Tank Filling Time: 6 Taps Vs. 9 Taps
Hey guys! Let's dive into a classic math problem about filling a tank with taps. This is a common type of problem that involves understanding rates and proportions. We're going to break it down step-by-step, so you can easily solve similar problems in the future. The question we're tackling today is: If 6 taps can fill a tank in 3 hours, how long will it take for 9 taps, all with the same flow rate, to fill the same tank?
Understanding the Problem
Before we start crunching numbers, let's make sure we really get what's going on. We have a tank, and we can fill it using taps. The more taps we have, the faster the tank should fill, right? That's the basic idea we need to keep in mind. So, in this problem, the key relationship we need to understand is the inverse relationship between the number of taps and the time it takes to fill the tank. More taps mean less time, and fewer taps mean more time. This is super important because it tells us we're dealing with an inverse proportion. When we say inverse proportion, we mean that if one quantity increases, the other quantity decreases proportionally. In our case, if we increase the number of taps, the time required to fill the tank decreases. Recognizing this relationship is the first step toward solving the problem correctly. Another crucial aspect is the concept of flow rate. The problem specifies that all taps have the same flow rate. This means each tap contributes equally to filling the tank. If the flow rates were different, the problem would become more complex, as we would need to consider each tap's individual contribution. However, since they are the same, we can treat the problem as a straightforward inverse proportion. Additionally, understanding the units is crucial. We are given the time in hours, so we need to make sure our final answer is also in hours or convert it appropriately if needed. Keeping track of the units helps avoid errors and ensures the answer is meaningful in the context of the problem. By carefully considering these aspects, we set a solid foundation for solving the problem accurately and efficiently.
Setting Up the Proportion
Okay, now that we understand the problem, let's set up a proportion to solve it. Since we know this is an inverse proportion, we need to set it up carefully. Let's use the following variables:
n1= number of taps in the first scenario (6 taps)t1= time it takes in the first scenario (3 hours)n2= number of taps in the second scenario (9 taps)t2= time it takes in the second scenario (what we want to find)
Because it's an inverse proportion, we have:
n1 * t1 = n2 * t2
This formula tells us that the product of the number of taps and the time it takes is constant. It's like saying the total work done (filling the tank) is the same, no matter how many taps we use. This equation is the heart of solving this problem. It encapsulates the inverse relationship we identified earlier and provides a direct way to calculate the unknown time t2. Understanding why this equation works is crucial. It's not just about memorizing a formula; it's about grasping the underlying principle of inverse proportionality. The equation essentially says that if you multiply the number of taps by the time it takes to fill the tank in the first scenario, you will get the same value as when you multiply the number of taps by the time it takes in the second scenario. This constant value represents the total amount of work required to fill the tank. By setting up the equation correctly, we ensure that we are accurately representing the relationship between the number of taps and the time it takes to fill the tank. This careful setup is essential for arriving at the correct answer. Moreover, double-checking the equation and the values assigned to each variable is always a good practice. This helps prevent errors and ensures that we are on the right track. By paying close attention to detail and understanding the underlying principles, we can confidently solve this problem and similar problems in the future.
Solving for the Unknown
Now, let's plug in the values we know and solve for t2:
6 * 3 = 9 * t2
That simplifies to:
18 = 9 * t2
To find t2, we divide both sides by 9:
t2 = 18 / 9
So:
t2 = 2 hours
Therefore, it will take 9 taps 2 hours to fill the same tank. Solving for the unknown involves basic algebraic manipulation. The key is to isolate the variable t2 on one side of the equation. We achieve this by performing the same operation on both sides of the equation, maintaining the equality. In this case, we divide both sides by 9 to isolate t2. This step is crucial, and it's important to perform it accurately. A simple mistake in this step can lead to an incorrect answer. After performing the division, we find that t2 = 2. This means that it takes 2 hours for 9 taps to fill the tank. It's always a good practice to double-check the calculations to ensure accuracy. Additionally, it's helpful to think about whether the answer makes sense in the context of the problem. Since we increased the number of taps from 6 to 9, we would expect the time required to fill the tank to decrease. Our answer of 2 hours is less than the original 3 hours, which aligns with our expectation. This sanity check helps confirm that our solution is reasonable. By carefully performing the algebraic manipulation and verifying the answer, we can confidently conclude that it will take 9 taps 2 hours to fill the tank.
Checking Our Answer
It's always a good idea to check our answer to make sure it makes sense. If 6 taps take 3 hours, then in one hour, 6 taps fill 1/3 of the tank. That means each tap fills 1/18 of the tank in an hour.
Now, if we have 9 taps, each filling 1/18 of the tank in an hour, then in one hour, 9 taps fill 9/18 (or 1/2) of the tank. So, it would take 2 hours for 9 taps to fill the entire tank. Yep, our answer checks out!
Checking the answer is a crucial step in problem-solving. It helps ensure that our solution is not only mathematically correct but also logically sound. There are several ways to check the answer, and we've used one method here. We started by calculating the fraction of the tank that 6 taps fill in one hour. This allowed us to determine the individual contribution of each tap. Then, we used this information to calculate the fraction of the tank that 9 taps fill in one hour. Finally, we used this fraction to determine the time it would take for 9 taps to fill the entire tank. Another way to check the answer is to use common sense and reasoning. Since we increased the number of taps, we would expect the time required to fill the tank to decrease. Our answer of 2 hours is less than the original 3 hours, which aligns with our expectation. Additionally, we can think about the relationship between the number of taps and the time it takes to fill the tank. If we double the number of taps, we would expect the time to be halved. While we didn't double the number of taps in this problem, we can still use this principle to check our answer. By using multiple methods to check the answer, we can increase our confidence in the correctness of our solution. This is a valuable skill that can help us avoid errors and improve our problem-solving abilities.
Another Approach: Unitary Method
Here’s another way to think about it, using the unitary method.
If 6 taps fill the tank in 3 hours, then 1 tap would fill the tank in 6 times the amount of time, because it's doing all the work alone. So, 1 tap fills the tank in 6 * 3 = 18 hours.
Now, if 1 tap takes 18 hours, then 9 taps would fill the tank in 1/9th of that time. So, 9 taps fill the tank in 18 / 9 = 2 hours.
Same answer, different way of thinking about it! The unitary method is a versatile approach to solving problems involving proportions. It involves finding the value of a single unit (in this case, one tap) and then using that value to find the value of multiple units (9 taps). This method can be particularly useful when dealing with problems where the relationship between the quantities is not immediately obvious. In this case, we started by finding the time it would take for one tap to fill the tank. Since one tap is doing all the work alone, it would take 6 times longer than it would take for 6 taps to fill the tank. This gave us a time of 18 hours for one tap. Then, we reasoned that if one tap takes 18 hours, then 9 taps would fill the tank in 1/9th of that time. This is because each tap is contributing equally to filling the tank, so the total time is divided among the 9 taps. This gave us a time of 2 hours for 9 taps. The unitary method provides a clear and logical way to solve this problem. It breaks down the problem into smaller, more manageable steps, making it easier to understand and solve. Additionally, it can be applied to a wide range of problems involving proportions, making it a valuable tool in problem-solving.
Conclusion
So, there you have it! If 6 taps can fill a tank in 3 hours, then 9 taps with the same flow rate will fill the same tank in 2 hours. Understanding inverse proportions can really help you solve a lot of different problems. Keep practicing, and you'll get the hang of it! Remember, the key to solving these problems is understanding the relationship between the quantities involved. In this case, it's the inverse relationship between the number of taps and the time it takes to fill the tank. The more taps you have, the less time it takes to fill the tank, and vice versa. By recognizing this relationship, you can set up the problem correctly and solve for the unknown. Additionally, it's important to check your answer to make sure it makes sense in the context of the problem. This can help you avoid errors and increase your confidence in your solution. Finally, remember that there are often multiple ways to solve a problem. We've explored two different approaches here: setting up a proportion and using the unitary method. By understanding these different approaches, you can choose the one that works best for you and apply it to a wide range of problems. So, keep practicing, and don't be afraid to try different approaches. With a little bit of effort, you'll become a pro at solving these types of problems!