Problem 47 - Prealgebra

Question

A ticket to a soccer game is \(\$ 12,\) a team pennant is \(\$ 7,\) and a T-shirt is \(\$ 15 .\) Write two equivalent expressions for the total cost of a group outing for 10 people if each person buys a ticket, a pennant, and a T-shirt. Then find the cost, lesson 3.1 )

Step-by-Step Solution

Answer

The total cost for 10 people is 340 dollars.

Step 1: Calculate Individual Cost

To find the expression for the total cost for one person, add the cost of the ticket, the pennant, and the T-shirt: \( 12 + 7 + 15 \)

Step 2: Simplify Individual Cost

Add the individual costs: \( 12 + 7 + 15 = 34 \)So, the cost for one person is \( 34 \) dollars.

Step 3: Write the Total Cost Expression - Method 1

For 10 people, multiply the individual cost by 10: \( 10 \times 34 \)

Step 4: Write the Total Cost Expression - Method 2

Alternatively, you can distribute the multiplication over addition: \( 10 \times (12 + 7 + 15) \)

Step 5: Calculate Total Cost

Calculate the total cost using the simplified expression: \( 10 \times 34 = 340 \)

Step 6: Verify with Second Expression

Verify using the second expression: First, check inside the parentheses: \( 12 + 7 + 15 = 34 \) Then multiply: \( 10 \times 34 = 340 \)

Key Concepts

Understanding Expressions in Prealgebra

In prealgebra, expressions are combinations of numbers, variables, and arithmetic operators like addition and multiplication. They represent quantities or values without stating an exact number. Understanding expressions is crucial because it helps you simplify and solve different types of algebraic problems.

For example, in a group outing scenario, the expression for a single person buying a ticket, pennant, and T-shirt is:

• Ticket: \(12\) dollars
• Pennant: \(7\) dollars
• T-Shirt: \(15\) dollars

Instead of repeating additions, combine these individual costs into one expression: \(12 + 7 + 15\). This expression succinctly represents the total cost for one person attending the event.

The Role of Multiplication in Group Cost Calculation

Multiplication is a key mathematical operation that helps in scaling numbers, making it highly useful in cost calculations for groups.

Once you know the cost of one item or one unit, multiplication allows you to quickly find the total cost for multiple items or units. For instance, if the total cost for each person at the soccer game is \(34\), multiplying by the number of people (\(10\)) gives you the total for the group:

• \(10 \times 34 = 340\)

Just like that, you see how multiplication makes calculating larger totals straightforward and efficient.

Utilizing Distribution in Algebraic Expressions

The distributive property is a helpful tool that allows you to multiply a single term by all parts within a parenthesis.

In a situation like the one in our group outing, it shows how you can apply distribution to simplify calculations effectively. For example, given the expression:

• \(10 \times (12 + 7 + 15)\)

yeah.This expression breaks down to distributing \(10\) across each individual cost:

• \(10 \times 12\)
• \(10 \times 7\)
• \(10 \times 15\)

Then, adding those results together, confirming that both methods of calculating can lead to the same result, \(340\). This property is critical in verifying the accuracy of algebraic calculations.

Accurate Cost Calculation for Group Activities

Calculating costs effectively for group activities involves creating logical expressions and applying arithmetic operations accurately.

This process is essential in ensuring that you budget correctly for events. In our example, once you know the costs involved per person, multiplying the total by the group size efficiently provides the complete cost of the outing.

• Cost for each person: \(34\) dollars
• Total for 10 people: \(34 \times 10 = 340\) dollars

Ensuring you've used the correct numbers and operations like addition and multiplication verifies you have an accurate total. This prevents financial mishaps and helps maintain control over expenses.