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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 8 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 19, 2023
 #1
avatar+758 
+1

This question makes no sense, and the question's answer should be no solution. This is because you should not have a higher multiplier when you are giving money away. She gets more and is 4x, while when she gives more, she is 8x her brothers money. 

 Jul 19, 2023
 #2
avatar+943 
0

 

@history ~~  That's what I thought, too.  But I decided to go through the motions, anyway, just to find out where the proposition broke down.  What I expected was a negative ratio as the answer.  Well, I got a negative all right, but it turned out to be in a surprising place.    

 

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 8 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)  =  4 • (B – n)   

                                         A + n    =  4B – 4n    

                                         A – 4B  =  –5n                   (1)    

 

                                         (A – n)  =  8 • (B + n)   

                                          A – n    =  8B + 8n   

                                          A – 8B  =  9n                    (2)   

Multiply both sides  

of (1) by 9                        9A – 36B  =  –45n              (3)    

 

Multiply both sides  

of (2) by 5                        5A – 40B  =    45n              (4)      

 

Add (3) and (4)               14A – 76B  =  0   

 

Add 76B to both sides               14A  =  76B   

 

Divide both sides by 76B   

                                                  14A          1    

                                                  ––––   =   ––    

                                                  76B           1                       

 

Multiply both sides by 76/14    

                                                      A          76                         38

                                                    –––   =   –––   reduces to   –––   

                                                      B          14                          7   

 

It works if you accept the  

concept of negative money.  

That is, n must equal –2 dollars.   

 

Say Alice has 38 and Bob has 7        

                                                     (38 + n)  =  4 • (7 – n)     

                                                      38 + n   =  28 – 4n    

                                                             5n  =  –10     

                                                               n  =  –2       

and               

                                                     (38 – n)  =  8 • (7 + n)    

                                                      38 – n   =  56 + 8n      

                                                          –9n   =  18    

                                                               n  =  –2       

.

 Jul 19, 2023
 #3
avatar+758 
+1

Nice Job! I did not want to do this question anyway, but the fact that you did it is cool. 

history  Jul 20, 2023

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