A and B together can do a work in 12 days. B and C together do it in 15 days. If A's efficiency is twice that of C, then the days required for B alone to finish the work is

Options

A. 60
B. 30
C. 20
D. 15

Answer

To solve the problem step by step, let's define the efficiencies of A, B, and C and use the information given in the question.

### Step 1: Define the efficiencies
Let the efficiency of C be \( c \). Since A's efficiency is twice that of C, we have:
- Efficiency of A = \( 2c \)
- Efficiency of B = \( b \)

### Step 2: Set up the equations based on the work done
From the problem, we know:
1. A and B together can complete the work in 12 days.
2. B and C together can complete the work in 15 days.

The work done in one day by A and B together is:
\[ \frac{1}{12} \text{ (work/day)} \]
So, we can write:
\[ 2c + b = \frac{1}{12} \quad \text{(Equation 1)} \]

The work done in one day by B and C together is:
\[ \frac{1}{15} \text{ (work/day)} \]
So, we can write:
\[ b + c = \frac{1}{15} \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( 2c + b = \frac{1}{12} \)
2. \( b + c = \frac{1}{15} \)

From Equation 2, we can express \( b \) in terms of \( c \):
\[ b = \frac{1}{15} - c \quad \text{(Equation 3)} \]

Substituting Equation 3 into Equation 1:
\[ 2c + \left(\frac{1}{15} - c\right) = \frac{1}{12} \]
This simplifies to:
\[ 2c - c + \frac{1}{15} = \frac{1}{12} \]
\[ c + \frac{1}{15} = \frac{1}{12} \]

### Step 4: Isolate \( c \)
Now, we need to isolate \( c \):
\[ c = \frac{1}{12} - \frac{1}{15} \]

Finding a common denominator (which is 60):
\[ c = \frac{5}{60} - \frac{4}{60} = \frac{1}{60} \]

### Step 5: Find \( b \)
Now that we have \( c \), we can find \( b \) using Equation 3:
\[ b = \frac{1}{15} - c = \frac{1}{15} - \frac{1}{60} \]
Finding a common denominator (which is 60):
\[ b = \frac{4}{60} - \frac{1}{60} = \frac{3}{60} = \frac{1}{20} \]

### Step 6: Calculate the days required for B alone to finish the work
Since \( b \) represents the work done by B in one day, we can find the time taken by B to complete the entire work:
\[ \text{Days required by B} = \frac{1}{b} = \frac{1}{\frac{1}{20}} = 20 \text{ days} \]

### Final Answer
Thus, the days required for B alone to finish the work is **20 days**.

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