# For richer for poorer

*For Richer for Poorer printable sheet*

In the rich country of Emerald, the mean income is £20,000.

In the poorer, neighbouring country of Flint, the mean income is £16,000.

Charlie lives in Emerald and earns £18,000 a year.

He moves to Flint and says "I've made both countries richer!"

What does he mean?**Charlie thinks that the mean income in both countries has increased.Can you convince yourself that Charlie is right?**

What other effects can moves between countries have on the mean incomes?

Describe what is required for moves to:

- increase the mean incomes in both countries
- decrease the mean incomes in both countries
- increase one whilst decreasing the other

**Extension:**

Would it be possible for someone to move from Emerald to Flint and reduce the mean income in Emerald and **double** the mean income in Flint?

The mean is a type of average of a set of numbers.

The mean is the total of the numbers divided by how many numbers there are.

Try starting with a specific example.

e.g.

Group A is composed of people aged 2, 7, 11, 14 and 16 (average 10)

Group B is composed of people aged 5, 11, 12, 13 and 19 (average 12)

Investigate what happens to the averages when different members move from one group to the other.

**Is Charlie right?**

John considered an extreme case:

Only one person lives on Emerald Isle, other that Charlie, and they earn £22,000

In this case, it is clear that the average income in Emerald increases when Charlie leaves. In general, even if there are many more people in Emerald, we can think of the average income as the amount of money each person would get if all of the money was shared out equally. So with less than the average income, Charlie is leaving with less than his 'fair share', and there is more to go around for everyone else.

Suryasnato Dasgputa said:

Charlie has increased the average income of both countries. In Emerald, he was a below average earner so he was decreasing the average. So once he moved, the average for Emerald increased. When he was in Flint, Charlie was an above average earner so he increased the average in Flint when he moved in.

**Increasing the average income in both countries**

Suryasnato continued:

In general to increase the average in both countries, someone with an income between £16,000 and £20,000 (but not exactly either) needs to move from Emerald to Flint.

**Decreasing the average income in both countries**

Suryasnato also explained how the average income in both countries could be decreased:

To decrease the average income in both countries, you must live in Flint and move to Emerald, and be an above average earner in Flint but a below average earner
in Emerald (with an income between £16,000 and £20,000).

**Increasing one whilst decreasing the other**

Jeremy from Longston Middle School wrote:

There are four ways of this happening.

1) Charlie earns over £20,000, and moves from Emerald to Flint. Then Emerald's average goes down and Flint's goes up.

2) Charlie earns over £20,000 and moves from Flint to Emerald. Then Flint's average goes down and Emerald's goes up.

3) Charlie earns less than £16,000 and moves from Emerald to Flint. Then Emerald's average goes up and Flint's goes down.

4) Charlie earns less than £16,000 and moves from Flint to Emerald. Then Flint's average goes up and Emerald's average goes down.

**Moving from Emerald to Flint, reducing the average in Emerald and doubling the average in Flint**

Thomas Hu from A Y Jackson school answered this:

The only way he could decrease the average income of Emerald and double that of Flint is to move from Emerald to Flint with an average income of $(2+n)\times a$, with $n =$ number of people in Flint, and $a$ being the average income in Flint. Now let the incomes of Emerald and Flint back to their original values, £20,000 and £16,000. Thus Charlie must have an
income of $(n+2)\times 16 000$

It is easy to see how this works by imagining average income as the amount of money that each person would get if the money were shared out equally. To double the average income in Flint, each person's share must be doubled, so increased by £16,000. This would require $ £16000\times n$, where there are $n$ people living in Flint.

The population of Flint will also increase by one when the new wealthy resident arrives. So the total amount of money earned in Flint must be enough for this new person to also get their fair share of $ £2\times16000= £32000$.

So the new person must earn $ £(16000n+32000)= £(n+2)\times16000$, as Thomas found.

If someone who earned $ £(16000n+32000)$ lived in Emerald, then they can't have been the only resident of Emerald, or the average income in Emerald would have been $ £(16000n+32000),$ which is at least £32,000 and so more than £20,000. So at least one other person lives in Emerald. If all the money earned in Emerald were shared out equally, each person in Emerald would get $ £20,000$, so this
wealthy person who earns $( £16000n+32000)$ contributes far more than their fair share. So the average income in Emerald will fall when they leave.

**Other interesting circumstances**

Thomas Hu from A Y Jackson school noticed some other interesting changes in average income that could occur:

To change the average income of one country, and not change the average income of the other, when moving either way, Charlie's income must be the same as the average income of one of the countries.

### Why do this problem?

This problem may support a deeper understanding of averages. The unexpected result may lead students to ask what else might be possible. Encouraging students to ask questions and organise their work in a systematic way in order to draw conclusions are all key mathematical skills that can be encouraged.

### Possible approach

*This problem featured in an NRICH webinar in December 2020.*

Before introducing the problem, revisit finding the mean through this simpler question:

Draw up the following table on the board:

Anna | Brin | Cayley | Dave | Erin |

2 | 8 | 10 | 14 | 16 |

The table shows the ages of five children. What is the average (mean) age?

If Anna leaves the group, what happens to the average?

If Cayley leaves the group instead, what happens to the average?

If Erin leaves the group instead, what happens to the average?

Ensure that students notice that the average can stay the same, go up or go down depending on whether the age of the child leaving is the same as, less than or greater than the average.This is the focus of the main problem.

Introduce the main problem.The first part should now be straightforward, so students can devote their thinking to the follow up question:

What other effects can moves between countries have on average incomes?

This question may need fleshing out -

### Key Questions

What could cause the mean to increase?

What could cause the mean to decrease?

### Possible support

You may choose to offer the following specific example.

Group 1: ages 2, 7, 11, 14 and 16 (average 10)

Group 2: ages 5, 11, 12, 13 and 19 (average 12).

Investigate what happens to the averages when different members move from one group to the other.

### Possible extension

Is it possible to double one country's average income whilst halving the other?

Interested students may also wish to consider whether there are contexts where this statistical manipulation may be used to advantage.