Twenty women can complete a piece of work in 16 days, while sixteen men can complete the same work in 15 days. What is the ratio of the working capacity (efficiency) of a man to that of a woman?

Introduction / Context:
This question compares the efficiencies of men and women workers based on how many of each are needed to complete the same job in different numbers of days. It requires forming equations using the concept of total work as a product of number of workers, time and individual efficiency, and then finding the ratio of efficiencies.

Given Data / Assumptions:

• 20 women can complete the work in 16 days.
• 16 men can complete the same work in 15 days.
• All women have the same efficiency as one another.
• All men have the same efficiency as one another.
• We are asked to find the ratio of the efficiency of a man to the efficiency of a woman.

Concept / Approach:
We denote the daily work rate of a woman by w and that of a man by m. The total work can be expressed in two different ways: once using women and once using men. Since both expressions represent the same amount of work, we can equate them and solve for the ratio m : w. The ratio of efficiencies is simply m divided by w, and we finally express it as a ratio of two integers.

Step-by-Step Solution:
Step 1: Let work done per woman per day be w units. With 20 women working for 16 days, total work W = 20 * 16 * w.Step 2: Let work done per man per day be m units. With 16 men working for 15 days, the same work W = 16 * 15 * m.Step 3: Equate the two expressions for total work: 20 * 16 * w = 16 * 15 * m.Step 4: Simplify the equation. First divide both sides by 16: 20 * w = 15 * m.Step 5: Rearranging for m in terms of w gives m = (20 / 15) * w = (4 / 3) * w.Step 6: Therefore the ratio of man's efficiency to woman's efficiency is m : w = (4 / 3) * w : w = 4 : 3.

Verification / Alternative check:
We can check this ratio by plugging back. If a woman does 3 units per day and a man does 4 units per day, then in 16 days, 20 women do 20 * 16 * 3 = 960 units. In 15 days, 16 men do 16 * 15 * 4 = 960 units. The total work is the same, which confirms that the ratio 4 : 3 is correct.

Why Other Options Are Wrong:

• 3:4: This is the inverse ratio and would claim that women are more efficient than men in this setting, which does not match the calculated relation.
• 5:3: This suggests men are much more efficient than indicated by the total work comparison and would not balance the two work expressions.
• 5:7: This ratio would not equalize the total work; substituting it back would give different amounts of work done in the two scenarios.

Common Pitfalls:
A common mistake is to compare only the numbers of workers and the days without introducing the efficiency variables, or to attempt to directly divide 20 by 16 and 16 by 15 in an incorrect way. Another error is inverting the ratio at the end, giving the ratio of women to men instead of men to women. Always double check which order the ratio is requested in the question.

Final Answer:
The ratio of the efficiency of a man to that of a woman is 4:3.