Question

A company rents moving vans for a rental fee of \(\$ 25.00\) per day with an additional charge of \(\$ 0.30\) per mile that the van is driven. Which of the following expressions represents the cost, in dollars, of renting a van for 1 day and driving it \(m\) miles? A. \(0.30 m+25\) B. \(25 m+30\) C. \(30 m+25\) D. 25.30\(m\) E. 55\(m\)

Short answer

Answer: The correct expression for the total cost is \(0.30m + 25\).

Step-by-step solution

Identifying cost components

First, let's identify the two components of the cost: 1. A daily rental fee of $25.00. 2. A per-mile fee of \(0.30 \times m\), where \(m\) is the number of miles driven. Now, we need to combine these two components to find the total cost.

Combining cost components

To find the total cost, we will add the daily rental fee to the per-mile fee. This will give us the following expression: Total cost = Daily rental fee + Per-mile fee Total cost = \(\$25.00\) + \(\$0.30 \times m\)

Simplifying the expression

To simplify the expression, we can rewrite it as: Total cost = \(0.30m + 25\)

Identifying the correct expression

Comparing our result with the given options, we can see that it matches option A: Total cost = \(0.30m + 25\) So, option A represents the correct expression for the cost of renting a van for one day and driving it \(m\) miles.

Key concepts

Algebraic Expressions

Algebraic expressions are a cornerstone of mathematics, particularly when it comes to translating real-world scenarios into a language that can be processed mathematically. They consist of variables, numbers, and operations that together form a mathematical phrase without an equal sign. In the context of the ACT Math Practice problem, the algebraic expression \(0.30m + 25\) succinctly represents the total cost of renting a moving van for a day plus the mileage fee.

An expression like this is valuable because it allows flexibility. Substituting any value for \(m\), the number of miles driven, quickly shows the user the corresponding cost. There is a direct link between the algebraic representation and the particularity of every individual rental scenario, making it an efficient tool to handle variations in use. When creating an algebraic expression from a word problem, students must carefully consider units (\$ per day, \$ per mile), operation symbols (plus for addition, etc.), and the order in which these costs must be combined according to the problem's context.

Word Problems

Word problems can often seem daunting as they require the translation of textual information into a mathematical format. To tackle them effectively, it's important to read the problem carefully, identify the key pieces of information, and understand what the question is asking. In our ACT Math Practice word problem, the key pieces of information are the daily rental fee and the per-mile charge when renting a van. By extracting these numerical values and understanding the relationship between them, one can construct an equation or expression that encapsulates the problem's scenario.

For instance, word problems often involve real-life scenarios like shopping, traveling, or planning events, where costs are calculated based on particular actions. It is important to identify the variables involved (e.g., \(m\) for miles) and the constants (\$25 rental fee), and then to use accurate mathematical operations to reflect the situation described. One of the most effective strategies is to break down the problem into smaller parts, as was done in the step-by-step solution, which aids comprehension and helps in building the final expression.

Cost Calculation

In problems regarding cost calculation, it is essential to differentiate between fixed costs and variable costs. Fixed costs, like the daily rental fee of \$25 in our exercise, do not change regardless of the level of activity. In contrast, variable costs, like the \$0.30 per mile driven, vary with usage. Understanding the distinction between these two types of costs allows students to construct accurate models of financial scenarios.

When calculating total costs, each type of cost must be accounted for appropriately. As seen in the solutions, the calculation method is a straightforward addition of the fixed cost and the total variable cost (which is a product of the per-unit cost and quantity).

General Formula for Total Cost

In general, the total cost (\(TC\)) can be calculated using the formula: \[ TC = FC + (VC \times Q) \] where \(FC\) represents the fixed cost, \(VC\) the variable cost per unit, and \(Q\) the quantity of units. In our word problem, \(TC\) is the total cost to rent the van for a day, \(FC\) is \$25, \(VC\) is \$0.30, and \(Q\) represents the miles driven (\(m\)). This simple yet powerful formula empowers students to deal with complex cost calculations in straightforward real-life situations.