# Doubling Investment for a Fee And Getting Zero

Here is a problem from an upcoming book Mathematics: Problem-Solving Challenges for Secondary School Students and Beyond (p 23, #143):

Willow owns a magic purse that doubles any sum of money placed into it. Willow charges $1.20 for each use of the purse. Alex starts with a certain sum of money, puts it in the purse to double it, and, after paying Willow his fee, reinvests all his money in the purse. After doubling his money again, and paying Willow the second fee, Alex reinvests all his money once more. After the money doubles and the third fee is paid, Alex finds that he has no money left at all. How much did Alex start with?

### References

- D. Linker, A. Sultan,
*Mathematics: Problem-Solving Challenges for Secondary School Students and Beyond*, Wold Scientific, 2016

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Copyright © 1996-2018 Alexander Bogomolny

### Solution

Willow owns a magic purse that doubles any sum of money placed into it. Willow charges $1.20 for each use of the purse. Alex starts with a certain sum of money, puts it in the purse to double it, and, after paying Willow his fee, reinvests all his money in the purse. After doubling his money again, and paying Willow the second fee, Alex reinvests all his money once more. After the money doubles and the third fee is paid, Alex finds that he has no money left at all. How much did Alex start with?

### Solution 1 (Algebra)

If Alex starts with x dollars, after doubling and paying Willow once, he has 2x-1.2,2x-1.2,x-2.4,x+1.2,2x+1.2 dollars. After doubling and paying again, he has 4x-3.6,4x-1.2,4x-2.4,4x-3.6 dollars. After the third time, he has 8x-8.4,8x-1.2,8x-8.4,4x-8.4 dollars. Since he's left with no money, we get an equation to solve 8x-8.4=0,8.4x-8=0,8.4x-1.2=0,8x-8.4=0 dollars. Solving which,

### Solution 2 (Arithmetic)

Imaging watching the process in reverse: three times Willow pays Alex $1.2 and then takes away half of his money,twice as much,half of his money,whatever Alex has. He starts with nothing, so the amounts of money he has after each step are $1.2, $.6,$.6,$1.2,$1.8,$2.4, $1.8, $.9, $2.1,$.9,$1.2,$1.8,$2.1 $1.05.

### Moral

By Einstein's definition, a person is crazy if he/she repeatedly does the same thing, every time expecting a different result. Alex should have noticed after the first step that the amount of money he held had decreased,increased,decreased and he should not have wasted all of it in the additional two attempts. Even better, he could have computed the expected outcome before doing anything stupid.

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