20 women can complete a work in 15 days. 16 men can complete the same work in 15 days. Find the ratio between the work efficiency of a man to a woman.

Steps to Solve

1. Calculate Total Work Done

Firstly, we need to determine the total work done in terms of "woman-days" and "man-days."

For 20 women completing the work in 15 days:

Total work = Number of women × Number of days

$$ \text{Total work} = 20 \times 15 = 300 \text{ woman-days} $$

2. Calculate Work Done by Men

Next, we also know that 16 men can complete the same work in 15 days.

Total work = Number of men × Number of days

$$ \text{Total work} = 16 \times 15 = 240 \text{ man-days} $$

3. Work Rate Comparison Between Men and Women

From the total work done, we can ascertain the work rate of a man compared to a woman.

Let the work done by one woman in one day be $W_w$ and by one man be $W_m$.

The total work done can also be expressed as:

For women: $$ W_w \times 20 \times 15 = 300 $$

So, $$ W_w = \frac{300}{300} = 1 \text{ work unit per day} $$

For men: $$ W_m \times 16 \times 15 = 240 $$

So, $$ W_m = \frac{240}{240} = 1 \text{ work unit per day} $$

4. Find Ratio of Work Efficiency

Now, we need to find the efficiency ratio of a man to a woman:

$$ \text{Efficiency Ratio} = \frac{W_m}{W_w} = \frac{1}{1} $$

This implies men and women have the same efficiency.

If we consider their work rates over the total days to complete the work, we then note that for every 20 women (300 woman-days), it takes 16 men (240 man-days).

Thus,

$$ \text{Ratio of Men's work to Women's work} = \frac{240}{300} = \frac{4}{5} $$

Thus, to find the ratio of man to woman:

$$ \text{Man : Woman} = 4 : 5 $$

Since we need the ratio of a man to a woman, we can invert this.

5. Final Ratio Calculation

Thus the ratio of work efficiency is:

$$ \text{Ratio} = \frac{4}{5} \longrightarrow 5:4 $$