Strange Bank Account
Stange Bank Account printable sheet
In Charlie's Bank you are only allowed to deposit £2 at a time and withdraw £3 at a time.
Imagine Alison deposits £2, another £2, another £2, another £2 and then finally withdraws £3. She now has an extra £5 in her account.
What other amounts of money is it possible for Alison to change her account balance by?
Charlie's Bank wants to change its rules whilst ensuring that its customers can still change their account balance by any total.
If Alison is only allowed to deposit £3 at a time and withdraw £7 at a time, will she be able change her account balance by any total?
Explore some other deposit and withdrawal amounts.
If Alison is allowed to deposit £x and withdraw £y, what has to be true about x and y to make sure it is possible to change the account balance by any total?
You might like to play the game Up, Down, Flying Around, and then take a look at Strange Bank Account (part 2).
With thanks to Don Steward, whose ideas formed the basis of this problem.
Part 1: Alison is only allowed to deposit £2 at a time and withdraw £3 at a time.
Many people sent in examples of amounts Alison could change her account by. However, this was quite a sneaky question, because the answer is that she can change her account by any whole number! Well done to Nathan from Mossbourne, Dare from Canada and Kaedan, Rebbeca and an unnamed student from St Stephens School Australia who realised this.
Here is Nathan's explanation:
If she wants to increase by $x$ pounds, she can deposit 2$x$ lots of £2 and withdraw $x$ lots of £3. This creates a input of £4$x-$ £3$x$= £$x$.
To decrease by £$x$ deposit £2 $x$ times and withdraw £3 $x$ times.
Dare noted that Alison can increase her bank account by £1 by depositing £2 twice and withdrawing £3. She can do this repeatedly to add any positive whole number. If she withdraws £3 and deposits £2 she can decrease her account by £1, then she can repeat this to decrease her account by any whole number.
Part 2: Alison is only allowed to deposit £3 at a time and withdraw £7 at a time.
Nathan noted:
Alison can still increase and decrease her balance by any amount of integer pounds.
To increase by £$x$, deposit 5$x$ lots of £3 and withdraw 2$x$ lots of £7.
To decrease by £$x$, deposit 2$x$ lots of £3 and withdraw $x$ lots of £7.
Part 3: Alison is only allowed to deposit £$x$ at a time and withdraw £$y$ at a time.
The solution to this might look complicated at first, but read it slowly and carefully. If you can understand it then you're developing excellent mathematical understanding!
Therefore, the amounts Alison can change her balance by are of the form $ax-by$ for any non-negative integers $a$ and $b$.
Positive integers are the numbers 1,2,3,...
Non-negative integers are the numbers 0,1,2,3,...
If $x$ and $y$ have a common factor $d$ then $d$ is also a factor of $ax-by$. This means that the only amounts Alison can change her balance by are multiples of $d$. So if $d$ is greater than 1 then Alison will never be able to change her balance by anything that is not a multiple of $d$.
Therefore, for Alison to be able to change her bank account by anything, $x$ and $y$ should have highest common factor 1.
But is it true that for any pair £$x$, £$y$ with HCF 1 Alison will be able to change her account by any amount?
Look back at Dare's solution to the first part. By depositing and withdrawing he managed to make £1 and - £1 and then he could make any amount by repeating this.
So if we can find positive integers $a$, $b$, $c$ and $d$ so that $$ax-by=1$$ and $$cx-dy=-1$$ then she can make any amount.
If we have found $a$ and $b$ then we can automatically find $c$ and $d$, because if we let $c=-a+ky$, $d=-b+kx$ where $k$ is any positive integer then $$\begin{align*} cx-dy &=(-a+ky)x-(-b+kx)y \\&=-ax+kyx+by-kyx \\&=-ax+by \\&=-(ax-by) \\&=-1 \end{align*}$$
and $k$ can be chosen large enough so that $c$ and $d$ are positive integers.
So all we need is to find positive integers $a$ and $b$ so that $$ax-by=1.$$
If we find any integers $u$ and $v$ so that $$ux+vy=1$$ then it must be the case that one is positive and the other is negative. (Because if they are both negative then $ux+vy$ will be negative, and if they are both positive then $ux+vy$ will be bigger than 1.)
If $u$ is positive and $v$ is negative then we can let $a=u$ and $b=-v$ and we have finished, if $u$ is negative and $v$ is positive then let $a=u+ny$ and $b=-v+nx$ where $n$ is some positive integer chosen large enough so that $a$ and $b$ are positive.
So all we need to be able to do is find integers $u$ and $v$ such that $ux+vy=1$.
Dare worked really hard on this problem, and at this stage had this insight:
It should not be too surprising, after working the first question, that in order to increase by any integer amount we need to be able to write the number 1 as a combination of $x$s and $y$s. More precisely, we can increase Alison's account balance by any positive integer if and only if there are integers $u$ and $v$ such that $$ 1 = ux + vy. $$
Two integers satisfying the above condition are called coprime, which is equivalent to the condition that $x$ and $y$ have highest common factor 1.
As Dare says, for every pairs of integers $x$ and $y$ that have highest common factor 1 you can find $u$ and $v$ which are both integers so that $$ 1 = ux + vy. $$ If you would like to know why this is the case you can read more about it here: An Introduction to Modular Arithmetic
Therefore, if $x$ and $y$ have HCF 1, Alison can make any amount, and if they have HCF greater than 1 Alison cannot make any amount.
Why do this problem?
This problem offers a context for the first explorations into the addition of positive and negative numbers, using the familiar idea of deposits and withdrawals from a bank account. It is intended to be used in a sequence of lessons together with the game Up, Down, Flying Around and Strange Bank Account (part 2), which introduce subtraction of positive and
negative numbers. The article Adding and Subtracting Positive and Negative Numbers offers a variety of models to refer to when teaching this topic.
Possible approach
Explain the rules for Charlie's Bank: "you are only allowed to deposit £2 at a time and withdraw £3 at a time" If necessary, show an example (perhaps the one in the video), and then invite students to choose a number of deposits and withdrawals and work out the combined effect.
Choose a few students to share their answers. Two possible lines of enquiry emerge:
Is it possible to change the account balance by any amount?
Can each amount be made in more than one way?
The second of these two questions is explored in the follow-up problem Strange Bank Account (part 2), so focus on the first question.
One possible way to explore is to invite students to try to make all possible amounts from £1 to £30 using only combinations of + £2 and - £3. Efficient notation naturally emerges; rather than writing down $+ £2 + £2+ £2+ £2+ £2$ it is much quicker to write $5 \times (+ £2)$.
Once students have had some time to work on this, take some time to discuss what they have found out. While students are working, you may wish to look out for insights into why it is possible to make every number. There are at least two nice methods for students to convince themselves that all totals are possible:
Method 1
All even totals are possible just by repeatedly adding + £2. I can make £1: (+ £2) + (+ £2) + (- £3) so then I can make all odd totals by repeatedly adding + £2 to my £1.
Method 2
I can make £1: (+ £2) + (+ £2) + (- £3)
I can make £2 by adding £1 and £1: {(+ £2) + (+ £2) + (- £3)} + {(+ £2) + (+ £2) + (- £3)}
So I can keep adding on (+ £2) + (+ £2) + (- £3) to increase the total by £1, and hence make any number.
Next: "The bank wants to change its rules, so that instead of depositing £2 and withdrawing £3, you need to decide the deposit and withdrawal amounts. You need to make sure that it's possible to make every possible total, so experiment with some different amounts. In a while, I'm going to choose a deposit and a withdrawal amount, and you need to be able to tell me straight away whether all
amounts will be possible."
You may wish to bring the class together for a mini-plenary to share what they have found out along the way, or collect together on the board observations about which pairs of deposit/withdrawal amounts give rise to all possible amounts and which ones don't.
Here are some pairs you might wish to offer students at the end of the lesson to check their understanding:
+5 and - £8
+8 and - £5
+5 and - £15
+ £6 and - £9
+ £4 and - £10
+11 and - £17
Possible support
For a similar investigation that only requires consideration of positive numbers, see How Much Can We Spend?
This problem is based on Don Steward's ideas. His resources can be found here.
Possible extension
Up, Down, Flying Around and Strange Bank Account (part 2) follow on from this problem.