Brain Teaser/Geometry Problem.?

Question

I am lost. I have no clue as to where I can even start this problem. If you have any idea as to what equations or how to solve this problem, please help. Thank you.


Tim spent all his money in five stores. In each store, he spent $1 more than half of what he had when he came in. How much money did Tim have when he entered the first store?

Answer 1

I don't know where geometry fits into all this... but when Tim enters a store with x dollars, he spends (x/2 + 1) dollars and exits the store with y dollars. This means
x - (x/2 +1) = y
x/2 - 1 = y
x = 2(y+1). This means if you take the money Tim had coming OUT of a store, add 1 to it and double the result, you get the amount Tim had BEFORE entering the store.

After Store 5 Tim had $0.
2(0+1) = 2. Before Store 5/After Store 4, Tim had $2.
2(2+1) = 6. Before Store 4/After Store 3, Tim had $6.
2(6+1) = 14. Before Store 3/After Store 2, Tim had $14.
2(14+1) = 30. Before Store 2/After Store 1, Tim had $30.
2(30+1) = 62. Before Store 1, Tim had $62.

Answer 2

If Tim walked into store 5 with 2 dollars, half of 2 dollar is 1 dollar. Spending a dollar more leaves him broke.

Working backwards, if Tim walked into store 4 with 6 dollars, half of 6 is 3, and spending a dollar more would leave him 2, which is what he went into store 5 with.

If Tim walked into store 3 with 14 dollars, spending a dollar more than half would leave him with 6.

If Tim walked into store 2 with 30 dollars, spending a dollar more than half would leave him with 14.

If Time walked into store 1 with 62 dollars, spending a dollar more than half would leave him with 30.

Tim started with 62 dollars.

Answer 3

You have to work backward. You have two equations:

money spent at store n = 1 + .5*total money after store n-1

total money after store n = total money after store n-1 - money spent at store n


He spent all his money in 5 stores. This means that in the last store, he spent all his money:

money spent at store 5 = total money after store 4

Let's call this quantity x. So the above equation for store 5 (n=5) becomes:

x=1+0.5*x
-> x=2
Therefore, he spent $2 in store 5, and he had $2 after store 4.

Suppose he started with y dollars. This is how much he has after each store:

1: 0.5*y-1
2: 0.5*(0.5*y-1)-1
3: 0.5*(0.5*(0.5*y-1)-1)-1

Just solve for store 4 and equate that with 2.

Answer 4

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