A is 50% as efficient as B. C does half of the work done by A and B together. If C alone does the work in 40 days, then A, B and C together can do the work in:

### Concept & Efficiency Mapping This problem tests the ability to link individual efficiencies relative to one another to form a unified ratio, which is then used to find the total work capacity. ### Step-by-Step Solution * **Given:** A is 50% as efficient as B. C does half the work of A and B together. C takes 40 days alone. * **Calculation:** Let B's efficiency be 2 units/day. * Since A is 50% as efficient as B, A's efficiency = 1 unit/day. * Combined efficiency of A + B = $1 + 2 = 3$ units/day. * C does half the work of A and B together, so C's efficiency = $1.5$ units/day. * The ratio of efficiencies is A : B : C = $1 : 2 : 1.5$. * We know C completes the work in 40 days. Total work = C's Efficiency $\times$ C's Time = $1.5 \times 40 = 60$ units. * Total efficiency of A + B + C together = $1 + 2 + 1.5 = 4.5$ units/day. * Time taken by A, B, and C together = Total Work / Combined Efficiency = $60 / 4.5$. * Simplify the fraction: $60 / 4.5 = 600 / 45 = 40 / 3 = 13\frac{1}{3}$ days. ### Exam Strategy & Shortcut To avoid decimals, assign an even number to the combined efficiency of A and B. Let B = 4 units, then A = 2 units. A+B = 6 units. Therefore, C = 3 units. Total work = $3 \times 40 = 120$ units. A+B+C efficiency = $2 + 4 + 3 = 9$ units. Time = $120 / 9 = 40/3 = 13\frac{1}{3}$ days. This calculation is much faster and avoids decimal arithmetic. ### Common Pitfall A frequent mistake is adding the days instead of the efficiencies. Time and efficiency are inversely proportional; you must always add the work done per day (efficiency), never the days directly. ### Final Answer Therefore, the correct answer is **$13\frac{1}{3}$ days**.