Chapter 7.3, Problem 89E

Solution

To state: The number of child, adult, and senior riders at the Pittsburgh zoo if children ride a train for 25 cents, adults pay $1.00 and senior citizens 75 cents. There were 250 more child riders than all other riders. On a given day, 1400 passengers paid a total of $740 for the rides.

There are 825 children, 410 adults, and 165 senior riders.

Given information:

At the Pittsburgh zoo, children ride a train for 25 cents, adults pay $1.00 and senior citizens 75 cents. There were 250 more child riders than all other riders. On a given day, 1400 passengers paid a total of $740 for the rides.

Explanation:

Let’s take x as the number of children riders, y as the number of adult riders, and z as the number of senior riders.

Now make equations using the given information.

On a given day, total number of passengers is 1400: $x+y+z=1400$

On a given day, 1400 passengers paid a total of $740 for the rides. Children ride a train for 25 cents on the other hand adults pay $1.00 and senior citizens 75 cents.

  $\begin{array}{c}\left(0.25\right)x+\left(1.00\right)y+\left(0.75\right)z=740\\ 0.25x+y+0.75z=740\end{array}$

There were 250 more child riders than all other riders:

  $\begin{array}{c}x=250+y+z\\ x-y-z=250\end{array}$

The three equations are:

  $\begin{array}{c}x+y+z=1400\cdots \left(1\right)\\ 0.25x+y+0.75z=740\cdots \left(2\right)\\ x-y-z=250\cdots \left(3\right)\end{array}$

Add equations (1) and (3),

  $\begin{array}{l}x+y+z=1400\\ \underset{¯}{x-y-z=250}\\ 2x=1650\\ x=825\end{array}$

Add equations (2) and (3),

  $\begin{array}{l}\underset{¯}{\begin{array}{c}0.25x+y+0.75z=740\\ x-y-z=250\end{array}}\\ 1.25x-0.25z=990\end{array}$

Substitute the value of x as -2 in $1.25x-0.25z=990$ ,

  $\begin{array}{c}1.25x-0.25z=990\\ 1.25\left(825\right)-0.25z=990\\ 1031.25-0.25z=990\\ -0.25z=-41.25\\ z=165\end{array}$

Now substitute the value of x and y as 825 and 165 in equation (1),

  $\begin{array}{c}x+y+z=1400\\ 825+y+165=1400\\ y+990=1400\\ y=410\end{array}$

Therefore, the resultant solution is $x=825,y=410,\text{and}z=165$ . It means the number of children is 825, number of adults is 410, and the number of seniors is 165.