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# i need help

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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 7 times as much money as Bob.  If, on the other hand, she gives n dollars to Bob, then she will have 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 1, 2023

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Let A be the amount of money Alice has and B be the amount of money Bob has.

From the first condition, we have:

A + n = 7(B - n)

Simplifying, we get:

A + n = 7B - 7n
A + 8n = 7B

From the second condition, we have:

A - n = k(B + n)

Simplifying, we get:

A - n = kB + kn
A - kn = kB + n
A = (k + 1)B + (k + 1)n

We want to find the ratio of A to B when neither gives the other any money, which means n = 0. Substituting n = 0 into the equations we derived earlier, we get:

A + 8n = 7B  becomes  A = 7B
A = (k + 1)B + (k + 1)n  becomes  A = (k + 1)B

Since both equations give us the value of A in terms of B, we can equate them to get:

7B = (k + 1)B

Solving for k, we get:

k = 6

Therefore, the ratio of the amount of money Alice has to the amount Bob has is:

A/B = (k + 1) = 7/1 = 7:1

Jun 1, 2023