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 8 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?
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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 8 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?
The first case:
(A + n) = (4) • (B – n)
A + n = 4B – 4n
A – 4B = –5n (1)
The second case:
(A – n) = (8) • (B + n)
A – n = 8B + 8n
A – 8B = 9n (2)
Multiply both sides of (1) by 9 9A – 36B = –45n
Multiply both sides of (2) by 5 5A – 40B = 45n
Add these two together 14A – 76B = 0
Add 76B to both sides
14A = 76B
Divide both sides by 76B
14A 1
–––– = ––––
76B 1
Multiply both sides by 76 / 14
76 14A 76 1
–––– • –––– = –––– • ––––
14 76B 14 1
The 76 and the 14 cancel out
on the left side, leaving
A 76
–––– = ––––
B 14
Reduce the fraction
A 38
–––– = ––––
B 7
.
