Chapter 5.2, Problem 33E

To determine

To describe: the number of songsplayed by the radio station

Answer to Problem 33E

Hence, the radio station play is:

  $144$ Pop songs

  $48$ Rock songs

  $80$ Hip-hop songs

Explanation of Solution

Given:

  $272$ Pop, rock, and hip-hop songsare played

The number of pop songs is $3$ times the number of rock songs.

The number of hip-hop songs is $32$ more than the number of rock songs.

Calculation:

Let:

  $x\to$ Number of pop songs

  $y\to$ Number of rock songs

  $z\to$ Number of hip-hop songs

Set up the equations.

The total number of songsis $272$ :

  $x+y+z=272\leftarrow \text{Equation 1}$

The number of pop songs is $\text{3}$ times the number of rock songs:

  $x=3y\leftarrow \text{Equation 2}$

The number of hip-hop songs is $\text{32}$ more than the number of rock songs:

  $z=y+32\leftarrow \text{Equation 3}$

Substitute eq. $2$ & $3$ to eq. $\text{1}$ :

  $3y+y+\left(y+32\right)=272$

Solve for $y$ :

  $5y=240\to y=48$

Solve for $x$ :

  $x=3\left(48\right)\to x=144$

Solve for $z$ :

  $z=48+32\to z=80$

Therefore, there are:

  $144$ Pop songs

  $48$ Rock songs

  $80$ Hip-hop songs

Conclusion:

Hence, the radio station play is:

  $144$ Pop songs

  $48$ Rock songs

  $80$ Hip-hop songs