A can paint a house in 45 days and B can do it in 15 days. Along with C, they did the job in 5 days only. Then, C alone can do the job in how many days?

Options

A. 12
B. 9
C. 15
D. 8

Answer

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

### Step 1: Determine the work rates of A and B
- A can paint a house in 45 days, so A's work rate is:
\[ \text{Efficiency of A} = \frac{1}{45} \text{ (houses per day)} \]
- B can paint a house in 15 days, so B's work rate is:
\[ \text{Efficiency of B} = \frac{1}{15} \text{ (houses per day)} \]

### Step 2: Calculate the combined efficiency of A and B
- To find the combined efficiency of A and B, we add their efficiencies:
\[ \text{Combined Efficiency of A and B} = \frac{1}{45} + \frac{1}{15} \]
- To add these fractions, we need a common denominator, which is 45:
\[ \frac{1}{15} = \frac{3}{45} \]
- Now, we can add:
\[ \text{Combined Efficiency of A and B} = \frac{1}{45} + \frac{3}{45} = \frac{4}{45} \]

### Step 3: Determine the combined efficiency of A, B, and C
- It is given that A, B, and C together can complete the job in 5 days. Thus, their combined efficiency is:
\[ \text{Combined Efficiency of A, B, and C} = \frac{1}{5} \text{ (houses per day)} \]

### Step 4: Find the efficiency of C
- We already calculated the combined efficiency of A and B as \(\frac{4}{45}\). We can now find C's efficiency by subtracting A and B's combined efficiency from the combined efficiency of A, B, and C:
\[ \text{Efficiency of C} = \text{Combined Efficiency of A, B, and C} - \text{Combined Efficiency of A and B} \]
\[ \text{Efficiency of C} = \frac{1}{5} - \frac{4}{45} \]
- To perform this subtraction, we need a common denominator, which is 45:
\[ \frac{1}{5} = \frac{9}{45} \]
- Now we can subtract:
\[ \text{Efficiency of C} = \frac{9}{45} - \frac{4}{45} = \frac{5}{45} = \frac{1}{9} \]

### Step 5: Calculate the time taken by C to complete the job alone
- Since C's efficiency is \(\frac{1}{9}\) (houses per day), the time taken by C to complete the job alone is the reciprocal of C's efficiency:
\[ \text{Time taken by C} = \frac{1}{\text{Efficiency of C}} = \frac{1}{\frac{1}{9}} = 9 \text{ days} \]

### Final Answer
C alone can complete the job in **9 days**.

---