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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?

 Jun 27, 2023
 #1
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Let the amount of Alice's original money = A

 

Let the amount of Bob's original money = B

 

So

 

(A + n )* 7 =  (B - n)     →  7A - 2B  = -5n   (1)

 

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

 

Multiply the first equation by 3  and the second equation by 4  and add them together

 

3A -  9B  =  -12n

4A  - 8B =    12n

 

7A  - 17B  =  0     add 17B to both sides

 

7A  = 17B          divide both sides by 7, B

 

A / B  =  17/7

 Jun 27, 2023
 #2
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it says its wrong

 Jun 27, 2023
 #3
avatar+274 
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A+n=7B

A-n=3B

2A=10B

 

Try to see if you can find the answer from there? If not, I will give you the solution. (I still think 17/7 is the correct answer though.)

 Jun 27, 2023
 #4
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i cant find an answer that works

 Jun 27, 2023
 #5
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Me neither. Do you know why it isn't 17/7?

 Jun 27, 2023
 #6
avatar+832 
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The error is in answer #1 at   

                                                      

(A + n )* 7 =  (B - n)     →  7A - 2B  = -5n   (1)   

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

 

The first one should be  

 

(A + n)  =  7 • (B – n)    (Since Alice has 7 times as much as Bob, you   

                                      have to multiply Bob's by 7 to make them equal.)   

 

I do like Guest's reasoning in answer #1 though. 

So let's continue that, but with these other numbers.     

 

                                         (A + n)  =  7 • (B – n)  

                                          A + n    =  7B – 7n  

                                          A – 7B  =  – 8n            (1)  

 

                                         (A – n)  =  3 • (B + n)  

                                          A – n   =  3B + 3n 

                                          A – 3B  =  4n                (2)  

 

Multiply both sides  

of (2) by 2 and then   

add (1) and (2)  

                                            A – 7B  =  – 8n  

                                          2A – 6B   =     8n    

                                          3A – 13B  =  0  

 

Add 13B to both sides                  3A  = 13B   

 

Divide both sides by 13B          

                                                     3A        1      

                                                     –––  =  –––   

                                                     13B       1  

Multiply both sides by 13/3   

                                                 13       3A          13        1  

                                                 –––  •  –––   =   –––  •  –––  

                                                   3       13B          3         1   

 

The digits cancel out   

of the left side, leaving                 

                                                  A         13   

                                                 –––  =  –––   

                                                  B           3   

.

 Jun 28, 2023

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