Solving Fuel Efficiency Equations Miles Per Gallon Calculation

In this article, we will explore the concept of fuel efficiency and how to determine the average miles a car travels on each gallon of gas. We'll break down the problem, identify the key components, and construct an equation that accurately represents the scenario. We will focus on Adam's car, which traveled 465.3 miles on 16.5 gallons of gas, and delve into the process of finding the right equation to calculate its fuel efficiency. Understanding this process is crucial for solving similar problems related to rates, ratios, and proportions, which are fundamental concepts in mathematics and everyday life.

Decoding the Problem

Before diving into the equations, let's first understand the problem statement clearly. Adam's car covered a total distance of 465.3 miles using 16.5 gallons of gasoline. The core question is: How many miles did the car travel on average for each gallon of gas? This question leads us to the concept of miles per gallon (MPG), a common metric for fuel efficiency. To calculate MPG, we need to find the ratio of the total distance traveled to the amount of fuel consumed. This involves setting up an equation that correctly represents this relationship. We need to identify the unknown variable, which in this case is the average number of miles traveled per gallon of gas, and represent it with a symbol, such as 'x'. Understanding the relationship between the total distance, the amount of fuel, and the miles per gallon is crucial for setting up the correct equation. We will explore different ways to represent this relationship mathematically and evaluate the options to determine which one accurately models the scenario. This step-by-step approach will help in not only solving this specific problem but also in developing a general problem-solving strategy for similar mathematical challenges. By the end of this section, we will have a clear understanding of what we are trying to find and the information we have at our disposal.

Identifying the Key Components

To formulate the correct equation, we need to identify the key components in the problem. These components are the total distance traveled, the amount of fuel consumed, and the unknown variable, which is the average number of miles per gallon. In this scenario, the total distance is 465.3 miles, and the amount of fuel used is 16.5 gallons. The unknown variable, which we will represent as 'x', is the miles per gallon (MPG). The relationship between these components can be expressed as: Miles per Gallon (x) = Total Distance / Amount of Fuel. This fundamental relationship is the basis for setting up the equation. Now, we need to translate this relationship into a mathematical equation using the given values and the variable 'x'. Understanding the units of measurement is also crucial. The total distance is in miles, and the amount of fuel is in gallons, so the resulting miles per gallon will be in miles per gallon, which aligns with our understanding of fuel efficiency. This step of identifying key components and their relationship is crucial for translating real-world problems into mathematical expressions. By clearly defining the variables and their connections, we can avoid errors in setting up and solving the equation. This methodical approach is a valuable skill in mathematical problem-solving.

Constructing the Equation

Now that we have identified the key components, we can construct an equation to represent the problem. We know that the average number of miles per gallon (x) is equal to the total distance (465.3 miles) divided by the amount of fuel used (16.5 gallons). This can be expressed mathematically as: x = 465.3 / 16.5. However, the given options do not directly present this equation. Instead, they offer variations that we need to evaluate. The key is to understand that division can also be represented as a multiplication relationship. The equation x = 465.3 / 16.5 can be rearranged to isolate the total distance. If we multiply both sides of the equation by 16.5, we get: 16. 5x = 465.3. This equation states that 16.5 times the average number of miles per gallon (x) is equal to the total distance of 465.3 miles. This is a crucial step in understanding how different equation formats can represent the same relationship. We can also analyze the other options provided to understand why they are incorrect. For example, an equation like x / 16.5 = 465.3 implies that the average miles per gallon divided by the amount of fuel equals the total distance, which is not the correct relationship. Similarly, 465.3x = 16.5 suggests that the total distance multiplied by the average miles per gallon equals the amount of fuel, which is also incorrect. By understanding how to manipulate and rearrange equations, we can confidently identify the one that accurately represents the problem scenario.

Evaluating the Options

With the equation 16.5x = 465.3 derived from the problem statement, we can now evaluate the given options to determine the correct one. Let's look at each option:

• Option 1: 16.5x = 465.3

  This equation matches the one we derived, where 16.5 times the unknown number of miles per gallon (x) equals the total distance of 465.3 miles. This option correctly represents the relationship between the fuel consumption, fuel efficiency, and total distance. Therefore, this is the correct equation.

• Option 2: x / 16.5 = 465.3

  This equation suggests that dividing the average miles per gallon (x) by the amount of fuel (16.5 gallons) gives the total distance (465.3 miles). This is an incorrect representation of the relationship, as it implies that fuel efficiency is inversely proportional to the total distance, which is not the case. Therefore, this option is incorrect.

• Option 3: 465.3x = 16.5

  This equation implies that multiplying the total distance (465.3 miles) by the average miles per gallon (x) equals the amount of fuel (16.5 gallons). This is also an incorrect representation, as it suggests that fuel efficiency, when multiplied by the total distance, yields the fuel consumption, which is the opposite of the actual relationship. Therefore, this option is incorrect.

• Option 4: x / 465.3 = 16.5

  This equation suggests that dividing the average miles per gallon (x) by the total distance (465.3 miles) gives the amount of fuel (16.5 gallons). This equation does not logically follow the relationship between fuel efficiency, total distance, and fuel consumption, making it an incorrect option.

By carefully evaluating each option and comparing it to the derived equation and the problem statement, we can confidently identify the correct equation that accurately represents the scenario.

Why 16.5x = 465.3 is the Correct Equation

The equation 16.5x = 465.3 is the correct representation of the problem because it accurately captures the relationship between the amount of fuel used, the average miles per gallon, and the total distance traveled. This equation is derived from the fundamental concept that the total distance traveled is equal to the product of the fuel efficiency (miles per gallon) and the amount of fuel consumed. In this case, 16.5 gallons of gas were used, and 'x' represents the average number of miles the car traveled on each gallon. Therefore, 16.5 multiplied by 'x' gives the total distance of 465.3 miles. This equation is a direct translation of the problem statement into mathematical terms. It isolates the unknown variable 'x', allowing us to solve for the average miles per gallon. The other options, as discussed earlier, misrepresent this relationship. They either suggest an inverse relationship or incorrectly place the variables, leading to an inaccurate representation of the problem. The equation 16.5x = 465.3 not only provides the correct setup but also aligns with the logical understanding of fuel efficiency. It reinforces the idea that fuel efficiency is a rate that, when multiplied by the quantity of fuel, gives the total distance covered. This understanding is crucial for solving similar problems involving rates, ratios, and proportions.

Conclusion

In conclusion, the correct equation to determine the average number of miles Adam's car traveled on each gallon of gas is 16.5x = 465.3. This equation accurately represents the relationship between the total distance, the amount of fuel consumed, and the average miles per gallon. Understanding how to set up and solve such equations is fundamental in mathematics and has practical applications in real-world scenarios. By breaking down the problem, identifying key components, constructing the equation, and evaluating the options, we can confidently arrive at the correct solution. This process not only helps in solving this specific problem but also in developing a strong problem-solving approach for various mathematical challenges. The ability to translate real-world scenarios into mathematical expressions is a valuable skill that enhances our understanding of the world around us.