Tank Filling Problem How Long Will It Take
Hey there, math enthusiasts! Let's dive into a classic problem involving pumps and tanks. This is a super common type of question you might see in math competitions or even real-life scenarios where you need to figure out how long it takes to fill something up with multiple sources working together. We'll break it down step-by-step, making sure everyone can follow along. So, grab your thinking caps, and let's get started!
The Tank Filling Problem
Okay, so here's the situation. We've got a tank, and two pumps, Pump A and Pump B, are ready to fill it. Pump A, working all by itself, is a bit of a slowpoke and takes 3 hours to completely fill the tank. Pump B, on the other hand, is much faster and can fill the same tank in just 2 hours. Now, here's the twist: We start with the tank completely empty. We turn on Pump A for an hour to get things going. After that hour, we switch on Pump B as well, so both pumps are working together to fill the tank. The big question is: How much total time will it take to fill the tank completely?
This problem might seem a little tricky at first, but don't worry! We'll break it down into smaller, manageable steps. We'll use the concept of rates to figure out how much of the tank each pump fills per hour. Then, we'll look at the combined rate when both pumps are working. By the end of this, you'll be a pro at solving these types of problems!
Understanding the Rates
The key to solving this problem lies in understanding the rates at which each pump fills the tank. Rate, in this context, simply means the fraction of the tank that a pump can fill in one hour. Let's figure out the rates for Pump A and Pump B.
Pump A's Rate
We know that Pump A takes 3 hours to fill the entire tank. This means that in one hour, Pump A fills 1/3 of the tank. Think of it this way: if you divide the tank into three equal parts, Pump A fills one of those parts every hour. So, Pump A's rate is 1/3 of the tank per hour. This is our first important piece of information.
Pump B's Rate
Now, let's look at Pump B. It's faster, filling the tank in just 2 hours. This means that in one hour, Pump B fills 1/2 of the tank. If you imagine dividing the tank into two equal parts, Pump B fills one of those parts each hour. Therefore, Pump B's rate is 1/2 of the tank per hour. We now have the rates for both pumps individually.
Understanding these individual rates is crucial because it allows us to figure out how much of the tank is filled when both pumps are working together. In the next section, we'll explore how to combine these rates to find the combined filling rate.
Pump A Works Alone: The First Hour
Before we get to the combined effort of both pumps, let's focus on that first hour when only Pump A is working. This is a crucial step in understanding the overall solution. Remember, Pump A has a rate of 1/3 of the tank per hour.
How Much is Filled?
Since Pump A works for one full hour at a rate of 1/3 of the tank per hour, it fills exactly 1/3 of the tank during that first hour. This is a pretty straightforward calculation: (1/3 tank/hour) * (1 hour) = 1/3 tank. So, after the first hour, 1/3 of the tank is full.
What's Left to Fill?
Now, the important question is: how much of the tank is still left to fill? If the tank represents a whole, or 1, and 1/3 of it is already full, then the remaining portion is 1 - 1/3. To subtract these, we need a common denominator, so we rewrite 1 as 3/3. Therefore, 3/3 - 1/3 = 2/3. This means that 2/3 of the tank remains to be filled after Pump A has worked for one hour. This is a critical piece of information because it sets the stage for the next phase, where both pumps are working together. We know the total task remaining, and we're about to figure out how quickly the two pumps can tackle it together.
Pumps A and B Work Together: Combined Rate
Alright, now we're at the exciting part where both pumps are working simultaneously! This is where we need to figure out their combined filling rate. Remember, Pump A fills 1/3 of the tank per hour, and Pump B fills 1/2 of the tank per hour.
Adding the Rates
To find their combined rate, we simply add their individual rates. So, we need to add 1/3 and 1/2. Just like with subtraction, we need a common denominator to add fractions. The least common denominator for 3 and 2 is 6. Let's convert our fractions:
- 1/3 = (1 * 2) / (3 * 2) = 2/6
- 1/2 = (1 * 3) / (2 * 3) = 3/6
Now we can add them: 2/6 + 3/6 = 5/6. This means that together, Pumps A and B fill 5/6 of the tank per hour. This combined rate is faster than either pump working alone, which makes sense!
The Power of Teamwork
This combined rate of 5/6 of the tank per hour is a crucial number. It tells us how quickly the two pumps, working together, can fill the remaining portion of the tank. In the next section, we'll use this combined rate to calculate the time it takes to fill the remaining 2/3 of the tank.
Calculating the Time to Fill the Remaining Tank
We've established that Pumps A and B, working together, fill 5/6 of the tank per hour. We also know that 2/3 of the tank remains to be filled after Pump A's initial hour of work. Now, we need to figure out how long it will take the two pumps, working at their combined rate, to fill that remaining 2/3 of the tank.
Using the Rate Formula
We can use a simple formula to solve this: Time = Work / Rate. In our case:
- Work = The amount of the tank remaining to be filled = 2/3
- Rate = The combined filling rate of Pumps A and B = 5/6 of the tank per hour
So, Time = (2/3) / (5/6). Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 5/6 is 6/5. So, we have:
Time = (2/3) * (6/5)
Solving for Time
Now, we multiply the fractions: (2 * 6) / (3 * 5) = 12/15. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3: 12/15 = 4/5. This means it will take 4/5 of an hour for Pumps A and B to fill the remaining 2/3 of the tank.
Converting to Minutes
Since we often think of time in minutes, let's convert 4/5 of an hour into minutes. There are 60 minutes in an hour, so (4/5) * 60 minutes = 48 minutes. Therefore, it takes 48 minutes for Pumps A and B to fill the remaining portion of the tank.
Finding the Total Time
We're almost there! We've calculated the time it takes for both pumps working together to fill the remaining portion of the tank. Now, we need to add that to the initial time Pump A worked alone to find the total time it takes to fill the entire tank.
Adding the Time Intervals
Pump A worked alone for 1 hour, and then Pumps A and B worked together for 4/5 of an hour (which we converted to 48 minutes). So, the total time is 1 hour + 48 minutes.
The Final Answer
Therefore, it takes a total of 1 hour and 48 minutes to fill the tank completely. We've successfully solved the problem!
Conclusion: Mastering Tank-Filling Problems
We've walked through a classic tank-filling problem step-by-step, and now you're equipped to tackle similar challenges! Remember the key concepts:
- Understanding Rates: Determine the individual rates of each pump or source.
- Combined Rates: Add the individual rates to find the combined rate when multiple sources work together.
- The Formula: Use the formula Time = Work / Rate to calculate the time it takes to complete a task.
- Breaking it Down: Divide complex problems into smaller, manageable steps.
By mastering these concepts, you'll be able to confidently solve a wide range of problems involving rates, work, and time. So keep practicing, and you'll become a math whiz in no time! Remember, math is like a muscle; the more you use it, the stronger it gets. So, keep those gears turning, and happy problem-solving!
