A and B together can complete a piece complete a piece of work in 12 days , B and C together can complete it in 15 days. If A is twice as good a workman as C, in many days will A alone complete the same work ?

Options

A. (A)45
B. (B)60
C. (C)75
D. (D)30

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript and break it down into clear steps:

### Step 1: Determine the Work Done by A and B Together
A and B together can complete the work in 12 days. Therefore, their combined work rate (efficiency) can be calculated as:
\[ \text{Efficiency of A + B} = \frac{1 \text{ work}}{12 \text{ days}} = \frac{1}{12} \]
This means A and B together can complete \(\frac{1}{12}\) of the work in one day.

### Step 2: Determine the Work Done by B and C Together
B and C together can complete the work in 15 days. Therefore, their combined work rate can be calculated as:
\[ \text{Efficiency of B + C} = \frac{1 \text{ work}}{15 \text{ days}} = \frac{1}{15} \]
This means B and C together can complete \(\frac{1}{15}\) of the work in one day.

### Step 3: Set Up the Equations for Efficiencies
Let the efficiency of A, B, and C be represented as \(a\), \(b\), and \(c\) respectively. From the information we have:
1. \(a + b = \frac{1}{12}\) (Equation 1)
2. \(b + c = \frac{1}{15}\) (Equation 2)

### Step 4: Use the Given Relationship Between A and C
We know from the problem that A is twice as good a workman as C, which can be expressed as:
\[ a = 2c \quad (Equation 3) \]

### Step 5: Substitute Equation 3 into Equations 1 and 2
Substituting \(a = 2c\) into Equation 1:
\[ 2c + b = \frac{1}{12} \quad (Equation 4) \]
Now, we have two equations (Equation 4 and Equation 2) with two unknowns \(b\) and \(c\).

### Step 6: Solve for \(b\) and \(c\)
From Equation 2:
\[ b + c = \frac{1}{15} \quad (Equation 2) \]
Now, we can express \(b\) from Equation 2:
\[ b = \frac{1}{15} - c \quad (Equation 5) \]
Substituting Equation 5 into Equation 4:
\[ 2c + \left(\frac{1}{15} - c\right) = \frac{1}{12} \]
This simplifies to:
\[ c + \frac{1}{15} = \frac{1}{12} \]
Now, we need to solve for \(c\):
\[ c = \frac{1}{12} - \frac{1}{15} \]
Finding a common denominator (60):
\[ c = \frac{5}{60} - \frac{4}{60} = \frac{1}{60} \]

### Step 7: Find \(b\) and \(a\)
Now that we have \(c\), we can find \(b\) using Equation 5:
\[ b = \frac{1}{15} - \frac{1}{60} \]
Finding a common denominator (60):
\[ b = \frac{4}{60} - \frac{1}{60} = \frac{3}{60} = \frac{1}{20} \]
Now, substitute \(c\) back into Equation 3 to find \(a\):
\[ a = 2c = 2 \times \frac{1}{60} = \frac{2}{60} = \frac{1}{30} \]

### Step 8: Calculate the Time Taken by A Alone
Now we know the efficiency of A:
\[ a = \frac{1}{30} \]
To find the time taken by A to complete the work alone, we take the reciprocal of A's efficiency:
\[ \text{Time taken by A} = \frac{1 \text{ work}}{a} = \frac{1}{\frac{1}{30}} = 30 \text{ days} \]

### Final Answer
A alone will complete the work in **30 days**.