Rates of men and boys in reaping hectares are different. If 5 men and 3 boys together reap 23 hectares in 4 days, and 3 men and 2 boys reap 7 hectares in 2 days, then how many boys must assist 7 men so that 45 hectares are reaped in 6 days?

Introduction / Context:
We have two groups with men and boys working at different rates. By forming linear equations for daily output, we can determine individual rates and then scale to the target combination (7 men plus x boys) to meet a specified area in a given time.

Given Data / Assumptions:

• (5 men + 3 boys) working 4 days ⇒ 23 hectares.
• (3 men + 2 boys) working 2 days ⇒ 7 hectares.
• Goal: (7 men + x boys) working 6 days ⇒ 45 hectares.
• Uniform rates per person per day; constant productivity.

Concept / Approach:
Let m = hectares per man per day, b = hectares per boy per day. Translate both statements into equations, solve for m and b, then find x such that (7m + x*b) * 6 = 45.

Step-by-Step Solution:

Verification / Alternative check:
Plug m = 1, b = 0.25 back into the original equations to confirm they satisfy both, ensuring consistency. They do, so x = 2 is valid.

Why Other Options Are Wrong:
6 boys: Would exceed 45 hectares within 6 days.

4 boys / 5 boys: Produce totals different from 45 hectares when evaluated; they do not match the target exactly.

3 boys: Still short of 45 hectares.

Common Pitfalls:
Forgetting to divide by days; mixing area with rate; or assuming men and boys contribute equally. Always solve for individual rates first.

Final Answer:
2 boys