Question

A can do a piece of work in 2 hours, \(B\) can do thrice the work. in 8 hours and \(C^{\prime}\) can do the same work as \(A\) in 8 hours. If a! of them work together, how long it would take them to complete the work: (a) 1 hour (b) 2 hours (c) 3 hours (d) 4 hours

Short answer

Answer: (b) 2 hours.

Step-by-step solution

Find the individual work rates of A, B, and C

Let's denote the work as W. Since A can finish the work in 2 hours, so in 1 hour, A can complete (1/2)W. B can do thrice the work in 8 hours, which means B can complete the entire work, W, in 8/3 hours. So, in 1 hour, B can complete (3/8)W. C can complete the work in 8 hours, so in 1 hour, C can complete (1/8)W.

Find the combined work rate

Since A, B, and C work together, their combined work rate is the sum of their individual work rates. So, (1/2 + 3/8 + 1/8)W = (4/8)W = (1/2)W. This means that together, they can complete half of the work in 1 hour.

Find the total time taken

Since their combined work rate allows them to complete half of the work in 1 hour, it would take them twice the time to finish the entire work together. So, the total time taken would be 2 × 1 hour = 2 hours. Therefore, the answer is (b) 2 hours.

Key concepts

Work and Time

Understanding the relationship between work and time is crucial for solving problems that involve completing a task. The concept revolves around determining how long it takes to accomplish a task based on the rate at which it's being done. In the context of our exercise, we consider 'work' to mean a specific amount of a task and time the duration spent to complete this task.

For example, if 'A' can do a piece of work in 2 hours, this implies that the entire task can be completed within that time frame. If more people are involved in the same task, the time taken may potentially reduce depending on their individual rates of working. However, bringing together the concept of work and time allows us to calculate how these individual efforts collectively affect the total time required to finish the task.

Rate Calculation

Rate calculation is used to determine the speed at which work can be performed. It's usually expressed as the amount of work done per unit of time. To establish a worker's rate, we divide the total work by the time it takes for the individual to complete that work. In the given exercise, we determined that 'A' has a work rate of \( \frac{1}{2} \) of the work per hour. That means for every hour 'A' works, half of the total work is completed.

Similarly, 'B' can complete thrice the amount of work in 8 hours, so 'B' can do the entire work in \( \frac{8}{3} \) hours. This yields a work rate of \( \frac{3}{8} \) of the work per hour for 'B'. For 'C', who can complete the task in 8 hours as 'A', their work rate is \( \frac{1}{8} \) of the work per hour. Rate calculation is a fundamental step in work rate problems as it provides a basis for figuring out how various work rates can be combined.

Combined Work Rate

When individuals are working together, their individual work rates contribute to a combined work rate, enabling them to complete a task more efficiently. To find the combined work rate, simply add the individual work rates of each person working together.

In our exercise, combining the rates of 'A', 'B', and 'C' gives us \( \frac{1}{2} + \frac{3}{8} + \frac{1}{8} = \frac{1}{2} \) of the work per hour. This combined work rate means that if 'A', 'B', and 'C' work simultaneously, they can complete half of the task in one hour. Understanding this combined work rate is vital as it directly determines the amount of time needed for the team to complete the entire task.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to handle numerical and problem-solving tasks. It involves mathematical understanding, analytical thinking, and the application of formulas and concepts to solve real-world problems. Work rate problems are typical questions in the quantitative aptitude realm.

They require individuals not only to understand individual and combined work rates but also to logically deduce the most efficient means to reach the solution. The problem given is an excellent example of quantitative aptitude in action, demanding the solver to identify a series of mathematical steps and understand the principles of work rate to arrive at the correct answer of 2 hours for the joint effort of 'A', 'B', and 'C'.