Ten women can complete a certain piece of work in 6 days, while six men can complete the same work in 5 days. What is the ratio of the efficiency (daily work rate) of a man to that of a woman?

Introduction / Context:
This question is a straightforward application of the Time and Work concept, focusing on comparing efficiencies of men and women when both groups complete the same job in different times and numbers. Efficiency here means the amount of work a person can do per day. Since the total work is the same, we can compare the total man-days and woman-days to obtain the ratio of their individual capacities.

Given Data / Assumptions:

• 10 women complete the work in 6 days.
• 6 men complete the same work in 5 days.
• All women have equal efficiency.
• All men have equal efficiency.
• Total work is taken as 1 unit.

Concept / Approach:
The central idea is that Work = (Number of workers) * (Efficiency per worker) * (Time). Since both groups complete the same work, their total worker-days represent the same total work. By equating these and simplifying, we can find the ratio of the efficiency of one man to that of one woman. We will treat the woman-day and man-day as units of work produced and then derive efficiencies from them.

Step-by-Step Solution:
Let the total work be W = 1 unit. Efficiency of one woman = Ew units per day. Efficiency of one man = Em units per day. Work done by 10 women in 6 days = 10 * Ew * 6 = 60Ew. Since this completes the job, 60Ew = 1. So, Ew = 1 / 60 units per day. Work done by 6 men in 5 days = 6 * Em * 5 = 30Em. This also completes the job, so 30Em = 1. Thus, Em = 1 / 30 units per day. Ratio of efficiency of one man to one woman = Em : Ew = (1 / 30) : (1 / 60). = (1 / 30) * (60 / 1) = 2 : 1.

Verification / Alternative check:
To verify, note that if one woman does 1 / 60 of the work in a day, 10 women do 10 / 60 = 1 / 6 per day, and hence finish in 6 days. Similarly, if one man does 1 / 30 of the work in a day, 6 men do 6 / 30 = 1 / 5 per day, which completes the work in 5 days. This confirms the consistency of our computed efficiencies and the ratio 2 : 1.

Why Other Options Are Wrong:
Option A (1 : 2) reverses the correct ratio, implying women are twice as efficient as men, which contradicts our calculations. Option C (2 : 3) and Option D (3 : 2) are arbitrary ratios that do not satisfy the given conditions when tested. Only 2 : 1 correctly reflects that a man's daily work is twice that of a woman under the data provided.

Common Pitfalls:
A common mistake is to compare times directly rather than computing efficiencies. For example, students might incorrectly think that since women take 6 days and men take 5 days, the ratio should simply be 5 : 6 or similar. However, the number of workers is different, so both time and number of workers must be considered. Always use Work = rate * time and then solve for individual rates before forming the ratio.

Final Answer:
The ratio of the efficiency of a man to that of a woman is 2 : 1.