Problem 27 - Prealgebra

Question

One movie ticket costs 7 dollars, and one small bag of popcorn costs 3 dollars. Write two equivalent expressions for the total cost of four movie tickets and four bags of popcorn. Then find the cost.

Step-by-Step Solution

Answer

Two equivalent expressions are \( 28 + 12 \) and \( 4(7 + 3) \). The total cost is 40 dollars.

Step 1: Define the Cost of Tickets and Popcorn

First, we identify the given values in the problem. The cost of one movie ticket is 7 dollars, and the cost of one small bag of popcorn is 3 dollars.

Step 2: Calculate Total Cost for Tickets

Multiply the cost of one movie ticket by the number of tickets. The total cost for four movie tickets is: \( 4 \times 7 = 28 \) dollars.

Step 3: Calculate Total Cost for Popcorn

Multiply the cost of one bag of popcorn by the number of bags. The total cost for four bags of popcorn is: \( 4 \times 3 = 12 \) dollars.

Step 4: Write the First Expression for Total Cost

Add the total cost of the movie tickets and popcorn together to find the first expression: \( 28 + 12 \).

Step 5: Write the Simplified Expression

Calculate \( 28 + 12 \) to simplify the expression: \( 40 \). So, \( 28 + 12 = 40 \).

Step 6: Write an Equivalent Expression

Factor the common factor out of the expression: \( 4(7 + 3) \). This is because both costs were multiplied by 4. First, calculate \( 7 + 3 = 10 \), so the expression inside the brackets is 10. Then, multiply 4 by this total, which equals 40.

Key Concepts

Understanding Equivalent Expressions

When you break down the problem of calculating costs, you come face to face with the concept of equivalent expressions. Equivalent expressions might look different initially, but after evaluating, they yield the same result.
In the exercise, we first find the total costs of movie tickets and popcorn separately:

• The cost of four tickets is calculated as \(4 \times 7 = 28\) dollars.
• The cost of four popcorn bags is \(4 \times 3 = 12\) dollars.

Adding these, we have the expression \(28 + 12\) which simplifies to 40. This is our first expression.
However, there's yet another way to represent the total cost. We can factor common values: both tickets and popcorn are multiplied by four. So, we can express this as \(4(7 + 3)\), which simplifies to \(4 \times 10 = 40\).
Both expressions, \(28 + 12\) and \(4(7 + 3)\), are equivalent, proving that even if expressions appear different, they can represent the same value when evaluated.

Cost Calculation in Real-life Scenarios

Cost calculation is a vital part of our daily lives—whether you’re buying groceries, planning a trip, or heading out for some entertainment.
Let’s take the example of going to a movie with snacks. You want to ensure that you have enough money for both the movie tickets and popcorn.

• First, determine the number of items you're purchasing and their individual costs.
• Calculate the total cost for each type of item by multiplying the number of items by the cost per item.
• Finally, add up all these individual totals to get the total expenditure.

In the given problem, this approach ensures you don't overspend and helps you keep track of your budget. It's the math that empowers you with financial insights!

Mastering Factoring in Algebra

Factoring is a fundamental algebraic technique that involves expressing a number or an expression as a product of its factors. This is especially useful when simplifying expressions to make calculations easier or when finding equivalent expressions.
In the movie ticket and popcorn example, the factoring process is demonstrated in the expression \(28 + 12\). Here both 28 and 12 can be expressed in terms of 4:

• 28 can be seen as \(4 \times 7\)
• 12 as \(4 \times 3\)

Recognizing 4 as a common factor, we can pull it out: \(4(7 + 3)\). This expression is not only simpler but highlights how each part can be viewed together, presenting a unified picture.
Factoring helps in solving equations, simplifying calculations, and understanding relationships between numbers or expressions in mathematics.