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# solve

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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?

Jul 27, 2023

#2
+901
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@tastybanana ~~  That's what I thought, too, no solution.  But I decided to go through the motions, anyway, to find out where the problem broke down.  What I expected was a negative ratio as the answer.  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

.

Jul 27, 2023
#3
+758
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Nice

history  Jul 27, 2023