Five men and three boys together can cultivate a 23 acre field in 4 days, while three men and two boys together can cultivate a 7 acre field in 2 days. How many boys must work together with 7 men to cultivate a 45 acre field in 6 days?

Introduction / Context:
This is a standard time and work question involving men and boys who work at different rates. We are given two scenarios that allow us to determine individual work rates of a man and a boy. Once those rates are known, we can compute how many boys must assist a group of men to finish a larger field within a given time.

Given Data / Assumptions:
• Five men and three boys cultivate 23 acres in 4 days.
• Three men and two boys cultivate 7 acres in 2 days.
• We assume each man has a constant daily work rate M acres per day.
• We assume each boy has a constant daily work rate B acres per day.
• We must find the number of boys required with 7 men to cultivate 45 acres in 6 days.

Concept / Approach:
We translate each condition into an algebraic equation using Work = Rate × Time. The first two pieces of information yield two linear equations with unknowns M and B. Solving these gives the individual productivity of a man and a boy. Then we express the condition involving 7 men and x boys working for 6 days and equate their total output to 45 acres to solve for x.

Step-by-Step Solution:
Let M = work done by one man per day, and B = work done by one boy per day. From the first condition: (5M + 3B) × 4 = 23, so 20M + 12B = 23. From the second condition: (3M + 2B) × 2 = 7, so 6M + 4B = 7. Solve the system: subtracting twice the second equation from the first gives (20M + 12B) − 2(6M + 4B) = 23 − 14. This simplifies to 8M + 4B = 9. From 6M + 4B = 7, subtract to get 2M = 2, so M = 1 acre per day. Substitute M = 1 into 6M + 4B = 7 to get 6 + 4B = 7, giving B = 1 / 4 acre per day. For the third scenario, let x be the number of boys working with 7 men for 6 days on 45 acres. Total work = (7M + xB) × 6 = 45. Substitute M = 1 and B = 1 / 4: (7 + x / 4) × 6 = 45. So 42 + (6x / 4) = 45, giving 42 + 1.5x = 45. Thus 1.5x = 3, and x = 2 boys.

Verification / Alternative check:
Check using daily output. Seven men produce 7 acres per day. Two boys produce 2 × 0.25 = 0.5 acre per day. Together they produce 7.5 acres per day. Over 6 days, total output is 7.5 × 6 = 45 acres, which matches the required area, confirming that the answer is correct.

Why Other Options Are Wrong:
If 4 or 6 boys were used, total daily output would be greater than necessary and the 45 acres would be finished in less than 6 days. If only 1 or 3 boys were used, the total cultivated area in 6 days would be less than 45 acres, so the work would not be completed on time.

Common Pitfalls:
Mistakes often occur when forming the initial equations, especially with multiplying by the number of days. Another common error is solving the equations incorrectly, leading to wrong values for M and B. Careful algebra and substitution avoid these issues.

Final Answer:
Two boys are needed together with seven men to cultivate 45 acres in 6 days.