Reservoir Water Level Calculation: Fraction Of Capacity?

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Hey guys! Let's dive into a classic math problem involving fractions and water reservoirs. This is the kind of question that might seem tricky at first, but once you break it down step-by-step, it becomes super manageable. We're going to explore how to calculate the water level in a reservoir after a series of additions and subtractions. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, so the core of the problem is understanding how the water level changes in the reservoir. We start with the reservoir half-full, which is 1/2 of its total capacity. Then, some water is consumed, specifically 1/4 of the water that's currently in the reservoir. This is a key detail – we're not taking 1/4 of the total capacity, but 1/4 of the existing water. After that, we add an amount of water equal to 1/3 of the total capacity. The question we need to answer is: what fraction of the total capacity is the reservoir filled with at the end?

To make this crystal clear, think of it like this: Imagine the reservoir can hold 1000 liters. Initially, it has 500 liters (1/2 capacity). Then, we remove 1/4 of that 500 liters, and add 1/3 of the 1000 liters. What's the final volume? This kind of mental picture can really help in visualizing the problem.

Key Steps to Solve This:

  1. Initial State: Identify the starting water level (1/2 capacity).
  2. Consumption: Calculate the amount of water consumed (1/4 of the current water).
  3. Addition: Calculate the amount of water added (1/3 of the total capacity).
  4. Final Calculation: Combine the initial state, consumption, and addition to find the final fraction of the reservoir's capacity.

We'll walk through each of these steps in detail, so you'll see exactly how to tackle this type of problem.

Step 1: The Initial State (1/2 Capacity)

The first thing we know is that the reservoir starts with 1/2 of its total capacity. This is our baseline. It's super important to remember that this is the starting point for all our calculations. We're going to use this as the foundation and then adjust it based on the consumption and addition of water.

Think of it like having a glass that's half-full. That's our starting point. We're not worried about the total size of the glass just yet; we just know that we're beginning with half of it filled. This is crucial because the next step involves taking a fraction of this existing amount, not of the total capacity.

To make this even clearer, let's use a visual. Imagine a rectangle representing the total capacity of the reservoir. If we divide that rectangle in half, one half is shaded, representing the initial 1/2 capacity. This visual representation can be really helpful, especially when we start dealing with fractions of fractions.

So, we've established our initial state: 1/2 capacity. Now, let's move on to the next step, which is calculating how much water is consumed from this initial amount. This is where we'll start playing with fractions within fractions, so stay tuned!

Step 2: Calculating the Water Consumed (1/4 of the Existing Water)

Alright, so we know the reservoir starts with 1/2 of its capacity. The next part of the problem tells us that 1/4 of the existing water is consumed. Now, this is a crucial point: we're not taking 1/4 of the total capacity, but 1/4 of the current amount, which is 1/2 of the capacity. This is a common trick in these types of problems, so it's super important to pay attention to the wording.

To figure out how much water was consumed, we need to calculate what 1/4 of 1/2 is. In mathematical terms, this means we need to multiply the two fractions: (1/4) * (1/2). Remember, when you multiply fractions, you multiply the numerators (the top numbers) and the denominators (the bottom numbers) separately.

So, (1/4) * (1/2) = (1 * 1) / (4 * 2) = 1/8.

This means that 1/8 of the total capacity was consumed. Think of it like this: if the reservoir could hold 8 equal parts of water, 1 part was taken out. We started with half the reservoir full (which would be 4 parts out of 8), and we removed one of those parts.

Now, we need to subtract this consumed amount from our initial amount. We started with 1/2 capacity and consumed 1/8 capacity. To subtract these fractions, we need a common denominator. The least common denominator for 2 and 8 is 8. So, we need to convert 1/2 to an equivalent fraction with a denominator of 8. 1/2 is the same as 4/8.

So, the calculation becomes: 4/8 - 1/8 = 3/8.

After the water is consumed, the reservoir has 3/8 of its total capacity remaining. This is a key intermediate result that we'll use in the next step. We've successfully navigated the tricky part of calculating a fraction of a fraction!

Step 3: Adding Water (1/3 of Total Capacity)

Okay, we've figured out that after consuming some water, the reservoir has 3/8 of its capacity remaining. Now, the problem tells us that 1/3 of the total capacity is added to the reservoir. Notice that this time, we're adding a fraction of the total capacity, not a fraction of the existing water. This makes the calculation a bit more straightforward.

So, we're adding 1/3 of the total capacity to the existing 3/8 capacity. To find the new water level, we simply need to add these two fractions together: 3/8 + 1/3.

To add fractions, we need a common denominator. The least common multiple of 8 and 3 is 24. So, we need to convert both fractions to equivalent fractions with a denominator of 24.

  • To convert 3/8 to a fraction with a denominator of 24, we multiply both the numerator and the denominator by 3: (3 * 3) / (8 * 3) = 9/24.
  • To convert 1/3 to a fraction with a denominator of 24, we multiply both the numerator and the denominator by 8: (1 * 8) / (3 * 8) = 8/24.

Now we can add the fractions: 9/24 + 8/24 = 17/24.

After adding the water, the reservoir has 17/24 of its total capacity. We're almost there! We've calculated the initial state, the consumption, and the addition. Now, we just need to put it all together to find the final answer.

Step 4: The Final Calculation (Combining the Steps)

Alright guys, we've done all the hard work! We know the reservoir started with 1/2 capacity, then had 1/4 of the existing water consumed, leaving it at 3/8 capacity. Finally, we added 1/3 of the total capacity, bringing the level to 17/24. So, the final answer to our problem is that the reservoir contains 17/24 of its total capacity at the end of the day.

Let's recap the whole process to make sure we've got it nailed down:

  1. Initial State: 1/2 capacity.
  2. Consumption: 1/4 of 1/2 = 1/8 capacity consumed. Remaining: 1/2 - 1/8 = 3/8 capacity.
  3. Addition: 1/3 of total capacity added. Total: 3/8 + 1/3 = 17/24 capacity.

So, by breaking down the problem into smaller steps and carefully calculating each change in the water level, we were able to arrive at the solution. Remember, the key to these types of problems is to pay close attention to what fraction you're taking a fraction of. Is it the total capacity, or the current amount? This makes all the difference!

Tips for Solving Similar Problems

Now that we've conquered this reservoir problem, let's talk about some general tips for tackling similar fraction-based word problems. These tips will help you approach these questions with confidence and avoid common pitfalls.

  1. Read Carefully and Identify Key Information: The most important thing is to read the problem very carefully. Pay attention to the details and identify the key information, such as the initial amounts, the fractions involved, and what you're being asked to find. Underlining or highlighting these key pieces of information can be super helpful.
  2. Break the Problem into Smaller Steps: Complex problems can feel overwhelming, but breaking them down into smaller, manageable steps makes them much easier to solve. Identify the different operations you need to perform (addition, subtraction, multiplication, division) and tackle them one at a time.
  3. Pay Attention to What Fraction You're Taking a Fraction Of: This is the most common mistake people make in these types of problems. Always double-check whether you're taking a fraction of the total amount or a fraction of the existing amount. This will significantly impact your calculations.
  4. Use Visual Aids: Drawing diagrams or using visual aids can be incredibly helpful, especially when dealing with fractions. You can draw rectangles to represent the total capacity and shade in portions to represent the fractions. This can make the problem more concrete and easier to understand.
  5. Check Your Answer: Once you've arrived at an answer, take a moment to check if it makes sense in the context of the problem. For example, if you started with a reservoir half-full and added some water, your final answer should be greater than 1/2. If it's not, you know you've made a mistake somewhere.

By following these tips, you'll be well-equipped to solve a wide range of fraction-based word problems. Remember, practice makes perfect! The more you work through these types of questions, the more comfortable and confident you'll become.

Real-World Applications of Fraction Calculations

Okay, so we've solved this reservoir problem, but you might be wondering,