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?

Answer 1

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.

Answer 2

Let's assume that Alice has x dollars, and Bob has y dollars.

According to the given information, if Alice receives n dollars from Bob, she will have 7 times as much money as Bob. This can be expressed as:
x + n = 7(y - n)

Similarly, if Alice gives n dollars to Bob, she will have 2 times as much money as Bob. This can be expressed as:
x - n = 2(y + n)

Now, let's solve these two equations to find the values of x and y.

Expanding the first equation, we get:
x + n = 7y - 7n

Rearranging terms, we have:
x + 8n = 7y

Expanding the second equation, we get:
x - n = 2y + 2n

Rearranging terms, we have:
x - 3n = 2y

Now we have a system of equations:
x + 8n = 7y
x - 3n = 2y

To eliminate x, let's multiply the second equation by 4:
4(x - 3n) = 4(2y)
4x - 12n = 8y

Now, subtract the first equation from this new equation:
(4x - 12n) - (x + 8n) = 8y - 7y
4x - x - 12n - 8n = y

Simplifying the equation, we have:
3x - 20n = y

Since Alice and Bob don't give each other any money, we can set n = 0 in the equation above:
3x - 20(0) = y
3x = y

So, we find that y = 3x, which means the ratio of the amount of money Alice has to the amount Bob has is 3:1.