If 2 men and 3 women can complete a piece of work in 8 days and 3 men and 2 women can complete the same work in 7 days, then in how many days will 5 men and 4 women working together complete the work?

Introduction / Context:
This time and work question involves two types of workers, men and women, each with their own unknown efficiency. We are given how long different combinations of men and women take to complete the same job. From this information, we must deduce the individual work rates and then calculate how long a new combination of 5 men and 4 women will take to complete the job.

Given Data / Assumptions:
Two men and three women together complete a piece of work in 8 days. Three men and two women together complete the same work in 7 days. We are asked for the time taken by 5 men and 4 women working together. The total work is taken as one unit and all men have the same efficiency, as do all women. Work rates remain constant throughout.

Concept / Approach:
We let the daily work done by one man be m and by one woman be w. The group rates for the given combinations become linear equations in m and w. Using these equations, we solve for m and w. Then we compute the combined rate for 5 men and 4 women and take the reciprocal to get the required time. The method involves solving a system of two linear equations with two unknowns.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Step 2: Let the rate of one man be m work per day, and the rate of one woman be w work per day. Step 3: Two men and three women together have a daily rate of (2m + 3w). They complete the work in 8 days, so 8(2m + 3w) = 1. Step 4: Simplify the first equation: 16m + 24w = 1. Step 5: Three men and two women together have a daily rate of (3m + 2w). They complete the work in 7 days, so 7(3m + 2w) = 1. Step 6: Simplify the second equation: 21m + 14w = 1. Step 7: We now solve the system: 16m + 24w = 1 and 21m + 14w = 1. Step 8: Multiply the first equation by 7 to get 112m + 168w = 7. Multiply the second equation by 8 to get 168m + 112w = 8. Step 9: Subtract the first new equation from the second: (168m + 112w) - (112m + 168w) = 8 - 7, giving 56m - 56w = 1. Step 10: Thus, m - w = 1 / 56. Also, using one of the original equations, we can solve explicitly to get m = 1 / 28 and w = 1 / 56. Step 11: Now compute the rate of 5 men and 4 women: rate = 5m + 4w = 5 * (1 / 28) + 4 * (1 / 56). Step 12: Simplify: 5 / 28 + 4 / 56 = 5 / 28 + 1 / 14 = 5 / 28 + 2 / 28 = 7 / 28 = 1 / 4 work per day. Step 13: Time taken is the reciprocal of the rate: time = 1 / (1 / 4) = 4 days.

Verification / Alternative check:
We can check the consistency by plugging m and w back into the original equations. For 2 men and 3 women, rate = 2 * (1 / 28) + 3 * (1 / 56) = 2 / 28 + 3 / 56 = 1 / 14 + 3 / 56 = 4 / 56 + 3 / 56 = 7 / 56 = 1 / 8, so they take 8 days. For 3 men and 2 women, rate = 3 * (1 / 28) + 2 * (1 / 56) = 3 / 28 + 1 / 28 = 4 / 28 = 1 / 7, so they take 7 days. This confirms that our m and w values and subsequent calculations are correct.

Why Other Options Are Wrong:
3 days and 2 days correspond to much higher combined rates than 1 / 4 per day and would imply inconsistencies with the original equations. 5 days and 6 days imply combined rates smaller than 1 / 4, again inconsistent with the derived values of m and w. Only 4 days aligns perfectly with the work rates inferred from the original combinations.

Common Pitfalls:
Common mistakes include treating the number of workers as a direct ratio to time without solving the underlying system of equations or miscalculating the combined rates. Some may attempt to average the days without accounting for the different numbers of men and women. Correct handling of simultaneous equations and consistent units is essential for success in this type of problem.

Final Answer:
Five men and four women together will complete the work in 4 days.