+16 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 $3$ 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?
Let A be the amount of money Alice has and B be the amount of money Bob has. If Alice receives n dollars from Bob, then she will have A+n=7B. If she gives n dollars to Bob, then she will have A−n=3B. Setting these two equations equal to each other, we get 7B−n=3B+n, so 4B=2n and B=n/2.
If neither gives the other any money, then the ratio of Alice's money to Bob's money is A:B=2:1.
+1277
Fundamental error in the attempted solution at
"If Alice receives n dollars from Bob, then she will have A+n=7B"
This should be
If Alice receives n dollars from Bob, then she will have A + n = 7(B – n)
etc.
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+1277
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 $3$ 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?
A + n = 7 • (B –n)
A + n = 7B – 7n
A – 7B = –8n (1)
A – n = 3 • (B + n)
A – n = 3B + 3n
A – 3B = 4n (2)
Repeat (1) here A – 7B = –8n (3)
Multiply both sides of (2) by 2 2A – 6B = 8n (4)
Add (3) and (4) 3A – 13B = 0
Add 13B to both sides 3A = 13B
Divide both sides by 13B
3A 1
––––– = –––
13B 1
Multiply both sides by 13 / 3
A 13
–––– = ––––
B 3
So the ratio of Alice's $ to Bob's $ is A : B = 13 : 3
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