Problem 28 - Prealgebra

Question

Mariko purchased three movie tickets for 24 dollars . Define a variable. Then write an equation that can be used to find how much Mariko paid for each ticket. What was the cost of each ticket?

Step-by-Step Solution

Answer

Each movie ticket cost $8.

Step 1: Define the Variable

Let's define the variable \( x \) to represent the cost of one movie ticket. By using this variable, we can set up an equation to represent the total cost.

Step 2: Set Up the Equation

Since Mariko purchased three tickets, the total cost of the tickets is 24 dollars. We can express this as an equation: \( 3x = 24 \). This equation means that three times the cost of one ticket is equal to 24 dollars.

Step 3: Solve for the Variable

To find the cost of one ticket, we need to solve the equation for \( x \). We do this by dividing both sides of the equation by 3: \( x = \frac{24}{3} \).

Step 4: Calculate the Cost

Now, perform the division: \( \frac{24}{3} = 8 \). This tells us that \( x = 8 \).

Step 5: Conclusion

Therefore, each ticket cost Mariko 8 dollars.

Key Concepts

Variable Definition

In mathematics, a variable is simply a symbol used to represent an unknown number. In our exercise, we need to determine what each movie ticket costs. To do so, we define a variable, usually represented by a letter such as \( x \). This variable serves as a placeholder for the unknown quantity we want to find. Defining \( x \) as the cost of one movie ticket provides us with a mathematical tool to work through the problem systematically. By defining the variable, we can express complex relationships in a mathematical form, making problem-solving easier.

Solving Equations

Once we have an equation set up, the next step is to solve it. The principle of solving equations involves finding the value of the variable that makes the equation true. This is a foundational skill in algebra.In our specific exercise, the equation is \( 3x = 24 \). Solving this equation involves isolating \( x \). To do this, we perform inverse operations to simplify the equation. We divide both sides by 3, which gives us \( x = \frac{24}{3} \). Finally, calculate the division to determine \( x = 8 \). Hence, the solution is found: every ticket costs 8 dollars. This systematic approach is vital for solving any algebraic equation.

Equation Setup

Setting up the equation is an essential part of solving problems in algebra. It involves translating words into mathematical statements. In our problem, Mariko bought three tickets for a combined total price of 24 dollars. When setting up the equation, we consider:

• The number of items: 3 movie tickets.
• The total cost: 24 dollars.

This results in the equation \( 3x = 24 \). This equation succinctly captures the relationship between the cost per ticket and the total amount paid. Correctly setting up the equation is critical, as it dictates the correctness of the solution. Practice this skill to become proficient in breaking down and understanding word problems.

Problem Solving Steps

Breaking down the problem-solving steps into smaller, manageable parts can make it much easier to understand complex problems. Here's a simple breakdown of our exercise:

• **Define the variable:** Start by identifying what the variable represents in the problem. In this case, it was the cost of one ticket, \( x \).
• **Set up an equation:** Use the information from the problem to create an equation. Here, we used \( 3x = 24 \).
• **Solve the equation:** Manipulate the equation to solve for the unknown variable by performing operations like addition, subtraction, multiplication, or division.
• **Calculate:** Perform any necessary calculations to find the solution, which was dividing 24 by 3 to get 8.
• **Review:** Double-check your work for accuracy. Ensure each step logically follows the previous one and that calculations are correct.

These structured steps facilitate systematic thinking and enhance problem-solving skills. By applying these steps consistently, solving algebraic problems becomes a straightforward task.