Problem 50 - Chemistry: An Atoms-Focused Approach

Question

A sport-utility vehicle has an average mileage rating of 18 miles per gallon. How many gallons of gasoline are needed for a 389 -mile trip?

Step-by-Step Solution

Answer

Answer: 22 gallons

Step 1: Identify the given values and what we need to find

The given values in this problem are: - Average mileage rating: 18 miles/gallon - Trip distance: 389 miles We need to find the number of gallons needed for a 389-mile trip.

Step 2: Set up a proportion

To find the number of gallons needed, we will set up a proportion as follows: \(\frac{18 \text{ miles}}{1 \text{ gallon}} = \frac{389 \text{ miles}}{x \text{ gallons}}\) Here, we are comparing the average mileage rating with the trip distance and the number of gallons needed for the trip.

Step 3: Solve for x (number of gallons)

To solve for x, we can cross-multiply: \(18 \text{ miles} \times x \text{ gallons} = 1 \text{ gallon} \times 389 \text{ miles}\) Now, divide both sides of the equation by 18 miles: \(x \text{ gallons} = \frac{389 \text{ miles}}{18 \text{ miles}}\)

Step 4: Calculate the number of gallons

Now, let's calculate the value of x: \(x \text{ gallons} = \frac{389}{18}\) \(x \approx 21.61\) Since we cannot have a fraction of a gallon, we will round up to the nearest whole number. Therefore, 22 gallons of gasoline are needed for a 389-mile trip.

Key Concepts

Proportion Method

The proportion method is a helpful and simple mathematical technique used to solve problems involving ratios. When you want to find a missing value that is directly proportional to another, the proportion method comes in handy.

• You set up two fractions that express the same relationship.
• The value you want to find is in one of these fractions as a variable.
• You solve the equation by multiplying across the diagonal (cross-multiplication) and dividing as necessary.

In our example, we're comparing the average mileage of a vehicle with the actual distance traveled.
The equation \[\frac{18 \text{ miles}}{1 \text{ gallon}} = \frac{389 \text{ miles}}{x \text{ gallons}}\]organizes this relationship and allows you to solve for the unknown variable, which is the amount of gasoline needed. Once the equation is set, solving for the missing part is straightforward using algebraic techniques.

Average Mileage

Understanding average mileage is crucial for calculating fuel efficiency. The average mileage, often referred to as fuel economy, tells you how many miles a vehicle can travel per gallon of fuel.

• This is expressed in miles per gallon (mpg).
• It's an important factor when calculating long-distance fuel needs.
• Average mileage can vary based on driving conditions, vehicle maintenance, and other factors.

In our example, the sport-utility vehicle has an average mileage of 18 miles per gallon. Thus, if you need to calculate the fuel needed for a specific trip, you use this average as your baseline. For a 389-mile trip, the consistency of these 18 miles per gallon helps determine that about 22 gallons are needed once we do the math.
This average is a simplified expression of the vehicle's efficiency under typical conditions.

Fuel Consumption

Fuel consumption describes how much gasoline a vehicle uses to travel a specific distance. It's an essential aspect of both cost management and environmental consideration.

• Efficient fuel consumption leads to savings in fuel costs.
• It also reduces environmental impact due to lower emissions.
• Consumption rates can be calculated for individual trips or average use over time.

In practical scenarios, calculating fuel consumption helps in planning road trips, budgeting for fuel expenses, and estimating the frequency of fuel refills.
In connection with our example, fuel consumption was determined through the equation\[x \text{ gallons} = \frac{389 \text{ miles}}{18 \text{ miles per gallon}}\]After calculation, we find out we need approximately 21.61 gallons, which means buying 22 gallons to complete the 389-mile journey without worrying about running out of fuel.