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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 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?

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Answer:

[tex]A:B=-8n:-27n\\\\=8:27[/tex]

Step-by-step explanation:

Let the A be the amount Alice initially has and B be Bob's initial amount.

#Alice receives $n from Bob:

[tex]4(A+n)=B-n\ \ \ \ \ \ \ \ \ i[/tex]

#Alice gives n to Bob:

[tex]3(A-n)=B+n\ \ \ \ \ \ \ \ \ \ ii[/tex]

#We simultaneously solve the two expressions:

[tex]4(A+n)=B-n\\\\3(A-n)=B+n\\\\=>from \ i\\4A+4n+n=B\\\\4A+5n=B\\\\\# substitute \ in \ ii\\\\3A-3n=4A+5n+n\\\\A=-8n\\\\B=4A+5n=4(-8n)+5n=-27n[/tex]

The initial amounts of Alice is -8n and Bob's is -27n

#We find the ratio of Alice:Bob:-

[tex]A:B=-8n:-27n\\\\=8:27[/tex]

Hence the ration of Alice to Bob is 8:27

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