Question

the manager of a movie theater is going over their receipts for the last few days. on wednesday, the theater sold 13 adult tickets and 10 child tickets and made $211.00. on friday, the theater sold 26 adult tickets and 17 child tickets and made $400.50. what are the admission prices for adults and children? a $12.00 per adult, $8.00 per child b $10.00 per adult, $7.50 per child c $10.50 per adult, $7.50 per child d $11.00 per adult, $8.00 per child

Explanation:

Let \( x \) be the adult ticket price and \( y \) be the child ticket price.

Step1: Set up equations from Wednesday's sales

On Wednesday, 13 adult tickets and 10 child tickets were sold, totaling $211.00. So, the equation is:
\( 13x + 10y = 211 \)

Step2: Set up equations from today's sales

Today, 26 adult tickets and 17 child tickets were sold, totaling $400.50. So, the equation is:
\( 26x + 17y = 400.5 \)

Step3: Solve the system of equations

First, multiply the first equation by 2: \( 26x + 20y = 422 \)
Now subtract the second equation from this new equation:
\( (26x + 20y) - (26x + 17y) = 422 - 400.5 \)
\( 26x + 20y - 26x - 17y = 21.5 \)
\( 3y = 21.5 \)
\( y = \frac{21.5}{3} \approx 7.17 \)? Wait, that's not matching. Wait, maybe I made a mistake. Wait, let's check the options. Let's test option B: \( x = 10 \), \( y = 7.5 \)
For Wednesday: \( 13*10 + 10*7.5 = 130 + 75 = 205 \)? No, not 211. Wait, option C: \( x = 10.5 \), \( y = 7.5 \)
\( 13*10.5 + 10*7.5 = 136.5 + 75 = 211.5 \)? Close. Wait, option B: \( x = 10 \), \( y = 7.5 \): 1310=130, 107.5=75, total 205. No. Option A: \( x=12 \), \( y=8 \): 1312=156, 108=80, total 236. No. Option D: \( x=11 \), \( y=8 \): 1311=143, 108=80, total 223. No. Wait, maybe I misread the numbers. Wait, the problem says: Wednesday: 13 adult, 10 child, total $211.00. Today: 26 adult, 17 child, total $400.50. Let's set up the equations correctly. Let \( x \) = adult price, \( y \) = child price.

Equation 1: \( 13x + 10y = 211 \)

Equation 2: \( 26x + 17y = 400.5 \)

Let's solve Equation 1 for \( 13x \): \( 13x = 211 - 10y \), so \( 26x = 422 - 20y \)

Substitute into Equation 2: \( 422 - 20y + 17y = 400.5 \)

\( 422 - 3y = 400.5 \)

\( -3y = 400.5 - 422 = -21.5 \)

\( y = \frac{21.5}{3} \approx 7.166... \). Wait, but the options have y=7.5, 8, etc. Wait, maybe the problem has a typo, but let's check option B: adult $10, child $7.5. Let's check Wednesday: 1310 + 107.5 = 130 + 75 = 205. Not 211. Option C: adult $10.5, child $7.5. 1310.5=136.5, 107.5=75, total 211.5. Close to 211. Maybe a rounding. Then today: 2610.5=273, 177.5=127.5, total 273+127.5=400.5. Yes! So 2610.5=273, 177.5=127.5, 273+127.5=400.5. Perfect. So adult price is $10.50, child price is $7.50. So option C.

Answer:

C. $10.50 per adult, $7.50 per child