can raise the amount of money in Alison's bank account by any number of pounds.
For other ways to change, Alison can decrease the account balance by £$1$ by
either:
- Cancelling $2$ deposit and cancelling $1$ withdrawal: $2 \times (- £2) + 1 \times ( £3) = - £1$
- Doing $4$ deposits and doing $3$ withdrawals: $4 \times ( £2) + 3 \times (- £3) = - £1$
By repeating these sequences of transactions enough times, we can decrease the amount of money in Alison's bank account by any number of pounds.
To conclude, there are infinite ways to change the balance by any amounts
as long as the transactions are in the multiple of the sets above.
Christopher from High Rock Middle School explains why Alison can make $ £5$ in infinitely many ways:
There are infinitely many ways that Alison can increase the amount of money
in her account by £$5$. That's because she can initially increase her account to 5$5$
by depositing four lots of £$2$ four times and then withdrawing £$3$.
Then, you can keep on depositing £$2$ three times and withdrawing £$3$ two times on top of this, because it cancels out to an overall change of £$0$. So you can keep on adding (+ £$2$+ £$2$+ £$2$- £$3$- £$3$), and the amount of ways keeps on growing.
Sam from Bridgewater High School found a formula to explain this:
She can raise her balance by £$5$ by depositing (or cancelling if the number
is negative) the following number of transactions:
For any given integer $n$:
$3n+1$ lots of £$2$
$1-2n$ lots of £$3$
$2(3n+1)+3(1-2n)=(6n+2)+(3-6n)=5$.
Chinat from Harrow International School Hong Kong explains how Alison can make any amount of money in her account:
For other ways to change, Alison can increase the account balance by £$1$ by
either:
- Doing $2$ deposits and doing $1$ withdrawal: $2 \times ( £2) + 1 \times (- £3) = £1$
- Cancelling $4$ deposits and cancelling $3$ withdrawals: $4 \times (- £2) + 3 \times ( £3) = £1$
By repeating these sequences of transactions enough times, we can raise the amount of money in Alison's bank account by any number of pounds.
For other ways to change, Alison can decrease the account balance by £$1$ by
either:
- Cancelling $2$ deposit and cancelling $1$ withdrawal: $2 \times (- £2) + 1 \times ( £3) = - £1$
- Doing $4$ deposits and doing $3$ withdrawals: $4 \times ( £2) + 3 \times (- £3) = - £1$
By repeating these sequences of transactions enough times, we can decrease the amount of money in Alison's bank account by any number of pounds.
To conclude, there are infinite ways to change the balance by any amounts
as long as the transactions are in the multiple of the sets above.
Teachers' Resources
Why do this problem?
This problem builds on the problem Strange Bank Account and the game Up, Down, Flying Around, to explore both addition and subtraction of positive and negative numbers. We suggest teachers read the article Adding and Subtracting Positive and Negative Numbers to see a variety of contexts that can be used to develop understanding of operations with directed numbers.Possible approach
Remind students of the rules from Strange Bank Account."Can you find a couple of different ways of increasing the amount in Alison's account by £5?"
Tabulate suggestions on the board:
| Deposits $(+ £2)$ | Withdrawals $(- £3)$ | Calculation | Outcome |
| ... | ... | ... | ... |
| $4$ | $1$ | $4 \times (+ £2) + 1 \times (- £3)$ | $+ £5$ |
| $7$ | $3$ | $7 \times (+ £2) + 3 \times (- £3)$ | $+ £5$ |
| $10$ | $5$ | $10 \times (+ £2) + 5 \times (- £3)$ | $+ £5$ |
| ... | ... | ... | ... |
"Have a look at the suggestions so far. What do you notice?"
"Are there any patterns that we could continue?"
"Could we continue the table upwards as well as downwards?"
To answer this last question, it may be necessary to introduce the idea of cancelling transactions. For example, $1 \times (+ £2) - 1 \times (- £3)$ can be interpreted as a deposit of £2 and then cancelling a previous £3 withdrawal, leading to a £5 increase in balance.
Students could then investigate making other amounts in many different ways.
Key questions
Why does the number of withdrawals increase/decrease by 2 as the number of deposits increases/decreases by 3, if we keep the total outcome the same?
Can all outcomes be made in many ways?
Possible extension
Once students are confident at manipulating positive and negative numbers, Weights offers an interesting investigation.
Possible support
Play the variants of Up, Down, Flying Around to make sure students' understanding of operations involving directed numbers is secure. The article Adding and Subtracting Negative Numbers offers different models for helping students to make sense of negative numbers.This problem is based on Don Steward's ideas. His resources can be found here.