A can do a work in 30 days B can complete same work in 20 days and A+B+C can do a work in 10 days . find in how many days C can complete the same work ?

Options

A. 60
B. 50
C. 40
D. 30

Explanation

To solve the problem step by step, we will first determine the work rates of A, B, and C, and then find out how long it will take for C to complete the work alone.

### Step 1: Determine the work rates of A and B
- A can complete the work in 30 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{30} \text{ (work per day)} \]

- B can complete the work in 20 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{20} \text{ (work per day)} \]

### Step 2: Determine the combined work rate of A, B, and C
- A, B, and C together can complete the work in 10 days. Therefore, their combined work rate is:
\[ \text{Combined work rate of A, B, and C} = \frac{1}{10} \text{ (work per day)} \]

### Step 3: Set up the equation for C's work rate
- The combined work rate can also be expressed as the sum of the individual work rates:
\[ \text{Work rate of A} + \text{Work rate of B} + \text{Work rate of C} = \text{Combined work rate} \]
\[ \frac{1}{30} + \frac{1}{20} + \text{Work rate of C} = \frac{1}{10} \]

### Step 4: Solve for C's work rate
- Let the work rate of C be represented as \( \frac{1}{C} \). The equation becomes:
\[ \frac{1}{30} + \frac{1}{20} + \frac{1}{C} = \frac{1}{10} \]

- To isolate \( \frac{1}{C} \), we first need to find a common denominator for the fractions on the left side. The least common multiple (LCM) of 30, 20, and 10 is 60.

### Step 5: Rewrite the fractions with a common denominator
- Converting each fraction:
\[ \frac{1}{30} = \frac{2}{60}, \quad \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{10} = \frac{6}{60} \]

- Substitute these values back into the equation:
\[ \frac{2}{60} + \frac{3}{60} + \frac{1}{C} = \frac{6}{60} \]

### Step 6: Combine the fractions
- Combine the fractions on the left:
\[ \frac{5}{60} + \frac{1}{C} = \frac{6}{60} \]

### Step 7: Isolate \( \frac{1}{C} \)
- Subtract \( \frac{5}{60} \) from both sides:
\[ \frac{1}{C} = \frac{6}{60} - \frac{5}{60} = \frac{1}{60} \]

### Step 8: Solve for C
- Taking the reciprocal gives us:
\[ C = 60 \]

### Conclusion
C can complete the work alone in **60 days**.