Alice and Bob each have a certain amount of money. If Alice receives n dollars from Bob, then she will have 7 times as much money as Bob. If, on the other hand, she gives n dollars to Bob, then she will have 2 times as much money as Bob. If neither gives the other any money, what is the ratio of the amount of money Alice has to the amount Bob has?

Question

Alice and Bob each have a certain amount of money.  If Alice receives n dollars from Bob, then she will have 7 times as much money as Bob.  If, on the other hand, she gives n dollars to Bob, then she will have 2 times as much money as Bob.  If neither gives the other any money, what is the ratio of the amount of money Alice has to the amount Bob has?

Bot · Answer

Let A be the amount of money Alice has and let B be the amount of money Bob has.
From the first statement, we have the equation A + n = 7(B - n)
Expanding and simplifying this equation gives A + n = 7B - 7n, or A + 8n = 7B
From the second statement, we have the equation A - n = 2(B + n)
Expanding and simplifying this equation gives A - n = 2B + 2n, or A - 2n = 2B
We can solve this system of equations for A and B by adding the two equations:
(A + 8n) + (A - 2n) = 7B + 2B
Simplifying, we get 2A + 6n = 9B
We can further simplify this equation to get A = (9/2)B - 3n
If neither gives the other any money, then n = 0 and we have A = (9/2)B.
Therefore, the ratio of the amount of money Alice has to the amount Bob has is 9:2.