Mixed men–women wage rates: 5 men and 6 women together earn ₹ 2160 in 4 days; 8 men and 10 women together earn ₹ 4400 in 5 days. In how many days will 4 men and 9 women together earn ₹ 1800?

Introduction / Context:
We are to determine per-day wages for a man and a woman using two combined scenarios. With those unit rates, we can then compute how long another group will take to earn a specified amount. This is a standard linear system setup.

Given Data / Assumptions:

• (5m + 6w) for 4 days = ₹ 2160 ⇒ per day = 2160/4 = 540
• (8m + 10w) for 5 days = ₹ 4400 ⇒ per day = 4400/5 = 880
• Find days d so that (4m + 9w) * d = ₹ 1800

Concept / Approach:
Let man's daily wage be M and woman's daily wage be W. Solve the linear equations: 5M + 6W = 540 and 8M + 10W = 880. Then compute (4M + 9W) and finally get d = 1800 / (4M + 9W).

Step-by-Step Solution:
5M + 6W = 5408M + 10W = 880Solve ⇒ W = 40, M = 604M + 9W = 4*60 + 9*40 = 240 + 360 = 600 per dayDays d = 1800 / 600 = 3

Verification / Alternative check:
Back-substitute M and W into original equations to confirm: 5*60 + 6*40 = 540; 8*60 + 10*40 = 880. Both hold true.

Why Other Options Are Wrong:
4, 5, 6 days do not match the computed daily earning capacity of 4 men and 9 women.

Common Pitfalls:
Arithmetic slips in elimination; forgetting to convert totals to per-day figures before forming equations; mixing wages with work rates (here wages are linear, not inverse-time rates).

Final Answer:
3