Quandaries and Queries

bob has a very rich aunt who is a mathematician. she wrote this letter to bob:

"Dear bob,
now that i am getting on (70 today). i want to give you some of my money, i shall give you a sum each year, starting now. you can choose which of the following schemes you would like me to use.

a) £100 now, £90 next year,£80 the yaer after and so on.
b) £10 now, £20 next year, £ 30 the next year and so on.
c) £10 now, 1.5 times as much next year, 1.5 times as much the year after that and so on.
d)£1 now, £2 next year, £ 4 the year after, £8 the year after and so on.

of course these schemes will only operate while i am alive. i look forward to hearing which scheme you choose and why!
Best wishes,
Aunt Lucy."

that was the letter and i need to know what scheme he should choose and why. please use trial and improvement.

my name is olivia and i am a student of level 5-8
please write back for i shall be very grateful. i am in secondary school.

Hi Olivia,

I like this question. Not only will it show you how the numbers grow under the different schemes but there is also a human element.

I would construct a table containing a column for each of the four schemes and a row for each year.

a b c d
Total after
year 1
£100 £10 £10 £1
Total after
year 2
£100+£90
= £190
£10+£20
=£30
£10+1.5£10
=£25
£2
Total after
year 3
£190+£80
= £270
£30+£30
=£60
£25+1.5£25
=£62.5
£4
Total after
year 4
... ... ... ...

Continue the table until Aunt Lucy is 100. Soon you will see that some of the schemes are clearly not the correct choice and you can stop calculating those columns. The ultimate choice depends on how long Aunt Lucy lives. The choice is yours but you have to explain why you make your choice.

What is the state of Aunt Lucy's health?
Does she come from a family that usually lives into their 90's?
How badly does Bob need the money now?

Penny

Consider just the following:
Plan a) £100 now, £90 next year,£80 the year after and so on.
Plan b) £ 10 now, £20 next year, £ 30 the next year and so on.

It these plans are followed for just a few years (say 3 years), plan a) is clearly the best; if they are followed for very many years (say 30 years), then plan b is the best. Between these two extremes, there will be a time where plan b) becomes better than plan a). Using algebra you can see that after 10 years, plan b) becomes better than plan a).

So, to decide between the two, you need to ask yourself whether your aunt is very healthy and should live many years or whether she is very sick and has less than 10 years to live. Of course in your original question you need to compare the four plans before making

- Actually, the question is a bit more complex, because you also need to take into account interest rates and expected returns based on a probability distribution for your aunt's life expectancy.

- In fact, it is still more complex, because you also need to consider whether you need money more today than you expect to need in 10 years or the other way around.

- Also, you need to know whether you will inherit your aunt's money when she passes away, or will she leave everything to her cat.

So all in all, it is not much of a math question. It is a question where you are asked to exercise your judgement and common sense. You can use the math you know to model the situation, and the more of it the better; but in the end, only you can devise and justify a decision you can live with.

Claude

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