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Difficulty: MediumSystems of Linear Equations

A community theater sold a total of 250250 tickets for a performance, consisting of adult tickets for $15\$15 each and student tickets for $10\$10 each. The total revenue collected from ticket sales was $3,100\$3,100. Let aa represent the number of adult tickets sold and ss represent the number of student tickets sold. Which of the following statements must be true? Select all such statements.

  1. The number of student tickets sold was 130130.Answer
  2. B
    The number of adult tickets sold was 130130.
  3. The total revenue generated from adult ticket sales was $1,800\$1,800.Answer
  4. D
    The ratio of adult tickets sold to student tickets sold was 1313 to 1212.
  5. E
    More adult tickets were sold than student tickets.

Answer

The correct statements are that the number of student tickets sold was 130, and the total revenue generated from adult ticket sales was $1,800.
The system of equations a+s=250a + s = 250 and 15a+10s=310015a + 10s = 3100 uniquely yields a=120a = 120 adult tickets and s=130s = 130 student tickets. Therefore, the statement that 130 student tickets were sold is true. Furthermore, multiplying 120 adult tickets by their price of 15perticketgivesatotaladultticketrevenueof15 per ticket gives a total adult ticket revenue of 1,800, making that statement true as well.

Step-by-Step Solution

1
Set up a system of two linear equations in terms of aa and ss.
Equation 1 (total tickets): a+s=250a + s = 250; Equation 2 (total revenue): 15a+10s=310015a + 10s = 3100.
The sum of the ticket quantities equals total tickets, and the sum of monetary contributions equals total revenue.
2
Express ss in terms of aa using Equation 1 and substitute into Equation 2.
s=250−a  ⟹  15a+10(250−a)=3100  ⟹  15a+2500−10a=3100s = 250 - a \implies 15a + 10(250 - a) = 3100 \implies 15a + 2500 - 10a = 3100.
Using substitution eliminates variable ss to solve for aa.
3
Solve for aa and then find ss.
5a=600  ⟹  a=1205a = 600 \implies a = 120; s=250−120=130s = 250 - 120 = 130.
Dividing 600600 by 55 gives a=120a = 120, and subtracting 120120 from 250250 gives s=130s = 130.
4
Evaluate the statements against the values a=120a = 120 and s=130s = 130.
s=130s = 130 is true. Revenue from adult tickets is 120×15=$1,800120 \times 15 = \$1,800, which is also true.
Comparing calculated values directly confirms which options state accurate quantitative properties.

Key Concept

Solving 2x2 Systems of Linear Equations using Substitution or Elimination in Word Problems
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Topics

Systems of Linear Equations

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