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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?

 

i got stuck on the 'she gives n dollars to bob, then she will have 3 times as much money as bob'... my english isn't very good.

 Jun 4, 2023
 #1
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Let's assume the amount of money Alice has is A dollars, and the amount of money Bob has is B dollars.

According to the given information, if Alice receives n dollars from Bob, she will have 4 times as much money as Bob. This can be expressed as:

A + n = 4(B - n) ---(1)

Similarly, if Alice gives n dollars to Bob, she will have 3 times as much money as Bob:

A - n = 3(B + n) ---(2)

We can solve these equations to find the values of A and B.

Expanding equation (1), we get:

A + n = 4B - 4n A + 5n = 4B

Expanding equation (2), we get:

A - n = 3B + 3n A - 3n = 3B

Now we have a system of equations:

A + 5n = 4B ---(3) A - 3n = 3B ---(4)

To eliminate A, we can subtract equation (4) from equation (3):

(A + 5n) - (A - 3n) = (4B) - (3B) 8n = B

Substituting this value back into equation (4):

A - 3n = 3(8n) A - 3n = 24n A = 27n

So, the ratio of the amount of money Alice has (A) to the amount Bob has (B) is:

A/B = (27n)/(8n) = 27/8

Therefore, the ratio of the amount of money Alice has to the amount Bob has is 27:8.

 Jun 4, 2023
 #2
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You set up your equations correctly, but somehow you didn't solve them correctly.

 

A + n = (4B - 4n)..............(1)    

A - n = (3B + 3n)..............(2), solve for A, B

 

From (1) above: A==4B - 4n - n

                              A==4B - 5n  {sub this into (2) above

 

4B - 5n - n ==3B + 3n

4B - 6n == 3B + 3n

4B - 3B ==3n + 6n

B ==9n  {sub this into (2) above}

 

A - n ==(3 * 9n) + 3n

A - n ==27n + 3n

A ==27n + 3n + n

A == 31n

 

Therefore, the ratio of their money BEFORE the exchange==A / B ==31n / 9n, or 31: 9

 Jun 4, 2023
 #3
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Let's assume the amount of money Alice has is denoted by A and the amount Bob has is denoted by B.

According to the given information:

If Alice receives n dollars from Bob, she will have 4 times as much money as Bob: A + n = 4(B - n).

If Alice gives n dollars to Bob, she will have 3 times as much money as Bob: A - n = 3(B + n).

Simplifying the equations, we have:

A + n = 4B - 4n

A - n = 3B + 3n

We can rewrite equation 1 as: A + 4n = 4B.

Adding equation 2 and the rewritten equation 1, we get: (A - n) + (A + 4n) = 3B + 3n + 4B 2A + 3n = 7B + 3n 2A = 7B

Dividing both sides by B, we have: 2A/B = 7

Therefore, the ratio of the amount of money Alice has to the amount Bob has is A/B = 7/2.

 Jun 4, 2023
 #4
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Let's assume the amount of money Alice has is denoted by A and the amount Bob has is denoted by B.

According to the given information:

If Alice receives n dollars from Bob, she will have 4 times as much money as Bob: A + n = 4B.

If Alice gives n dollars to Bob, she will have 3 times as much money as Bob: A - n = 3B.

We can solve this system of equations to find the values of A and B.

First, let's solve equation 1 for A: A = 4B - n.

Substituting this value of A into equation 2: 4B - n - n = 3B. 4B - 2n = 3B. B = 2n.

Now, substituting the value of B back into equation 1: A = 4(2n) - n, A = 8n - n, A = 7n.

Therefore, we have found that A = 7n and B = 2n.

The ratio of the amount of money Alice has to the amount Bob has is A/B = (7n)/(2n) = 7/2.

So, the ratio is 7:2.

 Jun 5, 2023

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