Haley's iPod has 425 more songs than Adam's.  Jason has 1890 songs on his iPod which is 132 more than Haley.  How many songs does Adam have?

 

J = the number of Jason's songs                                A = the number of Adam's songs

H = the number of Haley's songs

 

I know that J = 1890. So I can make the following chain of reasoning.

 

("" means "which means that")

 

                                                     so

                                      but

 

                                             so

                                  

 

Haley and Jason together have 315 more songs that Adam.  Haley has 1467 songs on her iPod.  Jason has _______ songs and Adam has ________ songs. 

 

Find 3 pairs of numbers that can go in the blanks so that everything works out. (Be sure you understand what it means that "everything works out"!)

 

"Everything works out" means that when I put numbers in the blanks, nothing contradicts the given conditions that Haley and Jason have 315 more songs than Adam and Haley has 1467 songs.

 

I'll use these variables:

 

J = the number of Jason's songs         H = the number of Haley's songs

A = the number of Adam's songs.

 

Let me see whether there are any limitations on the number of Jason's songs. It cannot be smaller than 0, unless he owes some songs. I don't see any constraints on how large it can be. I'll try J=2000.

 

If J=2000, then, knowing also that H = 1467, I can make this chain of reasoning:

 

              

                      

 

J=2000 and A=3152 work. H=1467, so H+J=3467, and 3467 is 315 larger than 3152. So the number of Haley's and Jason's songs combined is 315 more than the number of Adam's songs.

 

By the same reasoning pattern I found that J=2887 and A=4039 work and that J=29352 and A=30504 work.

 

What must be true about the number of songs that Jason has and the number of songs that Adam has so that everything works out?

 

Looking at these numbers, it seems that A must be 1152 more than J. This makes sense. Imagine that someone gives Jason one song at a time. Before Jason receives any songs, he and Haley start out with 1467 songs just with Haley's songs. Since Adam is always 315 behind Jason and Haley's total, then when Jason starts with 0 songs, Adam already has 1152. Whatever number of songs Jason receives, Adam must receive the same number to stay 315 behind Jason's and Haley's total. So Adam always stays 1152 songs ahead of Jason. So J = A + 1152. Hmmm. This also says that J cannot be smaller than 1152, so I lucked out by picking 2000 and not a number less than 1152.