A is thrice as productive as C. Together they can complete a job in 22.5 days. If B joins them after they have worked for 15 days then in how many days can they finish the rest of the job if B alone can do the job in 15 days?

Options

A. 6
B. 3
C. 9
D. 2

Answer

To solve the problem step by step, we will first determine the efficiencies of A, B, and C, and then calculate how much work they can complete together.

### Step 1: Determine the efficiencies of A, B, and C
Given that A is thrice as productive as C, we can denote the efficiency of C as \( x \). Therefore, the efficiency of A will be \( 3x \).

Now, the combined efficiency of A and C is:
\[ \text{Efficiency of A and C} = A + C = 3x + x = 4x \]

We know that A and C can complete the job in 22.5 days. Thus, their combined work rate (efficiency) can be calculated as:
\[ \text{Total work} = \text{Efficiency} \times \text{Time} \]
Let’s denote the total work as \( W \). Therefore:
\[ W = 4x \times 22.5 \]

### Step 2: Calculate the total work (W)
To find \( W \), we can express it in terms of \( x \):
\[ W = 4x \times 22.5 = 90x \]

### Step 3: Determine the efficiency of B
B can complete the job alone in 15 days. Thus, B's efficiency is:
\[ \text{Efficiency of B} = \frac{W}{15} = \frac{90x}{15} = 6x \]

### Step 4: Calculate the work done by A and C in 15 days
Now, we need to find out how much work A and C can complete in 15 days:
\[ \text{Work done by A and C in 15 days} = \text{Efficiency of A and C} \times \text{Time} = 4x \times 15 = 60x \]

### Step 5: Calculate the remaining work
Now, we can find the remaining work after A and C have worked for 15 days:
\[ \text{Remaining work} = W - \text{Work done by A and C} = 90x - 60x = 30x \]

### Step 6: Calculate the combined efficiency of A, B, and C
When B joins A and C, their combined efficiency becomes:
\[ \text{Combined efficiency of A, B, and C} = 4x + 6x = 10x \]

### Step 7: Calculate the time taken to finish the remaining work
Now, we need to find out how many days it will take for A, B, and C to finish the remaining work of \( 30x \):
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Combined efficiency}} = \frac{30x}{10x} = 3 \text{ days} \]

### Conclusion
Thus, A, B, and C can finish the remaining work in **3 days**.

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