A and B together can complete a work in 18 days, while A and C together can finish the same work in 30 days. Given that B is 200\% more efficient than C, find the number of days taken by A, B, and C working together to complete the work.

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Accepted Answer

**Solution:**

1. Let the efficiencies of A, B, and C be $a$, $b$, and $c$ units of work per day respectively.

2. Given that B is 200\% more efficient than C, which means:
   $$
   b = c + 2c = 3c
   $$

3. A and B together take 18 days to complete the work, so their combined efficiency is:
   $$
   a + b = \frac{1}{18} \text{ work per day}
   $$

4. A and C together take 30 days to complete the work, so their combined efficiency is:
   $$
   a + c = \frac{1}{30} \text{ work per day}
   $$

5. Substitute $b = 3c$ into the first equation:
   $$
   a + 3c = \frac{1}{18}
   $$

6. From the second equation:
   $$
   a + c = \frac{1}{30}
   $$

7. Subtract the second equation from the first:
   $$
   (a + 3c) - (a + c) = \frac{1}{18} - \frac{1}{30} \
   2c = \frac{1}{18} - \frac{1}{30} = \frac{5 - 3}{90} = \frac{2}{90} = \frac{1}{45}
   $$

8. Therefore:
   $$
   c = \frac{1}{90}
   $$

9. Using $a + c = \frac{1}{30}$, find $a$:
   $$
   a = \frac{1}{30} - c = \frac{1}{30} - \frac{1}{90} = \frac{3 - 1}{90} = \frac{2}{90} = \frac{1}{45}
   $$

10. Now, find $b = 3c = 3 \times \frac{1}{90} = \frac{1}{30}$.

11. The combined efficiency of A, B, and C is:
    $$
    a + b + c = \frac{1}{45} + \frac{1}{30} + \frac{1}{90}
    $$

12. Find the sum:
    $$
    \text{LCM of } 45, 30, 90 = 90
    $$
    $$
    \frac{1}{45} = \frac{2}{90}, \quad \frac{1}{30} = \frac{3}{90}, \quad \frac{1}{90} = \frac{1}{90}
    $$
    $$
    a + b + c = \frac{2 + 3 + 1}{90} = \frac{6}{90} = \frac{1}{15}
    $$

13. Therefore, the time taken by all three together to complete the work is:
    $$
    \frac{1}{\text{combined efficiency}} = 15 \text{ days}
    $$

**Answer:** 15 days

*Note:* The correct answer from the options is 15, but since 15 is option (e), the correct marking should be on option (e). The initial marking was incorrect; correcting it now.

**Correction:**
- Option (e) 15 is the correct answer.

Please consider option (e) as correct.

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