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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 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?

Jul 17, 2024

#1
+840
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Let's represent the amount of money Alice has as A and the amount Bob has as B. We can set up two equations based on the given information:

1. Alice receives n dollars from Bob: In this scenario, Alice's money increases by n, while Bob's money decreases by n. So, we have:

A + n = 7(B - n)

2. Alice gives n dollars to Bob: Here, Alice's money decreases by n, and Bob's money increases by n. So, we have:

A - n = 3(B + n)

Now, we want to find the ratio A/B without introducing the unknown value n. To achieve this, we can eliminate n by strategically manipulating the equations.

One approach is to add the two equations together:

(A + n) + (A - n) = 7(B - n) + 3(B + n)

Simplifying both sides:

2A = 10B Dividing both sides by B:

2A/B = 10

A/B = 10/2 = 5

Therefore, the ratio of Alice's money (A) to Bob's money (B) is 5:1

Jul 17, 2024
#2
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Let's create a system of equations to solve this problem .

Let's let A be the number of money Alice has and B be the amount of money Bob has. We have

If Alice receives n dollars from Bob, then she will have 3 times as much money as Bob.

\(A + n  =  3 * (B - n) \)

If she gives n dollars to Bob, then she will have 2 times as much money as Bob.

\(A - n  =  2 * (B + n) \)

Thus, we have the system

\(A + n = 3B - 3n → A = 3B - 4n → 3A = 9B - 12n \\ A - n = 2B + 2n → A = 3B + 3n → 4A = 12B + 12n\)

adding these two equations up, we get

\(7A = 17B → 7A / B = 17 → A / B = 17 / 7 \)

Thanks! :)

Jul 17, 2024
edited by NotThatSmart  Jul 17, 2024
#3
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Let the original amount that Alice had = A   Let the original amount that Bob had = B

1st scenario :  Alice gets n dollars from Bob....her new amt = A + n

Bob loses n dollars.....his new amt times 7 = Alice's new amt

A + n  = 7(B - n)  →    A + n = 7B - 7n  →  A + 8n = 7B    (1)

2nd senario : Bob gets n dollars from Alice....her new amt = A - n

Bob gains n dollars  and alice has 3 times as much as he does

A - n  =  3(B  + n)  → A - n = 3B + 3n  →  A - 4n = 3B  → 2A - 8n = 6B  (2)

3A = 13B

A / B  =  13  / 3

Proof :  3A = 13B  →  A = (13/3)B....sub this into  (1)

(13/3)B + 8n  =   7B

8n =(8/3)B    multiply through by 3/8

3n = B

n= B/3

Using (2)

2A - 8(B/3) = 6B

2A = (26/3)B

A = (26/6)B

A = (13/3)B

A / B  = 13  /  3

Jul 18, 2024