Problem 3 - Prealgebra

Question

The cost of 13 gallons of gasoline is 32.37 dollars. Write an equation relating cost to the number of gallons of gasoline. How much does 18.5 gallons of gasoline cost?

Step-by-Step Solution

Answer

The cost for 18.5 gallons is 46.065 dollars.

Step 1: Understanding the Problem

We need to relate the cost of gasoline to the number of gallons purchased. We know that 13 gallons cost 32.37 dollars. We define two variables: let \( C \) be the total cost in dollars, and \( g \) be the number of gallons of gasoline.

Step 2: Determining the Unit Cost

To find the cost per gallon, divide the total cost by the number of gallons. This gives the equation for the unit cost: \( \text{unit cost} = \frac{32.37}{13} \).

Step 3: Calculating the Unit Cost

Calculate \( \frac{32.37}{13} \) to find the cost per gallon: \( 32.37 \div 13 = 2.49 \). Thus, the cost per gallon is 2.49 dollars.

Step 4: Forming the Equation

Now that we know the cost per gallon, the equation relating the cost \( C \) to the number of gallons \( g \) is \( C = 2.49g \).

Step 5: Applying the Formula

Use the equation \( C = 2.49g \) to find the cost for 18.5 gallons. Substitute \( g = 18.5 \) into the equation: \( C = 2.49 \times 18.5 \).

Step 6: Solving for 18.5 Gallons

Calculate \( 2.49 \times 18.5 \): \( C = 46.065 \). Therefore, the cost for 18.5 gallons is 46.065 dollars.

Key Concepts

Unit Rate

The concept of a unit rate is essential in understanding and comparing different values or quantities. In this case, the unit rate is the cost per gallon of gasoline. When determining the unit rate, you divide the total cost by the total quantity. For example, if 13 gallons of gasoline cost $32.37, you would divide 32.37 by 13.

• This gives you the cost for each single unit (or in this case, a single gallon) of gasoline.

Computing the unit rate allows you to establish a baseline to make simple cost calculations for different quantities without having to calculate from scratch each time.

Cost Calculation

Once you have the unit rate, calculating cost becomes straightforward. You simply multiply the unit rate by the desired number of units.

• For example, if the unit cost for gasoline is $2.49 per gallon, and you want to know the cost for 18.5 gallons, you compute this by multiplying 2.49 (the unit rate) by 18.5 (the number of gallons).

This systematic approach makes it easy to adjust your cost calculations based on changing quantities or variables, ensuring accuracy in financial estimations.

Problem Solving

When faced with an algebraic problem like this one, it is important to strategically break it down into manageable steps. Start by clearly understanding what you need to solve.

• Identify the known information: In this case, 13 gallons cost $32.37.
• Define your goal: Determine the cost for 18.5 gallons of gasoline.

By organizing the information and systematically working through the steps, such as finding the unit rate and then calculating the total cost based on that rate, you enable a smooth problem-solving process.

Variable Definition

In algebra, defining variables is a crucial initial step in constructing equations to solve problems. For instance, when you see the variable \( C \), it represents the total cost in dollars, while \( g \) signifies the number of gallons purchased.

• Defining variables provides clarity and structure to mathematical modeling and reasoning.
• It also enables easier manipulation and calculation as you progress through the problem-solving process.

Clearly defined variables lead to successfully formulating equations and deriving solutions logically and efficiently.