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?
We can write two equations from this...
A+n = 4*(B-n) (Bob gives n dollars to Alice)
A - n = 3 * (B + n) (Alice gives n dollars to Bob)
We will get
A+n = 3B-3n
=A = 3B - 4n
=3A = 9B -12n
A-n = 2B+2n
=A = 3B + 3n
=4A = 12B + 12n
Then, we can add these together to get--
7A = 17B
=7A / B = 17
=A / B = 17 / 7
Let A and B be the amount of money Alice and Bob have, respectively, at the beginning. We know that
A + n = 4(B - n)
A - n = 3(B + n)
Simplifying, we have
A + 5n = 4B
A = 3B + 4n
Subtracting the first equation from the second gives 5n = B - 4n, so B = 9n. Substituing this into the first equation gives A + n = 4(9n - n), from which we get A = 31n.
Therefore, the desired ratio is A/B = 31n/9n = 31/9.
hope this helped !
I agree with the first part of Guest's answer; but then I got lost ...
A + n = 4(B - n) ---> A + n = 4B - 4n ---> 5n = 4B - A
(multiply by 4): 20n = 16B - 4A
A - n = 3(B + n) ---> A - n = 3B + 3n ---> 4n = A - 3B
(multiply by 5): 20n = 5A - 15B
Setting these two equations to each other: 16B - 4A = 5A - 15B
31B = 9A
31 / 9 = A / B