Answer:
Mark downloaded 1 pop song, 4 rock songs and 5 hip hop songs.
Let pop songs be represented by p.
Let rock songs be represented by r.
Let hip hop songs be represented by h.
For mark :
[tex]p+r+h=10[/tex] ------(1)
Jessica downloaded five times as many pop songs, twice as many rock, and three times as many hip hop songs as Mark. She downloaded 28 songs total.
Her equation becomes :
[tex]5p+2r+3h=28[/tex] -------(2)
If Nate downloaded 20 songs total with three times as many pop songs, three times as many rock songs, and the same number of hip hop songs as Mark.
His equation becomes:
[tex]3p+3r+h=20[/tex] -------(3)
Subtracting (1) from (3) we get
[tex]2p+2r=10[/tex] --------(4)
[tex]2(p+r)=10[/tex]
=> [tex]p+r=5[/tex] ------(5)
Now putting p+r=5 in (1)
[tex]5+h=10[/tex]
=> h=5
Now, using equation (2) and (3)
[tex]5p+2r+3(5)=28[/tex]
=> [tex]5p+2r+15=28[/tex]
=>[tex]5p+2r=13[/tex] ------(6)
Now subtracting (4) from (6)
[tex]3p=3[/tex]
=> p=1
Now putting the values of h and p in (1)
[tex]1+r+5=10[/tex]
[tex]6+r=10[/tex]
=> r=4
Therefore, Mark downloaded 1 pop song, 4 rock songs and 5 hip hop songs.
