### Why do this problem?

This problem gives an interesting problem solving challenge for students aged 16/17 with an interest in both chemistry and mathematics. It gives a good exercise in mathematical modelling: a physical process needs to be represented by equations which may then be analysed mathematically. It is not simply routine, nor is it especially difficult if considered in the right ways. However, it is a good example of a situation in which simply trying to 'do the algebra' will not work.

### Possible approach

This problem works well for individual use. It could be solved numerically, using trial and error, or graphically using many ideas from linear algebra.

If done as a group or set as a homework, it would be interesting to compare solutions, as there are several different possible routes to a conclusion. What do students think of each others answers? Which seem most powerful? Neatest?

### Key questions

Could the sample be a pure sample of a single isotope? Why?
How many pieces of information are given to us in the question?
Is there any way of distinguishing between the different isotopes for the timescales given?
Chemically, what is happening in the decay? How does decay affect the mass of the sample? Is this a significant effect?

### Possible extension

Chemical extension:
• Consider carefully the effects of the decay on the mass of the sample, taking into account that some matter will be lost.
• Energy is radiated from the system during decay. Energy is related to mass via Einstein's equation $E=mc^2$. Is this effect relevant or significant?
Mathematical Extension:
• Explore how many more data points would be needed to give a unique solution to the problem. What is the minimum number of extra data points needed to give a unique solution?

### Possible support

Encourage trial and error, starting with equal masses of each sample. Then raise or lower the proportions of two of the masses, using a spreadsheet to do the numerical calculations.