Solving For Gas Tank Capacity How Large Is Mrs. Hwangs Tank

In this article, we delve into a practical mathematical problem involving Mrs. Hwang and her vehicle's gas tank. Mrs. Hwang is planning to expand her vehicle's gas tank by 10 gallons. Currently, it costs her $42.56 to fill the tank completely. After the expansion, it will cost her $72.96 to fill the enlarged tank. The central question we aim to answer is: how large is her tank now? This problem requires us to utilize basic arithmetic and algebraic principles to arrive at the correct solution. We will break down the problem step by step, ensuring a clear and comprehensive understanding of the solution process.

Setting Up the Equations

To solve this problem effectively, it is crucial to set up the equations correctly. Let's denote the current capacity of Mrs. Hwang's gas tank as x gallons. After the expansion, the new capacity of the tank will be x + 10 gallons. The cost to fill the current tank is $42.56, and the cost to fill the expanded tank is $72.96. We can express the cost per gallon for both scenarios. The cost per gallon remains constant regardless of the tank size, which is the key to solving this problem.

1. Cost per gallon for the current tank: $42.56 / x
2. Cost per gallon for the expanded tank: $72.96 / (x + 10)

Since the cost per gallon is the same in both cases, we can equate these two expressions:

$42.56 / x = $72.96 / (x + 10)

This equation forms the foundation of our solution. By solving this equation for x, we will determine the current size of Mrs. Hwang's gas tank.

Solving the Equation

Now, let's proceed with solving the equation we established:

$42.56 / x = $72.96 / (x + 10)

To eliminate the fractions, we can cross-multiply:

$42.56 * (x + 10) = $72.96 * x

Next, distribute the terms on both sides:

$42.56x + $425.6 = $72.96x

Now, we need to isolate x on one side of the equation. Subtract $42.56x$ from both sides:

$425.6 = $72.96x - $42.56x

Simplify the right side:

$425.6 = $30.4x

Finally, divide both sides by $30.4$ to solve for x:

x = $425.6 / $30.4

x = 14

Therefore, the current capacity of Mrs. Hwang's gas tank is 14 gallons. This calculation provides a clear and precise answer to the problem.

Verifying the Solution

To ensure the accuracy of our solution, it's essential to verify the result. We found that the current tank size is 14 gallons. Let's plug this value back into our original equation to confirm its validity.

1. Cost per gallon for the current tank: $42.56 / 14 = $3.04 per gallon
2. Expanded tank capacity: 14 gallons + 10 gallons = 24 gallons
3. Cost per gallon for the expanded tank: $72.96 / 24 = $3.04 per gallon

The cost per gallon is consistent in both scenarios ($3.04 per gallon), which confirms that our solution is correct. Mrs. Hwang's current gas tank has a capacity of 14 gallons.

Deeper Insights and Implications

Understanding the problem beyond the numerical solution provides valuable insights. The problem illustrates a real-world application of basic algebraic principles. By setting up and solving equations, we can address practical scenarios such as calculating fuel costs and tank capacities. Moreover, this problem highlights the importance of verifying solutions to ensure accuracy.

Additionally, let's consider the implications of expanding the gas tank. Mrs. Hwang's decision to increase her tank's capacity by 10 gallons will allow her to travel longer distances between fill-ups. This can be particularly beneficial for long commutes or road trips. However, it's also important to consider the initial cost of the tank expansion and whether the long-term benefits outweigh this expense.

Real-World Applications

This type of problem has several real-world applications beyond just calculating gas tank capacities. Similar mathematical approaches can be used in various scenarios, such as:

• Budgeting and Finance: Calculating costs per unit, such as cost per gallon of fuel, cost per item in bulk purchases, or cost per service in a subscription.
• Engineering: Determining the capacity of containers, tanks, or storage units in various industries.
• Logistics: Optimizing fuel consumption for transportation fleets and planning fuel stops for long-haul journeys.
• Environmental Science: Calculating fuel efficiency and emissions for different vehicles or transportation methods.

By mastering the fundamental principles of algebra and problem-solving, individuals can make informed decisions in these and many other areas of life.

Conclusion

In conclusion, we successfully determined the current size of Mrs. Hwang's gas tank by setting up and solving an algebraic equation. The initial problem presented a scenario where Mrs. Hwang was expanding her gas tank by 10 gallons, which would increase the cost to fill it from $42.56 to $72.96. By equating the cost per gallon in both scenarios, we found that the current tank capacity is 14 gallons.

This problem not only provides a numerical answer but also demonstrates the practical application of mathematical concepts in everyday life. The steps involved in solving this problem, such as setting up equations, cross-multiplication, distribution, and verification, are valuable skills that can be applied to a wide range of real-world situations. Understanding these principles empowers individuals to tackle complex problems and make informed decisions.

Moreover, the exploration of the problem highlighted the importance of verifying solutions to ensure accuracy and the broader implications of decisions such as expanding a vehicle's gas tank. From budgeting and finance to engineering and logistics, the ability to apply mathematical reasoning is essential in numerous fields. By mastering these skills, we can better navigate the complexities of the world around us and make well-informed choices.

The problem-solving approach used in this scenario can serve as a template for tackling similar challenges. Whether it's calculating fuel costs, optimizing resource allocation, or making financial decisions, the ability to translate real-world situations into mathematical models is a powerful tool. By continuing to practice and refine these skills, we can enhance our problem-solving abilities and achieve greater success in various aspects of life.