A, B, and C can alone do a task in 40, 120 and 36 days, respectively. A and B together work for 20 days and leave the task incomplete. C resumes it an complete it alone. How many days did C take to complete it?

Options

A. A. 12
B. B. 20
C. C. 16
D. D. 18

Answer

To solve the problem, we need to determine how long it takes for C to complete the remaining work after A and B have worked together for 20 days. Here’s a step-by-step breakdown:

### Step 1: Determine the work rates of A, B, and C
- A can complete the task in 40 days, so A's work rate is \( \frac{1}{40} \) of the work per day.
- B can complete the task in 120 days, so B's work rate is \( \frac{1}{120} \) of the work per day.
- C can complete the task in 36 days, so C's work rate is \( \frac{1}{36} \) of the work per day.

### Step 2: Calculate the combined work rate of A and B
To find the combined work rate of A and B, we add their individual work rates:
\[ \text{Work rate of A and B} = \frac{1}{40} + \frac{1}{120} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 40 and 120 is 120.
\[ \frac{1}{40} = \frac{3}{120} \quad \text{and} \quad \frac{1}{120} = \frac{1}{120} \]
So,
\[ \text{Combined work rate} = \frac{3}{120} + \frac{1}{120} = \frac{4}{120} = \frac{1}{30} \]
This means A and B together can complete \( \frac{1}{30} \) of the work in one day.

### Step 3: Calculate the work done by A and B in 20 days
Now, we calculate how much work A and B can complete in 20 days:
\[ \text{Work done in 20 days} = 20 \times \frac{1}{30} = \frac{20}{30} = \frac{2}{3} \]
Thus, A and B together complete \( \frac{2}{3} \) of the work in 20 days.

### Step 4: Determine the remaining work
The total work is considered as 1 (the whole task). Therefore, the remaining work after A and B have worked is:
\[ \text{Remaining work} = 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 5: Calculate the time taken by C to complete the remaining work
Now we need to find out how long it will take for C to complete the remaining \( \frac{1}{3} \) of the work. Since C's work rate is \( \frac{1}{36} \) of the work per day, we can set up the equation:
\[ \text{Time taken by C} = \frac{\text{Remaining work}}{\text{Work rate of C}} = \frac{\frac{1}{3}}{\frac{1}{36}} = \frac{1}{3} \times 36 = 12 \text{ days} \]

### Final Answer
C takes **12 days** to complete the remaining work.

---