Question

Alice and Bob each have a certain amount of money. If Alice receives $n$ dollars from Bob, then she will have $4$ times as much money as Bob. If, on the other hand, she gives $n$ dollars to Bob, then she will have $3$ 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

31:9

Explanation:

Let $A$ and $B$ be the amount of money Alice and Bob have, respectively, at the beginning. We know that

\begin{align*}

A + n &= 4(B - n),\\

A - n &= 3(B + n).

\end{align*}

Simplifying, we have

\begin{align*}

A + 5n &= 4B, \\

A &= 3B + 4n.

\end{align*}Subtracting the first equation from the second gives $5n = B - 4n$, so $B = 9n$. Substituting this into the first equation gives $A + n = 4(9n - n)$, from which we get $A = 31n$.

Therefore, the desired ratio is $\frac{A}{B} = \frac{31n}{9n} = \boxed{\dfrac{31}{9}}$.

Answer 2

Answer: 31 : 9

Explanation:

Assume the following:

Alice's amount = P

Bob's amount = Q

Amount received = n

If Alice receives $n$ dollars from Bob ;then she will have $4$ times as much money as Bob.

P + n = 4(Q - n)

P + n = 4Q - 4n

P = 4Q - 4n - n

P = 4Q - 5n - - - - (1)

If, on the other hand, she gives $n$ dollars to Bob, then she will have $3$ times as much money as Bob

P - n = 3(Q + n)

P - n = 3Q + 3n

P = 3Q + 3n + n

P = 3Q + 4n - - - - - - (2)

Equating both equations - (1) and (2)

4Q - 5n = 3Q + 4n

4Q - 3Q = 4n + 5n

Q = 9n

Express P in terms of n, use either equation (1) or (2)

From equation 2:

P = 3Q + 4n

Substituting Q = 9n

P = 3(9n) + 4n

P = 27n + 4n

P = 31n

Alice's amount = P, Bob's = Q

Ratio = P:Q

31 : 9