Strange Bank Account (part 2)

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.