Gauthan and Karthik can do a piece of work in 10 days and 20 days respectively. With the help of Nilesh, they can complete the whole work in 5 days. In how many days can Nilesh alone complete the work ?

Options

A. 10
B. 20
C. 15
D. 30

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Determine the work done by Gauthan and Karthik
- Gauthan can complete the work in 10 days. Therefore, his work rate (efficiency) is:
\[ \text{Efficiency of Gauthan} = \frac{1 \text{ work}}{10 \text{ days}} = \frac{1}{10} \text{ work/day} \]

- Karthik can complete the work in 20 days. Therefore, his work rate (efficiency) is:
\[ \text{Efficiency of Karthik} = \frac{1 \text{ work}}{20 \text{ days}} = \frac{1}{20} \text{ work/day} \]

### Step 2: Calculate the combined efficiency of Gauthan and Karthik
- The combined efficiency of Gauthan and Karthik is:
\[ \text{Combined Efficiency} = \frac{1}{10} + \frac{1}{20} \]

- To add these fractions, find a common denominator (which is 20):
\[ \text{Combined Efficiency} = \frac{2}{20} + \frac{1}{20} = \frac{3}{20} \text{ work/day} \]

### Step 3: Include Nilesh's efficiency and set up the equation
- Together, Gauthan, Karthik, and Nilesh can complete the work in 5 days. Therefore, their combined efficiency is:
\[ \text{Combined Efficiency of all three} = \frac{1 \text{ work}}{5 \text{ days}} = \frac{1}{5} \text{ work/day} \]

- Let Nilesh's efficiency be \( x \) work/day. Thus, the equation becomes:
\[ \frac{3}{20} + x = \frac{1}{5} \]

### Step 4: Solve for Nilesh's efficiency
- Convert \(\frac{1}{5}\) to a fraction with a denominator of 20:
\[ \frac{1}{5} = \frac{4}{20} \]

- Now, substitute this back into the equation:
\[ \frac{3}{20} + x = \frac{4}{20} \]

- To isolate \( x \), subtract \(\frac{3}{20}\) from both sides:
\[ x = \frac{4}{20} - \frac{3}{20} = \frac{1}{20} \text{ work/day} \]

### Step 5: Calculate the number of days Nilesh can complete the work alone
- If Nilesh's efficiency is \(\frac{1}{20}\) work/day, then the number of days he would take to complete the work alone is:
\[ \text{Days} = \frac{1 \text{ work}}{x} = \frac{1}{\frac{1}{20}} = 20 \text{ days} \]

### Final Answer
Nilesh alone can complete the work in **20 days**.

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