@history ~~ That's what I thought, too. But I decided to go through the motions, anyway, just to find out where the proposition broke down. What I expected was a negative ratio as the answer. Well, I got a negative all right, but it turned out to be in a surprising place.
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?
(A + n) = 4 • (B – n)
A + n = 4B – 4n
A – 4B = –5n (1)
(A – n) = 8 • (B + n)
A – n = 8B + 8n
A – 8B = 9n (2)
Multiply both sides
of (1) by 9 9A – 36B = –45n (3)
Multiply both sides
of (2) by 5 5A – 40B = 45n (4)
Add (3) and (4) 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
A 76 38
––– = ––– reduces to –––
B 14 7
It works if you accept the
concept of negative money.
That is, n must equal –2 dollars.
Say Alice has 38 and Bob has 7
(38 + n) = 4 • (7 – n)
38 + n = 28 – 4n
5n = –10
n = –2
and
(38 – n) = 8 • (7 + n)
38 – n = 56 + 8n
–9n = 18
n = –2
.
