A can complete a certain work in 36 days. B is $$33\frac{1}{3}$$ % less efficient than A and C is 50% less efficient than B. All the three started the work together, but A left 9 days after the start of the work and B left 3 days before the completion of the work. In how many days was the whole work completed?

Correct option is D

Given:

​$$\text{A can complete the work in 36 days} \Rightarrow \text{A's 1 day work} = \frac{1}{36} \\\text{B is } 33\frac{1}{3}\% \text{ less efficient than A} \Rightarrow \text{B's efficiency} = \frac{2}{3} \text{ of A} \\\text{B's 1 day work} = \frac{2}{3} \times \frac{1}{36} = \frac{1}{54} \\\text{C is } 50\% \text{ less efficient than B} \Rightarrow \text{C's efficiency} = \frac{1}{2} \text{ of B} \\\text{C's 1 day work} = \frac{1}{2} \times \frac{1}{54} = \frac{1}{108} \\$$​​

Concept Used:

​$$\textbf{Let total work = LCM of 36, 54, 108 = 108 units} \\\ \\\text{A's efficiency} = \frac{108}{36} = 3 \text{ units/day} \\\ \\\text{B's efficiency} = \frac{108}{54} = 2 \text{ units/day} \\\ \\\text{C's efficiency} = \frac{108}{108} = 1 \text{ unit/day} \\\ \\$$​​
Solution:
​$$\text{Let total time to complete the work = } x \text{ days} \\\text{A worked for 9 days: } 9 \times 3 = 27 \text{ units} \\\text{B worked for } x - 3 \text{ days: } (x - 3) \times 2\\ = 2(x - 3) \text{ units} \\\text{C worked for } x \text{ days: } x \times 1 = x \text{ units} \\\textbf{Total work:} \\27 + 2(x - 3) + x = 108 \\27 + 2x - 6 + x = 108 \\3x + 21 = 108 \\3x = 87 \Rightarrow x = 29 \\$$​​

Final Answer:

(d) 29​​