Determining the Efficiency of R: Solving Work Problems Involving Multiple Employees

Introduction

Work rate problems involving multiple employees are common in both academic and practical settings. These problems require an understanding of how different individuals working together can impact the completion of a task. The process involves determining the efficiency of individuals and teams, often expressed as the fraction of work they can complete in a given unit of time (days, hours, etc.). This article will walk you through a typical problem and provide a step-by-step solution, ensuring you can manage such problems effectively.

Problem Statement

Two persons, P and Q, can finish a piece of work in 30 days. When a third person, R, is added to the team (P, Q, and R), they can complete the same work in 20 days. How many days will R alone take to finish half of the work?

Solution

Let's break down the problem step by step:

Step 1: Determine the Work Rate of P and Q

Since P and Q can complete the work in 30 days, their combined work rate is:

P   Q  1/30 (work per day)

Step 2: Determine the Work Rate of P, Q, and R

When all three, P, Q, and R, work together, they complete the work in 20 days, so their combined work rate is:

P   Q   R  1/20 (work per day)

Step 3: Determine R's Work Rate

To find R's work rate, subtract the combined work rate of P and Q from the combined work rate of P, Q, and R:

R  (P   Q   R) - (P   Q)  1/20 - 1/30

LCM of 20 and 30 is 60, so we can rewrite the equation as:

R  3/60 - 2/60  1/60

This means R can complete 1/60th of the work in one day.

Step 4: Calculate the Time for R to Complete Half the Work

Since R completes 1/60th of the work in one day, to complete half of the work (which is 1/2 of the total work), R will take:

T  (1/2) / (1/60)  60/2  30 days

Therefore, R alone will take 30 days to finish half of the work.

Additional Example

Two persons P and Q can finish a piece of work in 60 days while P, Q, and R can finish the same work in 40 days. How many days will R alone take to finish half of the work?

Using the same method, we can determine R's work rate:

P   Q  1/60

P   Q   R  1/40

R  (P   Q   R) - (P   Q)  1/40 - 1/60  1/120

R's work rate is 1/120th of the work per day. To complete half of the work, R will take:

T  (1/2) / (1/120)  120/2  60 days

Thus, R alone will take 60 days to finish half of the work.

Conclusion

Understanding work rate and efficiency problems is crucial for various applications, from scheduling tasks to planning projects. By breaking down the problem and using a methodical approach, you can solve these questions accurately and efficiently.