A can do a piece of work in 5 days, B can do it in 10 days. With the help of C, they finish the work in 2 days. In how many days C alone can do the whole work?

Options

A. 3
B. 4
C. 5
D. 6

Answer

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

### Step 1: Determine the work done by A and B
- A can complete the work in 5 days. Therefore, A's work rate (efficiency) is:
\[ \text{Efficiency of A} = \frac{1 \text{ work}}{5 \text{ days}} = \frac{1}{5} \]

- B can complete the work in 10 days. Therefore, B's work rate (efficiency) is:
\[ \text{Efficiency of B} = \frac{1 \text{ work}}{10 \text{ days}} = \frac{1}{10} \]

### Step 2: Determine the combined work rate of A and B
- The combined efficiency of A and B is the sum of their individual efficiencies:
\[ \text{Efficiency of A + B} = \frac{1}{5} + \frac{1}{10} \]

- To add these fractions, we need a common denominator. The least common multiple of 5 and 10 is 10:
\[ \text{Efficiency of A + B} = \frac{2}{10} + \frac{1}{10} = \frac{3}{10} \]

### Step 3: Determine the combined work rate of A, B, and C
- Together, A, B, and C complete the work in 2 days. Therefore, their combined efficiency is:
\[ \text{Efficiency of A + B + C} = \frac{1 \text{ work}}{2 \text{ days}} = \frac{1}{2} \]

### Step 4: Find the efficiency of C
- We can find C's efficiency by subtracting the combined efficiency of A and B from the combined efficiency of A, B, and C:
\[ \text{Efficiency of C} = \text{Efficiency of A + B + C} - \text{Efficiency of A + B} \]
\[ \text{Efficiency of C} = \frac{1}{2} - \frac{3}{10} \]

- To perform this subtraction, we need a common denominator. The least common multiple of 2 and 10 is 10:
\[ \text{Efficiency of C} = \frac{5}{10} - \frac{3}{10} = \frac{2}{10} = \frac{1}{5} \]

### Step 5: Determine the time taken by C to complete the work alone
- Since C's efficiency is \(\frac{1}{5}\), the time taken by C to complete the entire work alone is the reciprocal of his efficiency:
\[ \text{Time taken by C} = \frac{1}{\text{Efficiency of C}} = \frac{1}{\frac{1}{5}} = 5 \text{ days} \]

### Final Answer
C alone can complete the work in **5 days**.

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