Alice and Bob each have a certain amount of money. If Alice receives $n$ dollars from Bob, then she will have $10$ times as much money as Bob. If, on the other hand, she gives $n$ dollars to Bob, then she will have $4$ 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 $10$ times as much money as Bob.  If, on the other hand, she gives $n$ dollars to Bob, then she will have $4$ 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 GPT 3.5 · Answer

Let $x$ be the amount of money Alice has and $y$ be the amount of money Bob has. Then we have the system of equations \begin{align*} x+n &= 10(y-n)\ x-n &= 4(y+n) \end{align*}Expanding these equations gives \begin{align*} x+n &= 10y - 10n\ x-n &= 4y + 4n. \end{align*}Simplifying, we have \begin{align*} x &= 10y - 11n\ x &= 4y + 5n. \end{align*}Therefore, $10y - 11n = 4y + 5n,$ so $6y = 16n,$ and $y = \frac{8}{3} n.$ Thus, the ratio of the amount of money Alice has to the amount of money Bob has is $\frac{x}{y} = \frac{4y+5n}{y} = \frac{4\cdot \frac 83 n + 5n}{\frac 83 n} = \frac{32+15}{8} =\boxed{\frac{47}{8}}$.