Men and women together with partial information: 2 men and 3 women finish 25% of a job in 4 days. Also, 6 men and 14 women finish the whole job in 5 days. In how many days will 20 women finish the entire work?

Introduction / Context:
This problem provides two linear equations about the combined rates of men and women, one for a quarter of the job and one for the full job. We solve the system to find an individual woman’s daily rate and then scale up to 20 women to determine completion time.

Given Data / Assumptions:

• (2m + 3w) * 4 days = 0.25 job ⇒ 8m + 12w = 0.25.
• (6m + 14w) * 5 days = 1 job ⇒ 30m + 70w = 1.
• m = man’s daily rate; w = woman’s daily rate; rates are constant.

Concept / Approach:
Reduce the equations and solve simultaneously for m and w. Then compute the collective rate of 20 women (20w) and invert to find days for the whole job.

Step-by-Step Solution:
From 8m + 12w = 0.25 ⇒ divide by 4 ⇒ 2m + 3w = 0.0625. From 30m + 70w = 1 ⇒ divide by 10 ⇒ 3m + 7w = 0.1. Multiply the first by 3: 6m + 9w = 0.1875. Multiply the second by 2: 6m + 14w = 0.2. Subtract: (6m + 14w) − (6m + 9w) = 0.2 − 0.1875 ⇒ 5w = 0.0125 ⇒ w = 0.0025. Rate of 20 women = 20 * 0.0025 = 0.05 job/day. Days for 20 women = 1 / 0.05 = 20 days.

Verification / Alternative check:
Quick check with w = 0.0025 back in the equations gives consistent totals; m can be found but is not needed for the final answer.

Why Other Options Are Wrong:
25 and 24 assume different rates; 88 is not supported by the simultaneous equations.

Common Pitfalls:
Arithmetic slips with decimals or not reducing equations before solving. Keep fractions/decimals aligned when eliminating variables.

Final Answer:
20