Work Rate Calculation

This problem involves calculating the time taken by a combined group based on individual work rates.

Relating Men and Women's Work Rate

We are given that 15 men can reap a field in 22 days, and 25 women can reap the same field in 22 days. This means the total amount of work done by 15 men is equivalent to the work done by 25 women in the same amount of time.

Let $M$ be the work rate of one man (field per day) and $W$ be the work rate of one woman (field per day).

Total work = Rate $\times$ Time

Work done by 15 men in 22 days = $15 \times M \times 22$

Work done by 25 women in 22 days = $25 \times W \times 22$

Since the work is the same:

$15 \times M \times 22 = 25 \times W \times 22$

Simplifying this equation:

$15 M = 25 W$

$3 M = 5 W$

This establishes the relationship between the work rate of men and women: $M = \frac{5}{3} W$. One man's work rate is equivalent to $\frac{5}{3}$ women's work rate.

Calculating Combined Work Rate

We need to find the time taken by 9 men and 18 women.

First, convert the work of 9 men into the equivalent work of women:

$9 \text{ men} = 9 \times M = 9 \times \left( \frac{5}{3} W \right) = 15 W$

Now, find the total equivalent number of women:

Total equivalent women = Work of 9 men (in women's units) + Work of 18 women

Total equivalent women = $15 W + 18 W = 33 W$

So, the combined group of 9 men and 18 women works at the same rate as 33 women.

Calculating Time Taken

We know that 25 women can reap the field in 22 days.

Total work required (in woman-days) = $25 \text{ women} \times 22 \text{ days} = 550 \text{ woman-days}$

Let $D$ be the number of days required for 33 women (equivalent to 9 men and 18 women) to reap the field.

Total work = Rate $\times$ Time

$550 \text{ woman-days} = 33 \text{ women} \times D \text{ days}$

Solve for $D$:

$D = \frac{550}{33}$

Divide both numerator and denominator by 11:

$D = \frac{50}{3}$

Convert the improper fraction to a mixed number:

$D = 16 \frac{2}{3} \text{ days}$

Conclusion

Therefore, 9 men and 18 women will take $16 \frac{2}{3}$ days to reap the field.