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In Linguistics 101, the ratio of the number of juniors to the number of seniors is 3:2.  When six more juniors join the class, and one senior drops the class, the ratio of the number of juniors to the number of seniors becomes 2:3.  How many students are in the class after these changes?

 Jul 1, 2023
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In Linguistics 101, the ratio of the number of juniors to the number of seniors is 3:2.  When six more juniors join the class, and one senior drops the class, the ratio of the number of juniors to the number of seniors becomes 2:3.  How many students are in the class after these changes?   

 

I can tell you immediately that increasing the number     

of juniors will not invert the ratio.  But I'll go through    

the motions for you anyway.     

 

                                                       J          3   

                                                      ––   =   ––  

                                                       S          2     

 

                                                   J + 6          2   

                                                   ––––   =   –––   

                                                   S – 1          3        

 

Cross multiply both                            2J  =  3S            (eq 1)  

                                                  3J + 18  =  2S – 2      (eq 2)      

 

Get either J or S in terms of the   

other.  It doesn't matter which,   

but J would be simpler.                       J  =  3S/2      

 

Substitute this in eq 2                (3) • (3S/2) + 18  =  2S – 2    

 

                                                        9S/2 + 18  =  2S – 2     

 

Multiply both sides by 2                      9S + 36  =  4S – 4    

 

Subtract 4S from both sides               5S + 36  =  –4   

 

Subtract 36 from both sides                    5S  =  –40     

 

You cannot have a negative number of seniors    

so something is wrong somewhere.  The error     

is in the initial proposition.  This problem has ...... no solution.    

.

 Jul 7, 2023

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