Problem 28 - Prealgebra

Question

Car Rental A car rental company charges \(\$ 27\) a day and 18 cents per mile to rent their cars. If the total charge for a 1 -day rental was \(\$ 48.78,\) how many miles was the car driven?

Step-by-Step Solution

Answer

The car was driven 121 miles.

Step 1: Identify Given Values

The daily rental cost is given as \( \\(27 \), and the per mile charge is \( \\)0.18 \) (since 18 cents equals \( \\(0.18 \)). The total charge for a 1-day rental is \( \\)48.78 \). We need to find out how many miles the car was driven.

Step 2: Set Up the Equation

To find the number of miles driven, define the number of miles driven as \( x \). The total cost equation is: \[ 27 + 0.18x = 48.78 \]

Step 3: Isolate the Variable

Subtract the daily rental fee from the total cost to isolate the mileage cost: \[ 0.18x = 48.78 - 27 \]\[ 0.18x = 21.78 \]

Step 4: Solve for Miles Driven

Divide both sides by \( 0.18 \) to solve for \( x \): \[ x = \frac{21.78}{0.18} \]\[ x = 121 \]

Step 5: Verify the Solution

To ensure correctness, substitute \( 121 \) back into the cost equation: \[ 27 + 0.18 \times 121 = 48.78 \]Calculate: \[ 27 + 21.78 = 48.78 \], which is correct.

Key Concepts

Equation Solving

Equation solving is a fundamental part of prealgebra that involves finding the value of an unknown variable. In this car rental problem, we used an equation to understand the relationship between the total cost, the daily rental fee, and the cost per mile driven.

To solve such problems, you must first identify the unknown variable. Here, it's the number of miles driven, which we labeled as \( x \). We then expressed the problem in the form of an equation that balances costs and charges. The equation can be written as:

• \( 27 + 0.18x = 48.78 \), where \( 27 \) is the fixed daily cost, \( 0.18x \) is the variable cost depending on miles driven, and \( 48.78 \) is the total cost.

To solve it, you need to isolate \( x \) by performing arithmetic operations that keep the equation balanced. Simplifying both sides step-by-step lets you find the exact value of \( x \), which tells you how many miles were driven.

Linear Equations

Linear equations are equations of the first degree, meaning they involve no powers higher than one. They typically take the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants.

In our problem, the line equation is expressed as:

• \( 27 + 0.18x = 48.78 \)

This represents a direct proportional relationship between the number of miles driven and the total cost.

Whenever you deal with linear equations, the goal is to perform operations that simplify the equation by isolating the variable, \(x\) in this case. You start by eliminating constants from one side, as shown when subtracting \(27\) here, and then simplifying the expression, such as dividing by the coefficient \(0.18\). Successfully solving a linear equation like this allows you to determine the unknown variable's value, which is crucial for answering the word problem accurately.

Unit Rate Calculation

Unit rate calculation helps to understand costs per single unit, such as per mile or per item. In this exercise, 18 cents per mile is the unit rate, telling us the cost associated with each mile driven.

The skill of calculating unit rates is essential for breaking down complex costs and understanding the relationship between different types of fees. With the unit rate \(0.18\) identified, you multiply it by the number of miles, \(x\), to find the total variable cost contributed by mileage.

Using unit rates efficiently involves setting up a multiplication equation where the unit rate is multiplied by a quantity to find a total. Here:

• \( 0.18x \) gives the added cost from miles.

Practicing this concept makes calculations in both math problems and real-world scenarios simpler and clearer, as demonstrated in our rental exercise.