Question

the ratio of rock songs to dance songs on Jonathan's MP3 player is 5: 6 the total number of rock and dance songs Jonathan has is between 100 and 120 how many rock songs does he have

Answer 1

So I estimated to 110. And I got 50 rock songs and 60 dance songs. I used the ratio 5:6. 

Answer 2

Let the number of rock songs be r and let the number of dance songs be d. ratio of r to s is 5:6 just means r/s = 5/6.  we want to find r + d but it doesn't seem we have enough information.  Well, we aren't told this, but it is safe to assume that the number of rock and dance songs are both integers and this calls for some number theory.  

this is how ill approach the problem.  multiplying by d on both sides of the ratio equation gives r = (d)(5/6).  substituting that into r + d to write the total songs in terms of d gives:

d(5/6) + d and now we can combine like terms to get the total number of songs to be (11/6)d.  now we could solve a system of inequalities to get:

(11/6)d > 100 and (11/6)d <120 because in the problem we are told the total number of songs is between 100 and 120.

now we have d>54.54 and d<65.45.  now what do we do?  we remember the total number of songs has to be an integer which means d has to be divisible by 6.  the only integer between 54.54 and 65.45 that is divisible by 6 is 60.  Therefore, we know d = 60.  

remember, we want to find the TOTAL number of songs, not d, so we plug d into the expression we found for the total number of songs to get:
(11/6)(60) = 110 songs.

you may be wondering: what if instead i wrote the total number of songs in terms of r?  you'll find that you'll get the same answer.  this is the beauty of math: it doesn't matter how you do a problem you will always get the right answer if your method is logically sound.