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 the amount of Alice's original money = A
Let the amount of Bob's original money = B
So
(A + n )* 7 = (B - n) → 7A - 2B = -5n (1)
(A - n) = 3 * ( B + n) → A - 3B = 4n (2)
Multiply the first equation by 3 and the second equation by 4 and add them together
3A - 9B = -12n
4A - 8B = 12n
7A - 17B = 0 add 17B to both sides
7A = 17B divide both sides by 7, B
A / B = 17/7
+274 A+n=7B
A-n=3B
2A=10B
Try to see if you can find the answer from there? If not, I will give you the solution. (I still think 17/7 is the correct answer though.)
+788
The error is in answer #1 at
(A + n )* 7 = (B - n) → 7A - 2B = -5n (1)
(A - n) = 3 * ( B + n) → A - 3B = 4n (2)
The first one should be
(A + n) = 7 • (B – n) (Since Alice has 7 times as much as Bob, you
have to multiply Bob's by 7 to make them equal.)
I do like Guest's reasoning in answer #1 though.
So let's continue that, but with these other numbers.
(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)
Multiply both sides
of (2) by 2 and then
add (1) and (2)
A – 7B = – 8n
2A – 6B = 8n
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
13 3A 13 1
––– • ––– = ––– • –––
3 13B 3 1
The digits cancel out
of the left side, leaving
A 13
––– = –––
B 3
.
