Solving A Math Problem Time To Fill A Water Tank With Two Pipes

In this article, we will dissect a classic mathematical problem involving filling a tank with water through two pipes. This is a common type of problem that appears in various math curricula and standardized tests, and it's a great exercise in applying concepts related to rates and fractions. We'll break down the problem step by step, providing a clear and concise solution that can help you tackle similar problems in the future. This comprehensive guide aims to make the solution process accessible to everyone, regardless of their math background. We will delve deep into the concepts of rates, fractions, and how they interact when dealing with problems involving filling containers. By the end of this guide, you'll not only understand how to solve this specific problem but also gain a deeper appreciation for the underlying mathematical principles.

Problem Statement

The problem states that a tank can be filled by two pipes in 2 hours. The first pipe alone can fill the tank in 3 hours. The question we need to answer is: how long will it take the second pipe to fill 2/3 of the tank's volume? This type of problem is a practical application of mathematical concepts, mirroring real-world scenarios where understanding flow rates and capacities is crucial. Whether it's managing water resources, designing pipelines, or even planning industrial processes, the ability to calculate how long it takes to fill a container is invaluable. By carefully analyzing the given information and applying the right formulas, we can arrive at the solution and develop a deeper understanding of the dynamics at play.

Breaking Down the Problem

To solve this, we need to use the concept of rates. Let's represent the rate at which each pipe fills the tank. The key to solving this type of problem lies in understanding how rates combine when multiple sources contribute to the same task. In this case, the two pipes are working together to fill the tank, so their individual rates add up to the combined rate. We'll start by defining the rates of each pipe and then use the given information to establish a relationship between them. This approach allows us to translate the word problem into mathematical equations, which we can then solve using algebraic techniques. By systematically breaking down the problem into smaller, manageable parts, we can tackle even the most complex scenarios with confidence.

Let's denote the following:

• The rate of the first pipe as R1 (tanks per hour).
• The rate of the second pipe as R2 (tanks per hour).
• The combined rate of both pipes as R_combined (tanks per hour).

Calculating the Rates

From the problem, we know the first pipe fills the tank in 3 hours. Therefore, its rate is: R1 = 1/3 (tank per hour). This means that in one hour, the first pipe fills 1/3 of the tank. Similarly, the two pipes together fill the tank in 2 hours, so their combined rate is: R_combined = 1/2 (tank per hour). This indicates that when both pipes are working simultaneously, they fill half of the tank in one hour. Now, we can use these rates to determine the rate of the second pipe. Understanding the individual and combined rates is crucial for solving the problem, as it allows us to set up an equation that relates the rates of the two pipes.

Finding the Rate of the Second Pipe

We know that the combined rate is the sum of the individual rates: R_combined = R1 + R2. We can plug in the values we already know: 1/2 = 1/3 + R2. To find R2, we need to subtract 1/3 from 1/2. Finding a common denominator, we get: 1/2 - 1/3 = 3/6 - 2/6 = 1/6. Therefore, R2 = 1/6 (tank per hour). This means the second pipe fills 1/6 of the tank in one hour. This calculation is a crucial step in solving the problem, as it allows us to isolate the rate of the second pipe, which is essential for determining how long it takes to fill a specific portion of the tank.

Calculating the Time to Fill 2/3 of the Tank

Now that we know the rate of the second pipe (R2 = 1/6 tanks per hour), we can calculate how long it takes to fill 2/3 of the tank. Let T be the time in hours. We can set up the equation: (1/6) * T = 2/3. To solve for T, we multiply both sides of the equation by 6: T = (2/3) * 6. Simplifying, we get: T = 4 hours. This is the final step in solving the problem. We've used the rate of the second pipe to determine the time required to fill a specific fraction of the tank. This demonstrates the power of applying mathematical concepts to solve practical problems involving rates and proportions.

Answer

Therefore, it will take the second pipe 4 hours to fill 2/3 of the tank. This solution demonstrates the application of rates and fractions to solve a real-world problem. By understanding the concepts of individual and combined rates, we were able to break down the problem into manageable steps and arrive at the correct answer. This type of problem is a valuable exercise in mathematical reasoning and can help develop problem-solving skills that are applicable in various contexts.

This problem highlights the importance of understanding rates and how they combine when multiple entities contribute to a task. The concept of rate is fundamental in many areas of science, engineering, and everyday life. For instance, in physics, rate is used to describe velocity, acceleration, and flow rates. In chemistry, reaction rates are crucial for understanding chemical processes. In economics, interest rates and inflation rates play a significant role in financial planning. By mastering the concept of rate, you can gain a deeper understanding of the world around you and develop valuable problem-solving skills.

Applications in Real Life

Understanding how to calculate filling times has numerous practical applications. Imagine you're filling a swimming pool with multiple hoses, managing water flow in irrigation systems, or even calculating the time it takes to complete a task with multiple workers. These scenarios all involve the same underlying principles of rates and proportions. By being able to apply these concepts, you can make informed decisions and optimize processes in various situations. The ability to solve such problems is not only valuable in academic settings but also in professional and personal contexts.

Further Exploration

To further enhance your understanding, consider exploring similar problems with varying parameters, such as different tank sizes, pipe rates, or fractions of the tank to be filled. You can also explore more complex scenarios involving multiple pipes with different rates, or even pipes that both fill and drain the tank. By working through these variations, you can solidify your understanding of the concepts and develop your problem-solving skills even further. Additionally, exploring related concepts like work-rate problems and mixture problems can provide a broader understanding of how rates and proportions are applied in different contexts. This comprehensive approach will not only help you excel in mathematics but also equip you with valuable analytical skills for tackling real-world challenges.

This problem-solving exercise demonstrates the power of mathematics in addressing practical scenarios. By carefully analyzing the problem, defining rates, and applying algebraic techniques, we were able to determine the time required for the second pipe to fill a specific portion of the tank. The ability to solve such problems is a valuable skill that can be applied in various fields, from engineering to everyday life. We hope this step-by-step guide has provided you with a clear understanding of the solution process and inspired you to explore further mathematical challenges. Remember, the key to mastering these concepts is practice and perseverance. By working through various problems and exploring different approaches, you can develop a strong foundation in mathematics and build your confidence in problem-solving.

Practice Problems

To further solidify your understanding, try solving similar problems with different parameters. For example:

1. A tank can be filled by two pipes in 3 hours. The first pipe can fill the tank in 5 hours. How long will it take the second pipe to fill the tank alone?
2. Two pipes fill a tank together in 4 hours. The first pipe fills twice as fast as the second pipe. How long will it take each pipe to fill the tank individually?
3. A tank has a capacity of 1200 liters. One pipe fills it at a rate of 100 liters per hour, and another pipe drains it at a rate of 60 liters per hour. How long will it take to fill the tank if both pipes are open?

By attempting these problems, you can reinforce your understanding of the concepts and develop your problem-solving skills. Remember to break down the problems into smaller steps, define the rates, and use the appropriate formulas to arrive at the solution. With practice, you'll become more confident in tackling these types of problems.

Final Thoughts

Mathematics is not just about numbers and equations; it's about problem-solving and critical thinking. By engaging with challenging problems like the one we've discussed, you can develop valuable skills that will benefit you in various aspects of your life. So, embrace the challenge, explore the world of mathematics, and discover the power of problem-solving!