Ashok takes twice as much time as Arun and thrice as much time as Alok together finish the work in one day. Ashok can finish the work in

Options

A. 4 days
B. 2 days
C. 6 days
D. 3 days

Answer

To solve the problem, we need to determine how long Ashok takes to finish the work alone, given that he takes twice as much time as Arun and thrice as much time as Alok, and that together they finish the work in one day.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the time taken by Arun to complete the work be \( x \) days.
- Then, the time taken by Ashok will be \( 2x \) days (since Ashok takes twice as much time as Arun).
- Let the time taken by Alok be \( y \) days.
- According to the problem, Ashok takes thrice as much time as Alok, so \( 2x = 3y \).

2. **Express Alok's Time in Terms of Arun's Time**:
- From \( 2x = 3y \), we can express \( y \) in terms of \( x \):
\[ y = \frac{2x}{3} \]

3. **Calculate the Work Done**:
- The work done by each person in one day can be expressed as:
- Arun's work rate: \( \frac{1}{x} \)
- Ashok's work rate: \( \frac{1}{2x} \)
- Alok's work rate: \( \frac{1}{y} = \frac{3}{2x} \) (substituting \( y \) from above)

4. **Total Work Rate**:
- The total work rate of Ashok, Arun, and Alok together is:
\[ \text{Total Work Rate} = \frac{1}{x} + \frac{1}{2x} + \frac{3}{2x} \]
- Simplifying this:
\[ \text{Total Work Rate} = \frac{1 + \frac{1}{2} + \frac{3}{2}}{x} = \frac{1 + 2}{2x} = \frac{3}{2x} \]

5. **Set Total Work Rate to Complete Work in One Day**:
- Since they complete the work in one day, we set the total work rate equal to 1:
\[ \frac{3}{2x} = 1 \]

6. **Solve for \( x \)**:
- Rearranging gives:
\[ 3 = 2x \implies x = \frac{3}{2} \]

7. **Calculate Ashok's Time**:
- Since Ashok takes \( 2x \) days:
\[ \text{Ashok's time} = 2 \times \frac{3}{2} = 3 \text{ days} \]

### Final Answer:
Ashok can finish the work in **3 days**.

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