Problem 39 - Elementary Algebra Within Reach

Question

A movie rental costs \(\$ 3\) per day. A video game rental costs \(\$ 4\) per day. Write an algebraic expression that represents the total cost of renting \(m\) movies and \(v\) video games per day.

Step-by-Step Solution

Answer

The algebraic expression representing the total cost of renting `m` movies and `v` games per day is \(3m + 4v\).

Step 1: Calculate the Cost of renting movies

The cost of renting one movie is $3. So, for `m` number of movies, the total cost would be \(3m\). (Multiplying the cost of one movie with the number of movies)

Step 2: Calculate the Cost of renting video games

The cost of renting one video game is $4. Thus, for `v` number of video games, the total cost would be \(4v\). (Multiplying the cost of one game with the number of games)

Step 3: Calculate the Total cost

The total cost for renting both items would be the sum of the cost of renting movies and the cost of renting games. That is \(3m + 4v\) (Adding the product of cost of movies, `m`, and cost of games, `v`)

Key Concepts

Cost Calculation

When you're looking to calculate costs, especially for something like renting items, it's crucial to understand how to determine the total amount you will need to pay. In this scenario, we're dealing with the daily rental costs of movies and video games.

• The daily cost for one movie is \( \\(3 \).
• The daily cost for one video game is \( \\)4 \).

If you're renting multiple items, like several movies and video games, you'd want to calculate the entire cost accurately. You'll multiply the cost of each item by the number you plan to rent. Then, add these totals together to get the combined cost for all rentals. This way, you ensure that you've accounted for every item and computed the total correctly.
In algebraic terms, you express this entire process using an expression or equation that makes calculations repeatable and consistent.

Variables in Algebra

In algebra, we often use variables as symbols to represent numbers or quantities that can change. In the context of this exercise:

• The variable \( m \) stands for the number of movies rented.
• Similarly, \( v \) denotes the number of video games rented.

Variables help simplify problems by letting us create a general formula or expression that can adapt to different situations.
For example, whether you're renting 2 or 20 movies, by plugging the relevant number into the variable \( m \), your expression \( 3m + 4v \) remains valid and provides you with the result for any given number of rentals.
This flexibility is one of the powerful aspects of using algebraic variables, making it easier to handle dynamic problems, such as varying quantities and costs.

Basic Arithmetic Operations

Basic arithmetic operations form the backbone of algebra and number manipulation. In this exercise, we're using two core operations: multiplication and addition.

• **Multiplication:** This helps us find the cost for several items of the same type. If each movie costs \( \$3 \), then renting \( m \) movies will cost \( 3m \). Similarly, for video games, you calculate \( 4v \).
• **Addition:** Once individual costs are calculated for movies and games, they are summed up. This operation gives us the total cost when you add the expressions \( 3m \) and \( 4v \) to get \( 3m + 4v \).

Understanding these operations allows you to tackle varied problems that involve calculating totals, building equations, or even solving them. By combining multiplication and addition effectively, we get accurate and comprehensive solutions that support our cost calculations across different scenarios.