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# help 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?

Jun 6, 2023

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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?

The first case:

(A + n)  =  (4) • (B – n)

A + n  =  4B – 4n

A – 4B  =  –5n             (1)

The second case:

(A – n)  =  (8) • (B + n)

A – n  =  8B + 8n

A – 8B  =  9n               (2)

Multiply both sides of (1) by 9           9A – 36B  =  –45n

Multiply both sides of (2) by 5           5A – 40B  =    45n

Add these two together                    14A – 76B  =  0

14A            =  76B

Divide both sides by 76B

14A           1

––––  =   ––––

76B          1

Multiply both sides by 76 / 14

76         14A        76          1

––––  •  ––––  =  ––––  •  ––––

14         76B        14           1

The 76 and the 14 cancel out

on the left side, leaving

A           76

––––  =  ––––

B           14

Reduce the fraction

38

––––  =  ––––

B            7

.

Jun 30, 2023