Problem 63 - Prealgebra

Question

A ticket to a baseball game costs \(t\) dollars. A soft drink costs \(s\) dollars. Which expression represents the total cost of a ticket and soft drink for \(p\) people? A. \(p s t\) C. \(t(p+s)\) B. \(p+(t s)\) D. \(p(t+s)\)

Step-by-Step Solution

Answer

D. \( p(t+s) \)

Step 1: Define the Problem

The task is to express the total cost for purchasing tickets and drinks for several people. We have %t% dollars as the cost of one ticket, %s% dollars as the cost for one soft drink, and %p% is the number of people.

Step 2: Cost Calculation per Person

Each person needs a ticket and a drink. Therefore, the cost per person is the sum of the ticket price and the soft drink price, calculated by the expression %t + s%.

Step 3: Total Cost for Multiple People

Since we need to calculate the cost for %p% people, we multiply the individual cost per person by the number of people. This results in the expression %p(t + s)%.

Step 4: Identify the Correct Expression

The expression %D. p(t+s)% matches our derived formula from Step 3, hence it is the correct choice.

Key Concepts

Cost Calculation

Understanding cost calculation is essential when dealing with money matters, especially when you have multiple items to purchase. In this exercise, we need to find the total cost for a group of people who are going to buy both a ticket and a soft drink per person. Each ticket costs \( t \) dollars, and each soft drink costs \( s \) dollars.

• The cost for one individual is calculated by adding together the cost of the ticket and the drink: \( t + s \).
• Once we have the individual cost, we need the total cost for all \( p \) people attending. This is done by multiplying the individual cost by the number of people: \( p(t + s) \).

This process helps us ensure that we are accounting for all expenses systematically. It's a valuable skill when budgeting or planning for events.

Variables in Equations

Variables are symbols that represent changing values or quantities in mathematical expressions and equations. In this problem, we use variables to represent the costs for tickets, drinks, and the number of people.

• \( t \) represents the price of one ticket.
• \( s \) stands for the price of a single drink.
• \( p \) denotes the number of people attending the game.

When working with equations, you assign values to these variables. This lets you find concrete totals or make calculations that account for different scenarios without rewriting the entire expression. By defining the variables, we create a flexible and functional model of a real-world situation.

Multiplication of Terms

Multiplication of terms is a fundamental operation in algebra that allows us to scale values. In the context of cost calculation, it helps us determine the total cost based on quantity.
Consider that the cost for one person is \( t + s \). Now, if you want the total for multiple people, say \( p \), you multiply \( p \) by \( t + s \):\[ (t + s) \times p \]The product \( p(t + s) \) is achieved by multiplying each term inside the parentheses by \( p \) thereby scaling the cost of one person to the cost of \( p \) people. This shows the power of multiplication in managing and understanding collective cost in situations involving groups.