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 = Alice's original amount of money
B = Bob's original amount of money
(A + n) = (4) • (B – n)
(A – n) = (8) • (B + n) Just looking at these two equations, I have misgivings.
How can Alice with less have more times Bob with more?
A + n = 4B – 4n ——> A = 4B – 5n (eq 1)
A – n = 8B + 8n ——> A = 8B + 9n (eq 2) got to get rid of those n's somehow
multiply (eq 1) by 9 9A = 36B – 45n
multiply (eq 2) by 5 5A = 40B + 45n
add the two 14A = 76B
divide both sides by 14 A = (76/14) • B
divide both sides by B A / B = 76 / 14 76 / 14 will reduce to 38 / 7
ratio Alice 38
––––– = –––
Bob 7
check answer:
38 + n = 28 – 4n ——> 5n = – 10 ——> n = – 2
38 – n = 56 + 8n ——> – 9n = + 18 ——> n = – 2
It works out, to my surprise. It requires accepting the concept of negative money. I'm okay with that.
.
