$A$ can complete a wind-tunnel calibration task in 8 days, while $B$ can complete the same task in 12 days. With the assistance of an apprentice $C$, all three complete the entire task together in 4 days. If the total payment for the completed task is $\text{Rs. } 7{,}200$, the share of money earned by the apprentice $C$ (in Rupees) is:

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### 1. Strategy & Theoretical Principles Wages earned by individuals working together to complete a task are distributed in direct proportion to their individual work rates (fraction of total work contributed). ### 2. Analytical Formulation Let total work = 1 unit. - Rate of $A$: $r_A = \frac{1}{8}\text{ work/day}$ - Rate of $B$: $r_B = \frac{1}{12}\text{ work/day}$ - Combined rate of $A+B+C$: $r_{total} = \frac{1}{4}\text{ work/day}$ ### 3. Step-by-Step Calculation Find the work rate of $C$: $$r_C = r_{total} - (r_A + r_B) = \frac{1}{4} - \left(\frac{1}{8} + \frac{1}{12}\right)$$ Using common denominator 24: $$r_A = \frac{3}{24}, \quad r_B = \frac{2}{24}, \quad r_{total} = \frac{6}{24}$$ $$r_C = \frac{6}{24} - \left(\frac{3}{24} + \frac{2}{24}\right) = \frac{6 - 5}{24} = \frac{1}{24}\text{ work/day}$$ Ratio of rates of $A : B : C$: $$r_A : r_B : r_C = 3 : 2 : 1$$ Fraction of total wage earned by $C$: $$\text{Share of } C = \frac{1}{3 + 2 + 1} = \frac{1}{6}$$ $$\text{Payment to } C = \frac{1}{6} \times 7{,}200 = \text{Rs. } 1{,}200$$ ### 4. Distractor Forensics - Candidate dividing total wage equally into three parts gets $\text{Rs. } 2{,}400$, ignoring individual productivity. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Wages in joint work are distributed proportional to individual work done, not split equally. - **Standard Reference:** Aggarwal, R.S., *Quantitative Aptitude*.

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