Tap Problem Calculate Tank Filling Time
This article delves into a classic time and work problem, providing a step-by-step solution and exploring the underlying concepts. We'll analyze the scenario of three taps filling a water tank at different rates and determine the exact time the tank will be full, considering the opening and closing of taps. This problem is a staple in quantitative aptitude tests and demonstrates essential problem-solving techniques applicable in various real-life situations.
Understanding the Problem
In this filling the tank problem, we have three taps, A, B, and C, each capable of filling a water tank at different rates. Tap A takes 30 minutes, Tap B takes 20 minutes, and Tap C takes only 10 minutes to fill the tank independently. The problem introduces a time constraint: all three taps are opened simultaneously at 8:55 am, but Tap C is turned off at 8:59 am. The core question is to determine the precise time when the tank will be completely filled after Tap C is closed. To solve this, we'll utilize the concept of work rate, calculating how much of the tank each tap fills per minute and combining their efforts to find the overall filling time. The problem highlights the importance of understanding individual contributions and how they collectively impact the final outcome, a crucial skill in various analytical and practical contexts.
Step-by-Step Solution
To solve this time and work problem effectively, let's break it down into manageable steps, focusing on each tap's contribution and the combined effort over time. First, we need to determine the individual filling rates of each tap. Tap A fills 1/30 of the tank per minute, Tap B fills 1/20 of the tank per minute, and Tap C fills a significant 1/10 of the tank per minute. These fractions represent the proportion of the tank each tap can fill in a single minute. Next, we consider the period when all three taps are open simultaneously. From 8:55 am to 8:59 am, all taps work together for 4 minutes. During this time, their combined filling rate is the sum of their individual rates: (1/30) + (1/20) + (1/10). To add these fractions, we find a common denominator, which is 60. The combined rate then becomes (2/60) + (3/60) + (6/60) = 11/60. This means that in one minute, all three taps together fill 11/60 of the tank. Over the 4 minutes they are all open, they fill (11/60) * 4 = 44/60 of the tank. Now, we need to determine the remaining portion of the tank to be filled after Tap C is closed. If 44/60 of the tank is already full, then 1 - (44/60) = 16/60, which simplifies to 4/15, of the tank remains to be filled. After 8:59 am, only Taps A and B are working. Their combined filling rate is (1/30) + (1/20) = (2/60) + (3/60) = 5/60, which simplifies to 1/12. This means that together, Taps A and B fill 1/12 of the tank per minute. To find the time it takes for Taps A and B to fill the remaining 4/15 of the tank, we divide the remaining fraction by their combined rate: (4/15) / (1/12). Dividing by a fraction is the same as multiplying by its reciprocal, so we have (4/15) * 12 = 48/15 minutes. Simplifying this fraction, we get 3 and 3/15 minutes, which is 3 minutes and 12 seconds. Therefore, it takes an additional 3 minutes and 12 seconds to fill the tank after Tap C is closed. Adding this time to 8:59 am, we find that the tank will be completely filled at 9:02:12 am. This step-by-step approach, focusing on individual rates and combined efforts, is crucial for solving such problems accurately and efficiently.
Calculating Individual Filling Rates
Calculating individual filling rates is the cornerstone of solving time and work problems related to filling tanks or completing tasks. The filling rate of each tap represents the fraction of the tank it can fill in a single unit of time, typically one minute. In this scenario, Tap A fills the tank in 30 minutes, which means its filling rate is 1/30 of the tank per minute. Similarly, Tap B fills the tank in 20 minutes, so its filling rate is 1/20 of the tank per minute. Tap C, being the most efficient, fills the tank in just 10 minutes, giving it a filling rate of 1/10 of the tank per minute. These individual rates are crucial because they allow us to quantify the contribution of each tap to the overall task of filling the tank. Understanding these rates is the foundation for calculating combined filling rates and determining the time it takes to complete the task under various conditions. The concept of individual rates is not limited to tank-filling problems; it can be applied to any situation where individuals or machines work at different speeds to complete a task. For example, in construction, knowing the rate at which each worker can lay bricks or the speed at which each machine can excavate earth is essential for project planning and scheduling. Similarly, in manufacturing, understanding the production rate of each machine helps in optimizing the assembly line and meeting production targets. Therefore, mastering the calculation and application of individual rates is a valuable skill in a wide range of fields, from project management to resource allocation. In mathematical terms, the filling rate is the reciprocal of the time taken to complete the task. This inverse relationship is fundamental to understanding how time and work are related. A higher filling rate implies a shorter time to complete the task, and vice versa. By clearly defining and calculating these rates, we can accurately predict the outcome of combined efforts and solve complex problems involving multiple contributors working simultaneously or sequentially.
Determining the Combined Filling Rate
Determining the combined filling rate is a critical step in solving problems where multiple entities work together to complete a task. In the context of this problem, we need to calculate how much of the tank is filled per minute when all three taps, A, B, and C, are opened simultaneously. To do this, we simply add their individual filling rates, which we calculated earlier. Tap A fills 1/30 of the tank per minute, Tap B fills 1/20 of the tank per minute, and Tap C fills 1/10 of the tank per minute. So, the combined filling rate is (1/30) + (1/20) + (1/10). To add these fractions, we need to find a common denominator, which is the least common multiple (LCM) of 30, 20, and 10. The LCM is 60, so we convert each fraction to have a denominator of 60. This gives us (2/60) + (3/60) + (6/60). Now, we can easily add the numerators: 2 + 3 + 6 = 11. Therefore, the combined filling rate is 11/60 of the tank per minute. This means that when all three taps are open, they collectively fill 11/60 of the tank in a single minute. Understanding the combined filling rate allows us to calculate how much of the task is completed over a specific period when all entities are working together. In this case, since all three taps are open from 8:55 am to 8:59 am, a total of 4 minutes, we can multiply the combined filling rate by the time to find the fraction of the tank filled during this period. The concept of combined rates is applicable in numerous real-world scenarios. For instance, in project management, it's crucial to determine the combined productivity of a team working on a project. If each team member has a known work rate, their combined rate can be calculated to estimate how quickly the project will progress. Similarly, in manufacturing, the combined output rate of multiple machines working simultaneously can be determined by adding their individual production rates. In logistics, the combined delivery rate of several trucks can be calculated to estimate how quickly goods can be transported. The ability to accurately calculate and apply combined rates is essential for effective planning, scheduling, and resource allocation in various industries and applications.
Calculating the Portion of Tank Filled
Calculating the portion of the tank filled during a specific time interval is a crucial step in solving this problem. We know the combined filling rate of taps A, B, and C when they are all open, which is 11/60 of the tank per minute. The problem states that all three taps are opened from 8:55 am to 8:59 am, a duration of 4 minutes. To find the portion of the tank filled during these 4 minutes, we multiply the combined filling rate by the time. So, (11/60) * 4 = 44/60. This means that in the 4 minutes when all three taps are open, 44/60 of the tank is filled. It's often helpful to simplify fractions to make them easier to work with. The fraction 44/60 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives us 11/15. Therefore, 11/15 of the tank is filled during the first 4 minutes. Understanding how to calculate the portion of the task completed over a period is essential in many time and work problems. It allows us to track progress and determine how much work remains. In this case, after the first 4 minutes, we know that 11/15 of the tank is full, and we need to find out how much longer it will take to fill the remaining portion. The concept of calculating the portion of work completed is widely applicable in various fields. In construction, for example, it's essential to calculate the portion of a building that has been completed after a certain number of days to track progress against the project timeline. In software development, the portion of code written or the number of features implemented can be calculated to assess project status. Similarly, in manufacturing, the portion of products manufactured within a specific timeframe can be determined to monitor production efficiency. By consistently calculating the portion of the task completed, we can effectively manage projects, allocate resources, and ensure timely completion. In the context of problem-solving, this step bridges the gap between knowing the rate of work and understanding the amount of work accomplished over time, making it a fundamental skill for quantitative analysis.
Determining the Remaining Portion of the Tank
Determining the remaining portion of the tank to be filled is a critical step in calculating the total time required to fill the tank completely. After the first phase, where all three taps are open for 4 minutes, we calculated that 44/60 (or simplified, 11/15) of the tank is filled. To find out the remaining portion, we need to subtract the filled portion from the whole, which is represented as 1. So, the remaining portion is 1 - (44/60). To subtract fractions, we need a common denominator. In this case, we can express 1 as 60/60. Therefore, the remaining portion is (60/60) - (44/60) = 16/60. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives us 4/15. So, after Tap C is turned off, 4/15 of the tank remains to be filled. Understanding how to calculate the remaining work is essential in project management and task completion scenarios. It allows us to focus our efforts on the uncompleted portion and estimate the time and resources required to finish the job. In the context of this problem, now that we know the remaining portion of the tank, we can calculate how long it will take for the remaining taps, A and B, to fill it. This involves considering their combined filling rate and the amount of work left. The concept of determining the remaining portion is widely applicable in various real-world situations. For example, in a manufacturing process, after producing a certain number of units, it's crucial to calculate the remaining units to be produced to meet the target. Similarly, in construction, after completing a part of the building, determining the remaining work helps in planning the subsequent phases and estimating the project completion time. In software development, after implementing a set of features, calculating the remaining features helps in managing the project scope and timelines. The ability to accurately calculate the remaining portion of a task or project is a key skill for effective planning and execution. It allows us to adjust our strategies, allocate resources efficiently, and ensure that we stay on track towards achieving our goals. In mathematical terms, this step involves subtracting the completed portion from the total, providing a clear picture of the work that still needs to be done.
Calculating the Time to Fill the Remaining Portion
Calculating the time to fill the remaining portion of the tank is the final critical step in solving this problem. We know that after Tap C is closed, 4/15 of the tank remains to be filled. We also need to consider that only Taps A and B are working at this stage. Their combined filling rate was previously calculated as (1/30) + (1/20) = (2/60) + (3/60) = 5/60, which simplifies to 1/12 of the tank per minute. To find the time it takes for Taps A and B to fill the remaining 4/15 of the tank, we need to divide the remaining portion by their combined filling rate. So, we have (4/15) / (1/12). Dividing by a fraction is the same as multiplying by its reciprocal, so we rewrite the expression as (4/15) * 12. Multiplying these fractions, we get (4 * 12) / 15 = 48/15 minutes. Now, we need to simplify this fraction and convert it into a more understandable time format. The fraction 48/15 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us 16/5 minutes. To convert this improper fraction to a mixed number, we divide 16 by 5, which gives us 3 with a remainder of 1. So, 16/5 minutes is equal to 3 and 1/5 minutes. This means it will take 3 full minutes and 1/5 of a minute to fill the remaining portion of the tank. To convert 1/5 of a minute into seconds, we multiply it by 60 (since there are 60 seconds in a minute): (1/5) * 60 = 12 seconds. Therefore, it will take 3 minutes and 12 seconds to fill the remaining 4/15 of the tank. This final calculation allows us to determine the exact time when the tank will be completely filled. By dividing the remaining work by the rate at which it is being done, we can accurately predict the time required to complete the task. This concept is widely applicable in various fields, from project management to resource planning. In any situation where work needs to be completed and resources are allocated, understanding how to calculate the time to complete the remaining portion is essential for efficient execution and successful outcomes.
Determining the Final Time
Determining the final time the tank will be filled involves adding the time it takes to fill the remaining portion to the time when Tap C was closed. We know that Tap C was closed at 8:59 am, and it takes an additional 3 minutes and 12 seconds for Taps A and B to fill the remaining portion of the tank. To find the final time, we add 3 minutes and 12 seconds to 8:59 am. Adding 3 minutes to 8:59 am brings us to 9:02 am. Then, adding the remaining 12 seconds gives us a final time of 9:02:12 am. Therefore, the tank will be completely filled at 9:02:12 am. This final step is crucial as it provides the precise answer to the problem. It demonstrates the practical application of all the previous calculations and highlights the importance of accuracy in each step. By carefully adding the time intervals, we can confidently state the exact moment the tank will be full. The process of determining the final completion time is a common task in various real-world scenarios. In project management, it's essential to calculate the final project completion date by adding the remaining time to the current date. This helps in setting expectations, managing stakeholders, and ensuring that the project is delivered on time. Similarly, in manufacturing, determining the final production time allows for efficient scheduling and resource allocation. In logistics, calculating the final delivery time helps in planning routes and informing customers about expected arrival times. The ability to accurately determine the final time is a valuable skill in any field where time-sensitive tasks need to be managed. It requires careful attention to detail and a clear understanding of how time intervals add up. In this problem, we combined the time when all three taps were open with the time when only Taps A and B were open to arrive at the final filling time. This step-by-step approach ensures that all factors are considered, leading to a precise and reliable answer.
Conclusion
In conclusion, solving this time and work problem involved a systematic approach, breaking it down into smaller, manageable steps. We began by calculating the individual filling rates of each tap, then determined the combined filling rate when all taps were open. We calculated the portion of the tank filled during the time all taps were working together and subsequently found the remaining portion to be filled. Finally, we calculated the time required for the remaining taps to fill the rest of the tank and added it to the initial time to find the final time when the tank would be completely filled. This problem highlights the importance of understanding individual contributions, combined efforts, and the concept of rates in solving time-related problems. The skills and techniques used here are applicable in a wide range of real-world scenarios, from project management to resource allocation. By mastering these problem-solving methods, you can effectively tackle complex tasks and achieve accurate results. The step-by-step approach not only helps in solving the problem correctly but also enhances understanding of the underlying concepts, making it easier to apply them in different contexts. The ability to break down a complex problem into smaller, more manageable parts is a valuable skill in any field, and this problem serves as a good example of how that can be done effectively.
