Mrs. Hrang's Gas Tank Problem Solving Tank Capacity Calculation

This article delves into a practical mathematical problem involving gas tank capacity and cost calculations. We'll break down the steps to determine the original size of Mrs. Hrang's gas tank before she had it expanded. This problem combines basic arithmetic with real-world scenarios, making it a great exercise in problem-solving.

Understanding the Problem

Mrs. Hrang is upgrading her vehicle's gas tank, increasing its capacity by 10 gallons. Currently, filling her tank costs $42.56. After the expansion, it will cost $72.96 to fill the tank completely. Our goal is to determine the original size of her gas tank in gallons. This is a classic problem that involves understanding ratios and proportions, and we'll use a step-by-step approach to solve it.

To effectively tackle this, understanding the core relationship between the cost of filling the tank and its capacity is essential. The price of gasoline per gallon remains constant in this scenario, allowing us to set up a proportional equation. This equation will help us relate the initial cost and capacity to the new cost and capacity. By solving this equation, we can uncover the original size of Mrs. Hrang's gas tank. The journey to find the solution is not just about arriving at the correct number; it's about grasping the logical progression and the mathematical principles at play. Think of each piece of information as a puzzle piece that fits together to reveal the answer. It's not just about calculations, but understanding the problem's inherent structure. The steps to solve this problem will involve carefully dissecting the information, setting up the right equations, and then solving for the unknown variable. Each step is a building block, leading to the final solution. This approach will not only help solve this particular problem but will also enhance overall problem-solving skills.

Setting Up the Equations

The key to solving this problem lies in setting up the correct equations. We'll start by defining our variables:

• Let x represent the original size of the gas tank in gallons.
• The new size of the gas tank will be x + 10 gallons.

We know the cost to fill the original tank is $42.56, and the cost to fill the expanded tank is $72.96. The price per gallon of gasoline remains constant. Therefore, we can set up a proportion:

Cost of original tank / Original tank size = Cost of expanded tank / Expanded tank size

This translates to:

42.56 / x = 72.96 / (x + 10)

This equation forms the foundation for solving the problem. It reflects the relationship between the cost and the capacity of the tank, both before and after the expansion. Understanding this equation is crucial, as it encapsulates the core principle of the problem. The equation acts as a roadmap, guiding us through the mathematical steps required to find the solution. To effectively solve this problem, it is important to meticulously examine the information provided and translate it into mathematical expressions. This process involves not just identifying the relevant numbers but also understanding how they relate to each other. The careful setting up of the equations ensures that the subsequent steps are logically sound and will lead to the correct answer. It is not just about plugging numbers into a formula but about building a mathematical model that accurately represents the real-world scenario. By carefully setting up the equations, we pave the way for a clear and efficient solution.

Solving for the Unknown

Now that we have our equation, we can solve for x, the original size of the gas tank. The equation is:

42.56 / x = 72.96 / (x + 10)

To solve this, we'll cross-multiply:

42.  56 * (x + 10) = 72.96 * x

Expanding the left side gives us:

42.  56x + 425.6 = 72.96x

Now, we'll move all the x terms to one side and the constants to the other:

73.  96x - 42.56x = 425.6

31.  4x = 425.6

Finally, we divide both sides by 30.4 to isolate x:

x = 425.6 / 30.4

x = 14

Therefore, the original size of Mrs. Hrang's gas tank was approximately 14 gallons. The process of solving for the unknown involves a series of algebraic manipulations, each step bringing us closer to the answer. Accurate execution of these steps is crucial to arrive at the correct solution. Cross-multiplication is a key technique used here, transforming the proportion into a linear equation. Then, the equation is simplified by expanding terms and grouping like variables. The final step involves isolating the variable by performing the inverse operation, in this case, division. Each of these steps is a testament to the power of algebraic techniques in solving real-world problems. The solution is not just a number; it is the culmination of a logical process that transforms the problem from a statement into a concrete answer. It is this systematic approach to problem-solving that makes mathematics such a valuable tool in everyday life.

Checking the Solution

It's always a good practice to check our solution to ensure it's correct. We found that the original tank size was 14 gallons. The expanded tank size is therefore 14 + 10 = 24 gallons.

Let's calculate the price per gallon:

• Original price per gallon: $42.56 / 14 gallons = $3.04 per gallon
• New price per gallon: $72.96 / 24 gallons = $3.04 per gallon

The price per gallon is the same in both cases, which confirms our solution is correct. Checking the solution is an important step in the problem-solving process, providing assurance that the answer is accurate and makes sense within the context of the problem. This step reinforces the understanding of the underlying mathematical principles and ensures that no errors were made during the calculations. In this case, we verified our solution by calculating the price per gallon for both the original and expanded tanks. The fact that the price per gallon remained constant validated our result. This process not only confirms the numerical answer but also deepens the understanding of the relationship between the cost, capacity, and price per gallon. It is this thoroughness that separates a correct answer from a truly understood solution. The act of checking is not just a formality but an integral part of the learning process, solidifying the understanding of the problem and the solution.

The Answer

The original size of Mrs. Hrang's gas tank was 14 gallons. While this wasn't one of the multiple-choice options provided (20 gallons, 16 gallons, or 18 gallons), we've walked through the process to arrive at the correct answer. This exercise highlights the importance of understanding the problem-solving process rather than simply choosing from given options. This example emphasizes that the true value of problem-solving lies in the journey of understanding and applying mathematical principles, rather than just arriving at a final answer. The process of dissecting the problem, setting up the equations, solving for the unknown, and verifying the solution provides a deeper understanding of the mathematical concepts involved. It is this holistic approach to problem-solving that empowers individuals to tackle a wide range of challenges, both in academic settings and in real-life situations. The multiple-choice options provided in the original question serve as a reminder that sometimes the given choices may not include the correct answer, underscoring the importance of independent problem-solving skills. Ultimately, the goal is to develop a robust understanding of the underlying principles, enabling one to confidently navigate through problems and arrive at accurate solutions, regardless of the available options.

Conclusion

This problem demonstrates a practical application of mathematical principles in everyday life. By understanding the relationship between cost, capacity, and price per gallon, we were able to determine the original size of Mrs. Hrang's gas tank. This exercise reinforces the importance of problem-solving skills and the ability to translate real-world scenarios into mathematical equations. In conclusion, the problem of Mrs. Hrang's gas tank expansion serves as a valuable lesson in mathematical problem-solving. By carefully analyzing the information, setting up the equations, and systematically solving for the unknown, we arrived at the correct solution. This process underscores the importance of a methodical approach to problem-solving and highlights the practical applications of mathematical concepts in everyday life. The ability to translate real-world scenarios into mathematical models is a crucial skill that empowers individuals to make informed decisions and solve challenges effectively. This example demonstrates how mathematical reasoning can be used to solve practical problems, making it a valuable tool for navigating the complexities of the modern world. The key takeaway is that problem-solving is not just about finding the right answer; it's about developing a deep understanding of the underlying principles and applying them to new situations. It is this adaptability and critical thinking that makes mathematics such a powerful and essential skill.