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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?

 Aug 12, 2023
 #1
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Let A be the amount of money Alice has and B be the amount of money Bob has. If Alice receives n dollars from Bob, then she will have A+n=7B. If she gives n dollars to Bob, then she will have A−n=3B. Setting these two equations equal to each other, we get 7B−n=3B+n, so 4B=2n and B=n/2.

If neither gives the other any money, then the ratio of Alice's money to Bob's money is A:B=2:1​.

 Aug 12, 2023
 #3
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Fundamental error in the attempted solution at   

"If Alice receives n dollars from Bob, then she will have A+n=7B"   

 

This should be  

If Alice receives n dollars from Bob, then she will have A + n = 7(B – n)  

 

etc.   

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 Aug 12, 2023
 #4
avatar+850 
+1

 

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?    

 

                                                                    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)             

 

Repeat (1) here                                             A – 7B  =  –8n            (3)    

Multiply both sides of (2) by 2                      2A – 6B  =    8n            (4)    

 

Add (3) and (4)                                            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      

                                                                        A           13            

                                                                      ––––  =  ––––    

                                                                        B             3            

 

So the ratio of Alice's $ to Bob's $ is               A : B  =  13 : 3       

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 Aug 12, 2023

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