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?

Guest Jul 19, 2023

#1**+1 **

This question makes no sense, and the question's answer should be **no solution**. This is because you should not have a higher multiplier when you are giving money away. She gets more and is 4x, while when she gives more, she is 8x her brothers money.

history Jul 19, 2023

#2**0 **

@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

_{.}

Bosco Jul 19, 2023