Shashank started a work and left after 12 days, rest work done by Nitish in 15 days. Had Shashank left work, 2 days before, the rest work could have been done by Nitish in 18 days. If Shashank reduces his efficiency by 20% and Nitish increase his efficiency by 50%. Then in how many day working together they will finish the work?

Answer

To solve the problem step by step, we will first determine the efficiencies of Shashank and Nitish based on the information provided, and then calculate how long they will take to complete the work together after adjusting their efficiencies.

### Step 1: Determine the work done by Shashank and Nitish
Let Shashank's efficiency be \( S \) and Nitish's efficiency be \( N \).

- Shashank works for 12 days, so the work done by him is \( 12S \).
- Nitish completes the remaining work in 15 days, so the work done by him is \( 15N \).

The total work done can be expressed as:
\[ W = 12S + 15N \tag{1} \]

### Step 2: Analyze the scenario if Shashank left 2 days earlier
If Shashank had left after 10 days, he would have done:
\[ 10S \]
The remaining work would then be done by Nitish in 18 days, so:
\[ \text{Remaining work} = 18N \]

Setting the two expressions for the remaining work equal gives us:
\[ 12S + 15N = 10S + 18N \]

### Step 3: Simplify the equation
Rearranging the equation:
\[ 12S - 10S = 18N - 15N \]
\[ 2S = 3N \]
From this, we can express \( S \) in terms of \( N \):
\[ \frac{S}{N} = \frac{3}{2} \tag{2} \]

### Step 4: Assign values to efficiencies
Let’s assume \( N = 2x \) (for simplicity), then:
\[ S = 3x \]

### Step 5: Calculate total work done
Substituting \( S \) and \( N \) back into equation (1):
\[ W = 12(3x) + 15(2x) = 36x + 30x = 66x \]

### Step 6: Adjust efficiencies
Shashank reduces his efficiency by 20%:
\[ \text{New efficiency of Shashank} = S' = 3x - 0.2(3x) = 3x - 0.6x = 2.4x \]

Nitish increases his efficiency by 50%:
\[ \text{New efficiency of Nitish} = N' = 2x + 0.5(2x) = 2x + x = 3x \]

### Step 7: Calculate combined efficiency
Now, the combined efficiency when they work together is:
\[ \text{Total efficiency} = S' + N' = 2.4x + 3x = 5.4x \]

### Step 8: Calculate time to complete the work together
The total work is \( 66x \). The time taken to complete the work together is:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Total Efficiency}} = \frac{66x}{5.4x} = \frac{66}{5.4} = 12.22 \text{ days} \approx 12 \text{ days} \]

### Final Answer
Shashank and Nitish will finish the work together in approximately **12.22 days**.