Mrinal and Yukta together can paint half of a wall in 6 days. Working separately, Mrinal would need thrice as much time as Yukta would to paint half of the wall. In how many days can Yukta alone paint the whole wall ?

Options

A. 12
B. 20
C. 9
D. 16

Answer

To solve the problem step by step, let's break down the information given and derive the solution.

### Step 1: Understand the combined work rate
Mrinal and Yukta together can paint half of a wall in 6 days. Therefore, to paint the whole wall, they would take:
\[ \text{Time to paint the whole wall} = 6 \text{ days} \times 2 = 12 \text{ days} \]

### Step 2: Define the work rates
Let’s denote the time Yukta takes to paint half of the wall as \( y \) days. According to the problem, Mrinal takes three times as long as Yukta to paint half of the wall, so Mrinal takes \( 3y \) days to paint half of the wall.

### Step 3: Calculate the work rates
The work done is inversely proportional to the time taken. Therefore, the work rates (efficiencies) can be defined as follows:
- Yukta's work rate for half a wall: \( \frac{1}{y} \) (walls per day)
- Mrinal's work rate for half a wall: \( \frac{1}{3y} \) (walls per day)

### Step 4: Combine their work rates
When working together, their combined work rate is:
\[ \text{Combined work rate} = \frac{1}{y} + \frac{1}{3y} = \frac{3}{3y} + \frac{1}{3y} = \frac{4}{3y} \]

### Step 5: Relate the combined work rate to the total work
Since they can paint the whole wall in 12 days, their combined work rate can also be expressed as:
\[ \text{Combined work rate} = \frac{1 \text{ wall}}{12 \text{ days}} = \frac{1}{12} \]

### Step 6: Set the equations equal to each other
Now we can set the two expressions for the combined work rate equal to each other:
\[ \frac{4}{3y} = \frac{1}{12} \]

### Step 7: Solve for \( y \)
Cross-multiplying gives:
\[ 4 \times 12 = 3y \implies 48 = 3y \implies y = \frac{48}{3} = 16 \]

### Step 8: Find Yukta's time to paint the whole wall
Since \( y \) is the time taken by Yukta to paint half of the wall, the time taken by Yukta to paint the whole wall would be:
\[ \text{Time for Yukta to paint the whole wall} = 2y = 2 \times 16 = 32 \text{ days} \]

### Conclusion
Yukta alone can paint the whole wall in **32 days**.