Question

Alice and Bob each have a certain amount of money. If Alice receives n dollars from Bob, then she will have 3 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 \( B \) be the amount of money Bob has. We have two scenarios based on the transfers of \( n \) dollars:

1. **Alice receives \( n \) dollars from Bob:**
- After this transaction, Alice's new amount will be \( A + n \) and Bob's will be \( B - n \).
- According to the problem, we have:
\[
A + n = 3(B - n)
\]

2. **Alice gives \( n \) dollars to Bob:**
- After this transaction, Alice's new amount will be \( A - n \) and Bob's will be \( B + n \).
- According to the problem, we have:
\[
A - n = 2(B + n)
\]

Now we will simplify both equations.

### Simplifying the equations

**From the first equation:**
\[
A + n = 3(B - n)
\]
Expanding this gives:
\[
A + n = 3B - 3n
\]
Rearranging terms results in:
\[
A + n + 3n = 3B
\]
\[
A + 4n = 3B \quad (1)
\]

**From the second equation:**
\[
A - n = 2(B + n)
\]
Expanding this gives:
\[
A - n = 2B + 2n
\]
Rearranging terms results in:
\[
A - n - 2n = 2B
\]
\[
A - 3n = 2B \quad (2)
\]

### Solve the system of equations

Now we have the system:
1. \( A + 4n = 3B \)
2. \( A - 3n = 2B \)

We will express \( A \) in terms of \( B \) and \( n \) from both equations:

From equation (1):
\[
A = 3B - 4n
\]

From equation (2):
\[
A = 2B + 3n
\]

Setting these two expressions for \( A \) equal to each other:
\[
3B - 4n = 2B + 3n
\]

Now, solving for \( B \):
\[
3B - 2B = 4n + 3n
\]
\[
B = 7n
\]

Next, substituting \( B \) back into one of the expressions for \( A \):
Using \( A = 2B + 3n \):
\[
A = 2(7n) + 3n = 14n + 3n = 17n
\]

### Find the ratio \( \frac{A}{B} \):
Now we have \( A = 17n \) and \( B = 7n \):
\[
\frac{A}{B} = \frac{17n}{7n} = \frac{17}{7}
\]

Thus, the ratio of the amount of money Alice has to the amount Bob has is:
\[
\boxed{\frac{17}{7}}
\]