If $p$ men can complete a work in $q$ days and $q$ women can complete the same work in $p$ days, and if $20$ men and $16$ women work together, they can finish the work in $531\frac{1}{3}$ days. What is the ratio of the efficiency of a man to that of a woman?

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**Solution:**

1. Let the efficiency of one man be $m$ and that of one woman be $w$.

2. Given that $p$ men can complete the work in $q$ days, so total work = $p 	imes m 	imes q$ units.

3. Similarly, $q$ women can complete the same work in $p$ days, so total work = $q 	imes w 	imes p$ units.

4. Since both represent the same total work, equate:
   $$p 	imes m 	imes q = q 	imes w 	imes p$$
   Simplifying, we get:
   $$m = w$$

5. Therefore, the efficiency of a man and a woman are equal.

6. Now, $20$ men and $16$ women working together have combined efficiency:
   $$20m + 16w = 20m + 16m = 36m$$

7. The total work can be expressed as:
   $$	ext{Total work} = 	ext{combined efficiency} 	imes 	ext{time} = 36m 	imes 531\frac{1}{3} = 36m 	imes \frac{1600}{3} = 19200m$$

**Answer:** The ratio of efficiency of a man to a woman is $1:1$.

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