Problem 180 - Elementary Algebra

Question

If the number of \(\$ 8\) child tickets is seventeen less than three times the number of \(\$ 12\) adult tickets and the theater took in \(\$ 584,\) how many of each ticket were sold?

Step-by-Step Solution

Answer

20 adult tickets and 43 child tickets were sold.

Step 1: Define variables

Let the number of adult tickets be denoted by \( A \) and the number of child tickets be denoted by \( C \).

Step 2: Set up the equations

According to the problem, the relationship between child tickets and adult tickets is given by:\[ C = 3A - 17 \]Additionally, the total revenue from the tickets sold is given by:\[ 12A + 8C = 584 \]

Step 3: Substitute \( C \) into the revenue equation

Replace \( C \) in the total revenue equation with \( 3A - 17 \):\[ 12A + 8(3A - 17) = 584 \]Simplify this equation to:\[ 12A + 24A - 136 = 584 \]

Step 4: Solve for \( A \)

Combine like terms and solve for \( A \):\[ 36A - 136 = 584 \]Add 136 to both sides:\[ 36A = 720 \]Divide by 36:\[ A = 20 \]

Step 5: Calculate \( C \)

Substitute \( A = 20 \) back into the equation \( C = 3A - 17 \):\[ C = 3(20) - 17 \]Simplify to find \( C \):\[ C = 60 - 17 = 43 \]

Key Concepts

linear equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. In the given exercise, we have two main linear equations. The first equation relates the number of child tickets (C) to the number of adult tickets (A): C = 3A - 17. The second equation considers the total revenue received from selling the tickets: 12A + 8C = 584. Both equations have variables raised to the first power, which makes them linear.

variable substitution

Variable substitution is a method used to solve systems of equations, where you solve one equation for one variable and then substitute that expression into the other equation. In our exercise, we first rearranged the equation C = 3A - 17 to express C in terms of A. Then, we substituted 3A - 17 for C in the second equation: 12A + 8C = 584. By doing this, we transformed the revenue equation into: 12A + 8(3A - 17) = 584, allowing us to solve for A without needing to deal with C directly.

problem solving steps

Breaking down the problem into clear, manageable steps can make it easier to solve. Here are the steps taken in this exercise:

• Step 1: Define the variables. Let A represent the number of adult tickets and C represent the number of child tickets.
• Step 2: Set up the equations. Use the information provided to relate A and C and to set up the revenue equation.
• Step 3: Substitute one variable into the other equation. Solve for one variable in terms of the other to reduce the number of variables.
• Step 4: Solve for the first variable. Simplify the equation and solve for the remaining variable.
• Step 5: Solve for the second variable. Use the solution from step 4 to find the value of the other variable.

Following these steps methodically helps streamline the problem-solving process.

algebraic expressions

An algebraic expression is a mathematical phrase that involves numbers, variables, and operations. In our problem, we work with expressions like 3A - 17 and 12A + 8C. Algebraic expressions are used to model real-world situations mathematically. For example, 3A - 17 models the number of child tickets in relation to adult tickets, while 12A + 8C represents the total revenue from ticket sales. By manipulating these expressions, we can solve for unknown quantities and gain insight into the problem.