Matiu and Ariana have agreed to work for their Mum over the holidays.

The pay they get will vary.

Ariana will get $10 for the first day she works and two more dollars for every day she works after that.

Matiu will get $1 for the first day he works, but for each day he works from then on, his pay will be doubled.

Who would you rather be and why?

This problem poses a question that involves students in comparing two rates. In one a fixed number is added, and in the other the total is continually doubled.

It is worth noting that, regardless of the whole numbers you begin with, doubling will eventually win.

### The Problem

Matiu and Ariana have agreed to work for their Mum over the holidays. The pay they get will vary. Ariana will get $10 for the first day she works and two more dollars for every day she works after that. Matiu will get $1 for the first day he works, but for each day he works from then on, his pay will be doubled.

Who would you rather be and why?

### Teaching Sequence

- Read the problem to the class. Give them time to think about the problem by themselves.
- Have students restate the problem in their own words, identifying what it is asking them to find out.
- Ask how they might approach the problem and keep track of the totals.
- Have students work on the problem individually or in small groups.
- As they work ask questions that require them to
**compare**the daily totals. Ask them to**justify**the number operations that they are using and to explain the steps they are taking to find the answers. - Share and discuss solutions.

#### Extension to the problem

After how many days is Matiu’s Mum likely to run out of money?

### Solution

This problem doesn’t have a definite answer as it does not state how many days the two students work. If they work for less than 6 days then Ariana will earn most money. If they work for more than 6 days, then Matiu will earn the most.

#### Extension to the problem

There is no precise answer. However, the students should be able to see that doubling raises the number fairly quickly. After 20 days the amount exceeds one million dollars. Most mothers won’t be able to afford that!