Five boys and three girls together can cultivate a 23-acre field in 4 days, while three boys and two girls together can cultivate a 7-acre field in 2 days. How many girls must work together with seven boys so that they can cultivate a 45-acre field in 6 days?

Introduction / Context:
This is a time and work style problem framed in terms of area of field cultivated by boys and girls. You are given two scenarios involving different combinations of boys and girls and the corresponding areas they cultivate in given times. From this information, you must deduce the individual daily productivity of a boy and a girl and then determine how many girls are needed with seven boys to achieve a particular target area in a specified number of days.

Given Data / Assumptions:

• Five boys and three girls cultivate 23 acres in 4 days.
• Three boys and two girls cultivate 7 acres in 2 days.
• All boys have equal efficiency; all girls have equal efficiency.
• We need to find how many girls must work with seven boys to cultivate 45 acres in 6 days.
• Work rates are constant.

Concept / Approach:
Let the daily work of one boy be b acres per day and that of one girl be g acres per day. Convert each scenario into an equation involving b and g by using the formula Work done = (total daily rate) * days. With two equations from the two scenarios, we can solve for b and g. Then apply these values to the final scenario, where seven boys and x girls work for six days to cultivate 45 acres, and solve for x.

Step-by-Step Solution:
Let one boy cultivate b acres per day. Let one girl cultivate g acres per day. Scenario 1: (5b + 3g) * 4 = 23 ⇒ 5b + 3g = 23 / 4. Scenario 2: (3b + 2g) * 2 = 7 ⇒ 3b + 2g = 7 / 2. We now solve the simultaneous equations: 5b + 3g = 23 / 4 …(1) 3b + 2g = 7 / 2 …(2) Multiply (2) by 2 to ease elimination: 6b + 4g = 7. Multiply (1) by 4: 20b + 12g = 23. Now solve directly with a simpler method: from the solution it turns out b = 1 acre per day and g = 0.25 acre per day (you can verify by substitution). Check: 5 * 1 + 3 * 0.25 = 5 + 0.75 = 5.75, and 5.75 * 4 = 23 acres, correct. Also 3 * 1 + 2 * 0.25 = 3 + 0.5 = 3.5, and 3.5 * 2 = 7 acres, correct. Now let the required number of girls be x working with 7 boys for 6 days to cultivate 45 acres. Daily work in this scenario = 7b + xg = 7 * 1 + x * 0.25 = 7 + 0.25x acres per day. Total work = (7 + 0.25x) * 6 = 45. So 42 + 1.5x = 45 ⇒ 1.5x = 3 ⇒ x = 2.

Verification / Alternative check:
With 7 boys and 2 girls, daily work = 7 * 1 + 2 * 0.25 = 7 + 0.5 = 7.5 acres per day. Over 6 days, they would cultivate 7.5 * 6 = 45 acres, exactly matching the requirement. This confirms that x = 2 is correct.

Why Other Options Are Wrong:
If x = 1, daily work = 7.25 acres, so in 6 days only 43.5 acres are cultivated. If x = 3, daily work = 7.75 acres, so in 6 days they cultivate 46.5 acres, overshooting the target. Options 3 and 4 therefore do not satisfy the equation, and so only x = 2 is valid.

Common Pitfalls:
Typical errors include incorrect algebra when solving the system for b and g, or mixing up acres with days. Some students assume boys and girls have equal efficiency, which is not supported by the data and leads to contradictions. Always derive the individual rates from the given scenarios rather than assuming them.

Final Answer:
Seven boys must work together with 2 girls to cultivate 45 acres in 6 days.