Question

select the correct answer.
a theme park charges different prices for children under age 12, children over age 12, and adults. a family with 2 kids under age 12, 3 kids over age 12, and 3 adults pays $174 for entrance to the park. a mother and father bring their 4 kids under age 12 at a cost of $122. a grandmother who brings her teenage granddaughter to the park pays $46. how much does it cost for an adult to enter the theme park?
a. $25
b. $16
c. $18
d. $20

Explanation:

Step1: Define Variables

Let \( a \) = adult price, \( c \) = child (under 12) price, \( t \) = teen (over 12) price.

Step2: Set Up Equations

1. Family: \( 2c + 3t + 3a = 174 \)
2. Parents + 4 kids: \( 4c + 2a = 122 \) (simplify: \( 2c + a = 61 \) → \( 2c = 61 - a \))
3. Grandmother + teen: \( a + t = 46 \) → \( t = 46 - a \)

Step3: Substitute into First Equation

Substitute \( 2c = 61 - a \) and \( t = 46 - a \) into \( 2c + 3t + 3a = 174 \):
\( (61 - a) + 3(46 - a) + 3a = 174 \)

Step4: Solve for \( a \)

Simplify:
\( 61 - a + 138 - 3a + 3a = 174 \)
\( 199 - a = 174 \)
\( a = 199 - 174 = 25 \)? Wait, no—wait, miscalculation. Wait, \( 3(46) = 138 \), so \( 61 + 138 = 199 \). Then \( 199 - a + 3a - 3a = 199 - a \). So \( 199 - a = 174 \) → \( a = 25 \)? But let's check again. Wait, maybe I messed up the family equation. Wait, the family has 2 kids under 12, 3 kids over 12, 3 adults. So \( 2c + 3t + 3a = 174 \). Parents: 2 adults, 4 kids under 12: \( 2a + 4c = 122 \) → divide by 2: \( a + 2c = 61 \) → \( 2c = 61 - a \). Grandmother (adult) + teen: \( a + t = 46 \) → \( t = 46 - a \). Now substitute \( 2c = 61 - a \) and \( t = 46 - a \) into family equation:
\( (61 - a) + 3(46 - a) + 3a = 174 \)
\( 61 - a + 138 - 3a + 3a = 174 \)
\( 199 - a = 174 \)
\( a = 25 \). Wait, but option A is $25. Wait, but let's verify with the second equation. If \( a = 25 \), then \( 2c = 61 - 25 = 36 \) → \( c = 18 \). Then \( t = 46 - 25 = 21 \). Now check family: \( 2(18) + 3(21) + 3(25) = 36 + 63 + 75 = 174 \). Correct. Parents + 4 kids: \( 2(25) + 4(18) = 50 + 72 = 122 \). Correct. Grandmother + teen: \( 25 + 21 = 46 \). Correct. So \( a = 25 \).

Answer:

A. $25