The Gas Tank Of Dave's Car Has A Capacity Of 12 Gallons. The Tank Was 3/8 Full Before Dave Filled It To Capacity. It Cost Him $2.50 Per Gallon Of Gas. How Much Money Did Dave Spend, In Dollars, To Fill The Tank To Capacity?
Introduction
In this article, we will explore a practical mathematical problem involving the capacity of a car's gas tank, the fraction of gas already present, and the cost to fill the tank completely. This problem, which falls under the mathematics discussion category, is a great example of how math concepts are used in everyday situations. We will delve into the steps required to solve this problem, ensuring a clear understanding of the underlying principles. The scenario presented involves Dave's car, which has a gas tank with a capacity of 12 gallons. The tank is initially full, and Dave proceeds to fill it to its full capacity. Given that the cost of gas is $2.50 per gallon, our objective is to determine the total amount Dave spent to fill his car's gas tank. This exercise not only reinforces basic arithmetic skills but also highlights the importance of problem-solving in real-world contexts. By breaking down the problem into smaller, manageable steps, we can easily arrive at the solution. So, let's embark on this mathematical journey and unravel the cost Dave incurred to fill his gas tank.
Understanding the Problem
To effectively solve this problem, a thorough understanding of the given information is crucial. Let's break down the key elements: the car's gas tank has a capacity of 12 gallons. This is the total volume of gas the tank can hold. Initially, the tank is full, meaning it already contains a portion of its total capacity. The cost of gasoline is $2.50 per gallon, which is the price Dave has to pay for each gallon he adds to the tank. The central question we aim to answer is: how much did Dave spend, in dollars, to fill the tank to capacity? To tackle this, we need to determine the amount of gas required to fill the tank completely. This involves calculating the difference between the tank's full capacity and the amount of gas already present. Once we know the number of gallons needed, we can multiply it by the cost per gallon to find the total expenditure. This step-by-step approach ensures that we address all aspects of the problem and arrive at an accurate solution. Understanding the relationships between these elements is essential for a successful resolution. By carefully analyzing each piece of information, we can construct a clear path towards the final answer. This initial phase of comprehension sets the stage for the subsequent calculations and ensures that we are solving the correct problem.
Step-by-Step Solution
Now, let's embark on the step-by-step solution to determine how much Dave spent filling his car's gas tank. Our first step is to calculate the amount of gas already in the tank. Given that the tank has a capacity of 12 gallons and is full, we multiply these two values: gallons. This means Dave's tank already had 4.5 gallons of gas. Next, we need to find out how many more gallons Dave needs to fill the tank to its full capacity. We subtract the existing amount of gas from the total capacity: gallons. So, Dave needs to add 7.5 gallons of gas to fill the tank completely. Now that we know the number of gallons Dave needs to purchase, we can calculate the total cost. The cost per gallon is $2.50, so we multiply the number of gallons needed by the price per gallon: $7.5 \times $2.50 = . Therefore, Dave spent $18.75 to fill his car's gas tank to capacity. This step-by-step breakdown makes the solution clear and easy to follow, demonstrating how each calculation contributes to the final answer. By breaking the problem into smaller, manageable steps, we can confidently arrive at the correct solution.
Detailed Calculations
To provide a more detailed understanding of the solution, let's revisit the calculations with added clarity. First, we determined the initial amount of gas in the tank. The tank's capacity is 12 gallons, and it was full. To find the volume of gas already present, we multiply the fraction by the total capacity: gallons. This can be calculated as follows: . Simplifying the fraction, we get gallons. This confirms that the tank initially had 4.5 gallons of gas. Next, we calculated the amount of gas needed to fill the tank completely. We subtracted the initial amount of gas from the total capacity: gallons. This subtraction yields gallons. Therefore, Dave needed to add 7.5 gallons to fill the tank. Finally, we computed the total cost by multiplying the number of gallons needed by the cost per gallon. The cost per gallon is $2.50, so the total cost is $7.5 \times . Performing this multiplication: $7.5 \times $2.50 = . This detailed breakdown of each calculation reinforces the step-by-step solution and ensures a comprehensive understanding of the problem-solving process. By presenting the calculations in this manner, we aim to make the solution accessible and easy to follow for anyone, regardless of their mathematical background.
Real-World Applications
This mathematical problem, centered around calculating the cost to fill a car's gas tank, has numerous real-world applications. Understanding these applications highlights the practical relevance of mathematics in our daily lives. Firstly, the core concepts used in this problem – fractions, subtraction, and multiplication – are fundamental in personal finance. When budgeting for expenses, individuals often need to calculate the cost of gasoline, taking into account the tank's capacity and the current fuel price. This directly relates to the scenario presented in the problem. Secondly, the ability to estimate and calculate fuel costs is crucial for trip planning. Whether it's a short commute or a long road trip, knowing how much it will cost to fill the tank helps in making informed decisions about travel expenses. This problem-solving skill is valuable in everyday scenarios. Moreover, understanding fuel efficiency and consumption is essential for making environmentally conscious choices. Calculating how much gas is needed for a journey and the associated cost can encourage more efficient driving habits and the consideration of alternative transportation options. Furthermore, these mathematical skills are applicable in various professions, including transportation, logistics, and automotive services. Professionals in these fields often deal with calculations related to fuel consumption, costs, and efficiency. In conclusion, the problem of calculating the cost to fill a gas tank serves as a practical example of how mathematics is used in everyday life, from personal budgeting to professional applications. By mastering these fundamental concepts, individuals can make more informed decisions and navigate real-world challenges effectively.
Conclusion
In summary, the problem presented, which involves calculating the cost to fill Dave's car gas tank, provides a valuable exercise in applying basic mathematical principles to a real-world scenario. By breaking down the problem into manageable steps, we were able to determine the amount of gas needed to fill the tank and the total cost incurred. The initial step involved calculating the amount of gas already in the tank, which was found to be 4.5 gallons. Subsequently, we subtracted this amount from the total tank capacity of 12 gallons, revealing that 7.5 gallons were needed to fill the tank completely. Finally, multiplying the number of gallons needed (7.5) by the cost per gallon ($2.50) yielded the total cost of $18.75. This step-by-step approach not only provides a clear solution but also reinforces the importance of methodical problem-solving. The problem highlights the practical application of mathematics in everyday situations, such as budgeting for fuel expenses. Understanding and applying these concepts can empower individuals to make informed financial decisions. Furthermore, the problem serves as a reminder that mathematical skills are essential in various aspects of life, from personal finance to professional endeavors. By mastering these fundamental principles, we can confidently tackle real-world challenges and make well-informed choices. Thus, the exercise of calculating the cost to fill a gas tank underscores the relevance and utility of mathematics in our daily lives.
Original Question
The gas tank of Dave's car has a capacity of 12 gallons. The tank was full before Dave filled it to capacity. It cost him $2.50 per gallon of gas. How much did Dave spend, in dollars, to fill the tank to capacity?