Work and Time Calculation

This problem involves calculating the time required for a combined group of men and women to complete a work, given the time taken by individual groups.

Equating Work Capacity

We are given that 10 men can complete the work in 170 days, and 15 women can complete the same work in 170 days. This implies that the work done by 10 men is equivalent to the work done by 15 women.

Let 'M' represent the work rate of one man and 'W' represent the work rate of one woman.

From the problem statement:

• \( 10 \times M \times 170 = \text{Total Work} \)
• \( 15 \times W \times 170 = \text{Total Work} \)

Equating the work done:

\( 10 M \times 170 = 15 W \times 170 \)

This simplifies to:

\( 10 M = 15 W \)

Dividing both sides by 5:

\( 2 M = 3 W \)

This relationship tells us the relative work efficiency of men and women. From this, we can express the work rate of a woman in terms of a man's work rate:

\( W = \frac{2}{3} M \)

Calculating Combined Workforce Efficiency

We need to find the time taken by 20 men and 20 women working together.

First, let's express the combined workforce in terms of a single unit, say, equivalent men.

The combined group consists of 20 men and 20 women. Substitute the equivalent work rate of women:

\( \text{Total Equivalent Men} = 20 M + 20 W \)

\( = 20 M + 20 \left( \frac{2}{3} M \right) \)

\( = 20 M + \frac{40}{3} M \)

\( = \left( \frac{60}{3} + \frac{40}{3} \right) M \)

\( = \frac{100}{3} M \)

So, 20 men and 20 women together have the work capacity equivalent to \( \frac{100}{3} \) men.

Calculating Total Time Required

The total amount of work can be calculated from the initial condition. Let's use the men's data:

\( \text{Total Work} = 10 \text{ men} \times 170 \text{ days} = 1700 \text{ man-days} \)

Now, we can find the time required for the combined group (\( \frac{100}{3} \) men) to complete this work:

\( \text{Time} = \frac{\text{Total Work}}{\text{Number of Equivalent Men}} \)

\( \text{Time} = \frac{1700 \text{ man-days}}{\frac{100}{3} \text{ men}} \)

\( \text{Time} = 1700 \times \frac{3}{100} \text{ days} \)

\( \text{Time} = 17 \times 3 \text{ days} \)

\( \text{Time} = 51 \text{ days} \)

Alternatively, we could convert the combined workforce to equivalent women:

\( 2 M = 3 W \implies M = \frac{3}{2} W \)

\( \text{Total Equivalent Women} = 20 M + 20 W = 20 \left( \frac{3}{2} W \right) + 20 W \)

\( = 30 W + 20 W = 50 W \)

\( \text{Total Work} = 15 \text{ women} \times 170 \text{ days} = 2550 \text{ woman-days} \)

\( \text{Time} = \frac{2550 \text{ woman-days}}{50 \text{ women}} = \frac{255}{5} \text{ days} = 51 \text{ days} \)

Conclusion

Both methods confirm that 20 men and 20 women will take 51 days to complete the same work.