Question

The admission fee at an amusement park is $2.75 for children and $4.80 for adults. On a certain day, 312 people entered the park, and the admission fees collected totaled $1104. How many children and how many adults were admitted?

Answer 1

Let the number of children admitted be x and let the number of adults admitted be y.

It is given that the total number of 312 people were admitted, this implies that the sum of the number of children and adults is 312:

`x+y=312`

It is given that the admission fee for children is $2.75, this implies that the total fees collected for x children are:

`x\cdot2.75=2.75x`

It is also given that the admission fee for adults is $4.80, this implies that the total fees collected for y children are:

`y\cdot4.80=4.80y`

Since the admission fees collected totaled $1104, it follows that:

`2.75x+4.80y=1104`

Hence, the following system of equations is formed:

`\begin{cases}x+y=312{} \\ 2.75x+4.80y={1104}\end{cases}`

Solve the system of equations:

Solve the first equation for x:

`x=312-y`

Substitute this for x in the second equation:

`\begin{gathered} 2.75\left(−y+312\right)+4.8y=1104 \\ \text{ Simplify both sides of the equation:} \\ \Rightarrow-2.75y+858+4.8y=1104 \\ \Rightarrow-2.75y+4.8y+858=1104 \\ \Rightarrow2.05y+858=1104 \\ \text{ Subtract }858\text{ from both sides:} \\ \Rightarrow2.05y+858−858=1104−858 \\ \Rightarrow2.05y=246 \\ \text{ Divide both sides by }2.05: \\ \Rightarrow(2.05y)/(2.05)=(246)/(2.05) \\ \Rightarrow y=120 \end{gathered}`

It follows that the number of adults admitted is 120.

Substitute y=120 into the equation x=312-y to find the value of x:

`\begin{gathered} x=312-120 \\ \Rightarrow x=192 \end{gathered}`

Hence, the number of children admitted is 192.

Answer: 192 children and 120 adults were admitted.