Problem 16 - Algebra for College Students

Question

For a high school production of Hello, Dolly!, student tickets cost \(\$ 5\) each and nonstudent tickets cost \(\$ 8\) each. If 480 tickets were sold and a total of \(\$ 2895\) was collected, how many tickets of each type were sold?

Step-by-Step Solution

Answer

315 student tickets and 165 nonstudent tickets were sold.

Step 1: Define Variables

Let the number of student tickets sold be denoted by \( s \), and the number of nonstudent tickets sold be denoted by \( n \).

Step 2: Set Up the System of Equations

Create equations based on the given conditions:1. The total number of tickets: \( s + n = 480 \)2. The total amount of money collected: \( 5s + 8n = 2895 \)

Step 3: Solve the First Equation for One Variable

Solve the first equation for \( s \):\[ s = 480 - n \]

Step 4: Substitute into the Second Equation

Substitute \( s = 480 - n \) into the second equation:\[ 5(480 - n) + 8n = 2895 \]

Step 5: Simplify and Solve for \( n \)

Expand and simplify the equation:\[ 2400 - 5n + 8n = 2895 \]Combine like terms:\[ 3n = 495 \]Solve for \( n \):\[ n = 165 \]

Step 6: Solve for \( s \)

Use the value of \( n \) to find \( s \):\[ s = 480 - 165 = 315 \]

Step 7: Verify the Solution

Verify the solution by checking both conditions:1. Total tickets: \( 315 + 165 = 480 \), which is correct.2. Total amount: \( 5 \times 315 + 8 \times 165 = 1575 + 1320 = 2895 \), which is correct.

Key Concepts

Linear Equations

Linear equations are mathematical statements that describe a line and involve variables raised to the power of one. In this exercise, we deal with two linear equations that represent the total number of tickets sold and the total amount collected. Linear equations are important as they help us find the relationship between different quantities. The standard form looks like this: a\(x\) + b\(y\) = c\, where a, b, and c are constants and x & y are variables. Here, we set up our variables and equations simply:

• The number of student tickets: s
• The number of nonstudent tickets: n
• Total tickets: s + n = 480
• Total amount collected: 5s + 8n = 2895

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves solving one equation for one variable in terms of the other. Then, we substitute this expression into the second equation. In our problem, we solved the first equation for s: s = 480 - n Next, we substituted this result into the second equation: 5(480 - n) + 8n = 2895 By simplifying, we focused on solving for one variable at a time, making the equations easier to manage. The substitution method helps break down complex problems into simpler steps.

Problem-Solving in Algebra

Algebra is a powerful toolkit for solving various problems. This exercise illustrates practical problem-solving steps:

• First, define variables to represent unknown quantities.
• Next, set up equations based on the problem’s conditions.
• Solve the equations using methods like substitution or elimination.
• Finally, verify the solution to check correctness.

Algebraic problem-solving develops critical thinking and makes it easier to tackle real-world challenges through mathematical models.

Verification of Solutions

Verification ensures that our solutions satisfy the original problem. After finding s and n, we verified both conditions:1. The total number of tickets: 315 + 165 = 480 2. The total amount collected: 5\(315\) + 8\(165\) = 1575 + 1320 = 2895 Both checks confirmed our solution. Verification provides confidence in our answers and helps identify mistakes that may have occurred during calculations.