Sarthak can fill a sand-pit with sand in 36 days while Vivan takes 90 days to fill it. Ali can take the entire sand of filled sand-pit out in 60 days. If all three start working when the pit is empty, in how many days will the sand-pit be full again?
(a)45
(b)54
(c)48
(d)50

Options

A. 45
B. 54
C. 48
D. 50

Answer

To solve the problem, we will calculate the work done by each individual and then find out how long it will take for all three to fill the sand-pit when working together.

### Step 1: Determine the work rates of Sarthak, Vivaan, and Ali.

1. **Sarthak's work rate**:
- Sarthak can fill the sand-pit in 36 days.
- Work done by Sarthak in one day = \( \frac{1}{36} \) of the sand-pit.

2. **Vivaan's work rate**:
- Vivaan can fill the sand-pit in 90 days.
- Work done by Vivaan in one day = \( \frac{1}{90} \) of the sand-pit.

3. **Ali's work rate**:
- Ali can empty the sand-pit in 60 days.
- Work done by Ali in one day = \( -\frac{1}{60} \) of the sand-pit (negative because he is removing sand).

### Step 2: Calculate the combined work rate of Sarthak, Vivaan, and Ali.

- Combined work rate = Sarthak's work rate + Vivaan's work rate + Ali's work rate

\[ \text{Combined work rate} = \frac{1}{36} + \frac{1}{90} - \frac{1}{60} \]

### Step 3: Find a common denominator and simplify.

- The least common multiple (LCM) of 36, 90, and 60 is 180.

Now, convert each fraction:

1. \( \frac{1}{36} = \frac{5}{180} \)
2. \( \frac{1}{90} = \frac{2}{180} \)
3. \( -\frac{1}{60} = -\frac{3}{180} \)

Now, add them together:

\[ \text{Combined work rate} = \frac{5}{180} + \frac{2}{180} - \frac{3}{180} = \frac{4}{180} \]

### Step 4: Simplify the combined work rate.

\[ \text{Combined work rate} = \frac{4}{180} = \frac{1}{45} \]

This means that together, they can fill \( \frac{1}{45} \) of the sand-pit in one day.

### Step 5: Calculate the total time to fill the sand-pit.

If they fill \( \frac{1}{45} \) of the sand-pit in one day, then to fill the entire sand-pit (1 unit of work):

\[ \text{Time} = \frac{1 \text{ unit}}{\frac{1}{45} \text{ units/day}} = 45 \text{ days} \]

### Final Answer:

The sand-pit will be full again in **45 days**.