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mark, jessica, and nate each downloaded music from the same website. mark downloaded 10 songs in total consisting of pop, rock, and hip hop. 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. 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, how many songs of each type did mark download?

mark downloaded 2 pop songs, 5 rock songs, and 3 hip hop songs.
mark downloaded 3 pop songs, 2 rock songs, and 5 hip hop songs.
mark downloaded 4 pop songs, 2 rock songs, and 4 hip hop songs.
mark downloaded 1 pop songs, 4 rock songs, and 5 hip hop songs.

Mark downloaded 1 pop songs, 4 rock songs, and 5 hip hop songs.

Step-by-step explanation:

Let us assume that the number of pop songs be x .

Let us assume that the number of rock songs be y .

Let us assume the the number of hip hop songs be z .

As given

Mark downloaded 10 songs in total consisting of pop, rock, and hip hop.

Than the equation becomes

x + y + z = 10

Jessica downloaded five times as many pop songs, twice as many rock, and three times as many hip hop songs as Mark.

Number of pop songs jessica download = 5x

Number of rock songs jessica download = 2y

Number of hip hop songs jessica download = 3z

She downloaded 28 songs total.

Than the equation becomes

5x + 2y + 3z = 28

As given

If Nate downloaded 20 songs total .

Three times as many pop songs, three times as many rock songs, and the same number of hip hop songs as Mark .

Number of pop songs Nate download = 3x

Number of rock songs Nate download = 3y

Number of hip hop songs Nate download = z

Than the equation becomes

3x + 3y + z = 20

Thus the three equations are .

x + y + z = 10

5x + 2y + 3z = 28

3x + 3y + z = 20

Multiply x + y + z = 10 by 2 and subtracted from 5x + 2y + 3z = 28 .

5x - 2x + 2y - 2y + 3z - 2z = 28 - 20

3x + z = 8

Multiply x + y + z = 10 by 3 and subtracted from 3x + 3y + z = 20  .

3x - 3x + 3y - 3y + z - 3z = 20 -30

-2z = -10

z = 5

Put in the equation 3x + z = 8 .

3x + 5 = 8

3x = 8 - 5

x = 1

Put x = 1 , z = 5 in the equation x + y + z = 10 .

1 + 5 + y = 10

6 + y = 10

y = 10 - 6

y = 4

Thus

Number of pop songs mary download = 1

Number of rock songs mary download = 4

Number of hip hop songs mary download = 5

Therefore Mark downloaded 1 pop songs, 4 rock songs, and 5 hip hop songs.

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step-by-step explanation:

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