P, Q, and R can each complete the same piece of work at different rates. P can complete one quarter of the work in 10 days, Q can complete 40 percent of the work in 40 days, and R can complete one third of the work in 13 days. Based on these rates, which worker would finish the entire work first if each worked alone from start to finish?

Introduction / Context:

This problem compares the efficiencies of three workers based on partial work data given in different fractions and percentages. It tests the candidate's ability to convert fractional work completed over given time periods into full completion times and then directly compare those times. Problems of this type are common in the time and work section of quantitative aptitude tests.

Given Data / Assumptions:

• P completes 1/4 of the work in 10 days.
• Q completes 40 percent (which is 2/5) of the work in 40 days.
• R completes 1/3 of the work in 13 days.
• Work rate is constant for each person and scales linearly with time.
• We assume the total work is one unit for calculation convenience.

Concept / Approach:

We convert each person's given partial work performance into a daily work rate. For example, if someone completes a certain fraction of the job in a set number of days, we divide that fraction by the number of days to obtain their per day rate. The reciprocal of that rate gives the total time that person would need to complete the entire work alone. Once the completion times for P, Q, and R are found, we identify the smallest one, which corresponds to the fastest worker.

Step-by-Step Solution:

Verification / Alternative Check:

The rates can be checked quickly. Because R completes almost the same fraction as P (one third versus one quarter) but in slightly more days compared with P's 10 days versus R's 13 days, we still need to compare full completion times. Our calculations show that R is slightly faster, needing 39 days versus P's 40 days. This aligns well with the relative rates and confirms the logic is consistent.

Why Other Options Are Wrong:

P is slightly slower than R because P needs 40 days while R needs 39 days, so option P alone is not correct. Q is clearly the slowest with 100 days, so Q is not a valid choice. Both P and R together as an answer would imply equal times, which is not the case because their completion times differ by one day. Therefore, only R is correct.

Common Pitfalls:

Typical mistakes include treating percentages and fractions inconsistently, misreading 40 percent as 40 over 100 without converting it to a fraction of the job, or forgetting to take reciprocals to move from rate to total time. Another error is to compare partial times directly instead of first finding the implied total time for each worker. A systematic rate based approach avoids these issues and leads to the correct comparison.

Final Answer:

The worker who would complete the work first when working alone is R.