Problem 32 - College Algebra with Modeling and Visualization

Question

One thousand tickets were sold for a baseball game. There were one hundred more adult tickets sold than student tickets, and there were four times as many tickets sold to students as to children. How many of each type of ticket were sold?

Step-by-Step Solution

Answer

100 children's tickets, 400 student tickets, and 500 adult tickets sold.

Step 1: Define Variables

Let's define the variables for the number of each type of ticket sold. Let \( x \) be the number of children tickets sold, \( y \) be the number of student tickets sold, and \( z \) be the number of adult tickets sold.

Step 2: Set Up Equations

We are given: 1. The total number of tickets is 1000: \[ x + y + z = 1000 \] 2. There were 100 more adult tickets than student tickets: \[ z = y + 100 \] 3. There were 4 times as many student tickets as children tickets: \[ y = 4x \]

Step 3: Substitute Equations

First, substitute the expression for \( z \) from the second equation into the first equation: \[ x + y + (y + 100) = 1000 \] Simplify this to get: \[ x + 2y + 100 = 1000 \] Subtract 100 from both sides: \[ x + 2y = 900 \]

Step 4: Substitute Further

Next, substitute the expression for \( y \) from the third equation into \( x + 2y = 900 \): \[ x + 2(4x) = 900 \] Simplify: \[ x + 8x = 900 \] Combine like terms: \[ 9x = 900 \]

Step 5: Solve for Children's Tickets

Solve for \( x \): \[ x = \frac{900}{9} \] \[ x = 100 \] So, 100 children tickets were sold.

Step 6: Solve for Student Tickets

Substitute \( x = 100 \) back into the equation \( y = 4x \): \[ y = 4(100) \] \[ y = 400 \] So, 400 student tickets were sold.

Step 7: Solve for Adult Tickets

Substitute \( y = 400 \) into the equation \( z = y + 100 \): \[ z = 400 + 100 \] \[ z = 500 \] So, 500 adult tickets were sold.

Key Concepts

System of Equations

In the realm of algebra word problems, one frequent task is to analyze situations using a system of equations. This involves finding the unknowns using multiple equations that are interconnected. In this exercise, each equation represents a condition given in the word problem.

Key benefits of using a system of equations include:

• Ability to solve for multiple variables simultaneously.
• Use of algebraic expressions to model real-world problems.
• Help in verifying solutions by checking if they satisfy all equations.

For the baseball game ticket problem, our system comprises three linear equations derived from the descriptions:

• Equation for total tickets: \(x + y + z = 1000\)
• Equation relating adult and student tickets: \(z = y + 100\)
• Equation relating student and children tickets: \(y = 4x\)

Using this system, we can uncover the values for each variable by methodically solving these linked equations.

Variable Definition

The initial step in tackling any algebra word problem is defining variables. Variables act as placeholders for unknown quantities that we need to find. In our exercise, defining the variables is crucial to translate the word problem into mathematical equations.

For the problem on ticket sales:

• Let \(x\) represent the number of children tickets sold.
• Let \(y\) represent the number of student tickets sold.
• Let \(z\) represent the number of adult tickets sold.

By defining variables clearly, we establish a clear path from the problem's narrative to its numerical resolution. This clarity ensures we correctly formulate each equation needed to solve the problem. Whenever dealing with word problems, always start by considering what each unknown is and clearly denote it with a variable.

Substitution Method

The substitution method is a common strategy used to solve systems of equations. It involves solving one equation for a single variable and substituting this expression into another equation. This process reduces the number of variables in an equation, making it simpler to solve.

In our ticket sales exercise, the substitution method is used effectively:

• First, we express \(z\) in terms of \(y\) from the equation \(z = y + 100\) and substitute it into the equation for total tickets.
• This reduces the problem to an equation with two variables: \(x + 2y = 900\).
• Next, express \(y\) in terms of \(x\) from \(y = 4x\), and substitute again, simplifying to the equation \(9x = 900\).

The substitution method continues to whittle down the equation until it reveals clear values for \(x\), then \(y\), and finally \(z\). It's a step-by-step approach that brings clarity to complex systems, making it easier to find each unknown. Always be sure to systematically substitute and simplify, verifying that solutions fulfill all initial equations.