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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?

 Jun 6, 2023
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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?      

 

The first case:   

                                                         (A + n)  =  (4) • (B – n)  

                                                          A + n  =  4B – 4n  

                                                          A – 4B  =  –5n             (1)  

The second case:   

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

                                                          A – n  =  8B + 8n  

                                                          A – 8B  =  9n               (2)  

 

Multiply both sides of (1) by 9           9A – 36B  =  –45n  

Multiply both sides of (2) by 5           5A – 40B  =    45n  

Add these two together                    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  

                                                         76         14A        76          1   

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

                                                         14         76B        14           1      

The 76 and the 14 cancel out  

on the left side, leaving  

                                                          A           76  

                                                        ––––  =  ––––  

                                                          B           14  

Reduce the fraction  

                                                                    38 

                                                        ––––  =  ––––  

                                                          B            7 

.

 Jun 30, 2023

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