To solve this problem, we need to find the regular price of an adult's ticket and from there determine the regular price of a child's ticket. Let's break it down step by step.
1. Define the Variables:
Let [tex]\( x \)[/tex] be the regular price of an adult ticket in dollars.
2. Develop the Equation:
According to the problem, the regular price of a child's ticket is [tex]$9 less than the adult ticket, so the regular price of a child's ticket is \( x - 9 \).
3. Account for the Discounts:
During the off-peak season, all tickets are sold at half price. Therefore:
- The price of an adult ticket during the off-peak season is \( \frac{1}{2}x \).
- The price of a child's ticket during the off-peak season is \( \frac{1}{2}(x - 9) \).
4. Set Up the Equation:
The Sandlers paid a total of \$[/tex]132 for 1 adult ticket and 4 children's tickets. Therefore, the total cost equation is:
[tex]\[
\frac{1}{2}x + 4 \left( \frac{1}{2}(x - 9) \right) = 132
\][/tex]
5. Simplify the Equation:
Simplify the terms inside the equation:
[tex]\[
\frac{1}{2}x + 4 \left( \frac{1}{2}x - \frac{1}{2} \cdot 9 \right) = \frac{1}{2}x + 4 \left( \frac{1}{2}x - 4.5 \right)
\][/tex]
[tex]\[
= \frac{1}{2}x + 4 \left( \frac{1}{2}x - 4.5 \right) = \frac{1}{2}x + 2x - 18 = 132
\][/tex]
6. Combine Like Terms:
Combine [tex]\(\frac{1}{2}x\)[/tex] and [tex]\(2x\)[/tex]:
[tex]\[
\frac{1}{2}x + 2x - 18 = 132
\][/tex]
[tex]\[
2.5x - 18 = 132
\][/tex]
7. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], first add 18 to both sides of the equation:
[tex]\[
2.5x = 150
\][/tex]
Then, divide both sides by 2.5:
[tex]\[
x = 60
\][/tex]
8. Find the Regular Price of a Child's Ticket:
The regular price of a child's ticket is:
[tex]\[
x - 9 = 60 - 9 = 51
\][/tex]
Therefore, the regular price of a child's ticket is \[tex]$51. The correct choice is:
\$[/tex]51
