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# algebra

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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 17, 2023

#1
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Let us say that Alice = a, and Bob = b

a + n = 4 ( b - n )  → this would be the equation for the first situation, now let us simplify.

a + n = 4b - 4n  →. a  = 4b - 5n  → this would be the simplified version

Substitute  ( a = 4b -5n ) into the equation

a - n = 8 ( b + n )  →  4b - 5n - n = 8b + 8n  → 4b - 6n = 8b + 8n.  → 4b + 2n = 8b   → 2n = 4b   n = 2b

Substitute ( n = 2b ) into the first equation

a = 4b - 5 ( 2b )

a = 4b - 10b

a = -6b

thus the ratio would be a: b = -6: 1

this answer does not seem correct, however if anyone wants to correct me they can Jul 17, 2023
#2
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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?

(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 17, 2023