Question

If one pipe \(A\) can fill a tank in 20 minutes, then 5 pipes, each of \(20 \%\) efficiency of \(A\), can fill the tank in : (a) \(80 \mathrm{~min}\) (b) 100 min (c) \(20 \mathrm{~min}\) (d) \(25 \mathrm{~min}\)

Short answer

Answer: 20 minutes

Step-by-step solution

Find the efficiency of one 20% efficient pipe.

First, we need to find the efficiency of one pipe with 20% efficiency of pipe A. Since pipe A can fill the tank in 20 minutes, its efficiency is \(\frac{1}{20}\) of the tank per minute. A pipe with 20% efficiency would have \(\frac{20}{100} \times \frac{1}{20}\) efficiency, which is equal to \(\frac{1}{100}\) of the tank per minute.

Find the combined efficiency of five 20% efficient pipes.

Now, we need to find the combined efficiency of the five 20% efficient pipes. Each pipe can fill \(\frac{1}{100}\) of the tank per minute, so their combined efficiency would be \(5 \times \frac{1}{100}\), which is equal to \(\frac{1}{20}\) of the tank per minute.

Calculate the time required to fill the tank.

The combined efficiency of five 20% efficient pipes is \(\frac{1}{20}\) of the tank per minute, which is the same as pipe A. Therefore, these five pipes can fill the tank in 20 minutes, just like pipe A. So, the correct answer is (c) \(20 \mathrm{~min}\).

Key concepts

Quantitative Aptitude

Quantitative aptitude refers to the ability to handle numerical and logical reasoning questions, often found in competitive exams and aptitude tests. It includes a broad range of topics, such as algebra, statistics, geometry, and time, speed, and distance calculations.

When tackling pipe and cistern problems, quantitative aptitude involves understanding the concept of rates and how they combine. For example, if a pipe fills a tank at a certain rate, then finding the time it takes to fill the tank requires a grasp of basic division and multiplication. Applying these skills to calculate the fill rate of multiple pipes, each at a different efficiency rate, tests one's quantitative abilities. Students are encouraged to focus on the logical application of mathematical operations, rather than just memorizing formulas.

Efficiency Rate

The efficiency rate concept is crucial when solving problems related to pipes and cisterns. It is a measure of how quickly a pipe can fill or empty a tank, usually given in the form of a part of a tank per unit of time.

For instance, the efficiency rate of a pipe can be expressed as the tank's capacity it can fill in one minute. If a pipe's efficiency rate is low, it will take longer to fill the same tank compared to a pipe with a higher efficiency. The cumulative effect of multiple pipes working together is directly related to the sum of their individual efficiencies, as demonstrated by the solution steps to the exercise provided. Understanding this concept allows students to solve more complex problems involving multiple pipes and varying efficiency rates.

Time and Work

The 'time and work' concept in mathematics relates to the amount of work done by individuals or machines within a certain time. This idea extends seamlessly to pipe and cistern problems, where 'work' equates to the amount of water filled or emptied.

In the context of our exercise, the 'work' to be done is filling the tank. The time it takes to complete this work is determined by the efficiency of the pipes involved. In these problems, understanding the inverse relationship between time and efficiency (rate) is essential. If one pipe can fill a tank in 20 minutes (its rate of work), five pipes with equal efficiency would fill the tank five times as fast, but since the efficiency of each is reduced to 20%, the original time is retained.

Problem-Solving

Problem-solving skills are a composite of analytical thinking, understanding the problem at hand, and applying appropriate mathematical principles to find a solution. In our pipe and cistern problem, problem-solving involves breaking down the question into smaller, more manageable parts.

Initially, one must understand what is asked – in this case, how long it would take for five pipes with a certain efficiency to fill a tank. The next step is to establish the efficiency of one pipe and then extend this to the cumulative efficiency of all five. Problem-solving is not only about arriving at the correct answer but also about comprehending the process, which allows students to tackle similar problems with confidence. To enhance problem-solving skills, students should practice different scenarios by altering the variables such as the number of pipes or their efficiency rates.