Tank Capacity Problem: Solving For Total Volume

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Tank Capacity Problem: Solving for Total Volume

Hey everyone! Today, we're diving into a classic math problem involving a tank that's being emptied over a few days. It's a fun puzzle that combines fractions and a bit of algebra. So, let's break it down and figure out how to solve it together. Stick with me, and you'll see how easy it can be!

Understanding the Problem

Okay, so here’s the deal. We have a tank, right? And it's totally full. On day one, we use up half of the tank. Day two rolls around, and we find that three-quarters of the original amount is still left. Then, on the third day, we use up a third of what's remaining, and that amount is exactly 60 liters. The big question is: what was the total capacity of the tank to begin with?

This problem might seem a bit confusing at first, but don't worry! We're going to take it step by step. First, let’s define what we’re trying to find. We need to find the total volume of the tank, which we will call "V." The problem gives us a series of events or actions that affect the tank’s volume over three days. On the first day, the tank loses half its contents, which means it retains half of the total volume. On the second day, the tank retains three-quarters of its original volume. This is a key piece of information, as it relates the volume on day two back to the initial volume, and not to the volume on day one. On the third day, the tank loses a third of the volume it had at the end of day two. This loss is explicitly given as 60 liters. We can start building equations representing each day’s events. The equation for day one is simple: remaining volume = V / 2. The equation for day two also is pretty straightforward: remaining volume = (3 / 4) * V. For day three, we know that the tank loses a third of its volume from day two, so we can write an equation involving the volume lost on day three, which is given as 60 liters. By carefully constructing these equations, we can link the actions each day to the tank’s total volume, allowing us to eventually solve for the unknown, V. This methodical approach helps us tackle complex problems by breaking them down into smaller, manageable parts. We'll organize the information, define the variables, and use equations to represent the relationships. Ready? Let's dive into the step-by-step solution!

Setting Up the Equations

Alright, let's get down to the nitty-gritty. To solve this problem effectively, we need to translate the word problem into mathematical equations. This will help us to visualize the changes in the tank's volume each day and set up a clear path to find the total capacity.

  • Let's use 'V' to represent the total capacity of the tank. This is what we're trying to find!
  • Day 1: The tank loses half its contents. This means the remaining volume is V / 2.
  • Day 2: The tank has three-quarters of the original amount left. So, the remaining volume is (3/4) * V.
  • Day 3: Here's where it gets a bit trickier. We know that a third of the remaining contents from Day 2 is consumed, and that amount is 60 liters. So, (1/3) * (remaining volume from Day 2) = 60 liters. Which translates to (1/3) * ((3/4) * V) = 60.

So, our main equation is: (1/3) * ((3/4) * V) = 60. This equation tells us that one-third of what was left in the tank on Day 2 equals 60 liters. This is our key to unlocking the solution. Now, let's simplify this equation. First, multiply the fractions: (1/3) * (3/4) equals 3/12, which simplifies to 1/4. So, the equation becomes (1/4) * V = 60. This equation is much easier to work with. It tells us that one-fourth of the tank's total capacity is 60 liters. To find the total capacity, we need to isolate V. This means we must get V alone on one side of the equation. Since V is being divided by 4, we can do the opposite operation to both sides of the equation to isolate V. The opposite of dividing by 4 is multiplying by 4. By multiplying both sides of the equation by 4, we maintain the equality and solve for V. This allows us to determine the full capacity of the tank by extrapolating from the given information about the portion consumed on day three. Algebraic manipulation like this helps us turn complex word problems into straightforward calculations. Let’s move on to solving this equation and finding the value of V!

Solving for the Total Capacity

Now comes the fun part – solving for 'V'! We've already got our equation set up: (1/4) * V = 60. To find 'V', we need to get it all by itself on one side of the equation. Right now, 'V' is being multiplied by 1/4. To undo that, we can multiply both sides of the equation by 4. This keeps the equation balanced and helps us isolate 'V'.

So, we multiply both sides by 4:

4 * (1/4) * V = 4 * 60

On the left side, 4 * (1/4) cancels out, leaving us with just 'V'. On the right side, 4 * 60 equals 240.

Therefore, our equation simplifies to:

V = 240

This means that the total capacity of the tank is 240 liters. We found that the total volume of the tank is 240 liters by following a series of logical steps. First, we expressed the word problem using mathematical symbols and notations. Next, we translated the descriptive information into mathematical equations. Finally, we simplified the equation step by step until we isolated the variable representing the tank’s total volume. At each step, we made sure to keep the equation balanced, applying the same operations to both sides to preserve the equality. This allowed us to solve for V confidently. It is often useful to check the solution by inserting it back into the original equations or conditions to see if it holds true. This helps to ensure accuracy and validates the correctness of the solution. It also reinforces our understanding of the relationships described in the problem. The value V = 240 liters makes logical sense given the conditions of the problem. This reinforces our confidence in the solution.

Checking Our Answer

Okay, so we've found that the tank's total capacity is 240 liters. But, just to be super sure, let's check our answer to make sure it fits with all the information we were given in the problem. Checking our work is a great way to ensure we didn't make any mistakes along the way.

  • Original Capacity: 240 liters
  • Day 1: Half the tank is emptied, so 240 / 2 = 120 liters remain.
  • Day 2: Three-quarters of the original amount is left, so (3/4) * 240 = 180 liters. This is what's remaining before Day 3.
  • Day 3: A third of the remaining contents from Day 2 is used, and that's 60 liters. So, (1/3) * 180 = 60 liters. Yep, that matches the problem!

Since everything checks out, we can be confident that our answer is correct. The initial volume of the tank is indeed 240 liters. By confirming that our solution aligns with each condition of the problem, we solidify our understanding and demonstrate the accuracy of our calculations. This checking process is critical in mathematical problem-solving, ensuring that the solution not only makes sense theoretically but also fits practically within the context of the problem. Furthermore, it enhances our problem-solving skills by prompting us to re-evaluate each step and logical connection we've made. This ensures that our thought process is sound and reliable. Through consistent practice of such verification methods, we become more adept at identifying potential errors and refining our solutions, ultimately strengthening our competence in mathematics. Always remember to double-check your answers and assumptions when working through such problems. It can save you a lot of headaches. So, there you have it, folks!

Conclusion

So, there you have it! The total capacity of the tank is 240 liters. We successfully solved this problem by breaking it down into smaller, manageable steps, setting up clear equations, and carefully solving for the unknown. Remember, when faced with a tricky math problem, don't be intimidated. Just take it one step at a time, and you'll get there!

I hope this explanation was helpful and clear. Keep practicing, and you'll become a pro at solving these types of problems in no time. Happy problem-solving, guys! You can apply the problem-solving strategy we have used to other mathematical problems, so you should keep it in your toolbox. Remember to take it one step at a time, translate the word problem into a mathematical expression, and double-check your work!