'A' is 3 times as good a workman as 'B' and therefore is able to complete a job in 36 days less than B'. In how many days will they finish it working together?

Options

A. `12 1/2`
B. `15 1/2`
C. `14 1/2`
D. `13 1/2`

Answer

To solve the problem step by step, let's define the efficiencies and the time taken by A and B to complete the job.

### Step 1: Define the efficiencies
Let the efficiency of B be \( x \). Since A is 3 times as good a workman as B, the efficiency of A will be \( 3x \).

### Step 2: Define the time taken to complete the job
Let the time taken by B to complete the job be \( T_B \) days. Therefore, the time taken by A to complete the job will be \( T_A = T_B - 36 \) days (since A completes the job 36 days less than B).

### Step 3: Relate efficiency and time
The work done can be expressed as:
- Work done by A in \( T_A \) days: \( 3x \times T_A \)
- Work done by B in \( T_B \) days: \( x \times T_B \)

Since both A and B complete the same job, we can set the work done equal to each other:
\[ 3x \times T_A = x \times T_B \]

### Step 4: Substitute \( T_A \)
Substituting \( T_A = T_B - 36 \) into the equation:
\[ 3x \times (T_B - 36) = x \times T_B \]

### Step 5: Simplify the equation
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 3(T_B - 36) = T_B \]
Expanding this gives:
\[ 3T_B - 108 = T_B \]

### Step 6: Solve for \( T_B \)
Rearranging the equation:
\[ 3T_B - T_B = 108 \]
\[ 2T_B = 108 \]
\[ T_B = 54 \text{ days} \]

### Step 7: Find \( T_A \)
Now, substituting \( T_B \) back to find \( T_A \):
\[ T_A = T_B - 36 = 54 - 36 = 18 \text{ days} \]

### Step 8: Calculate total work
The total work can be calculated using either A's or B's efficiency:
Using B's efficiency:
\[ \text{Total Work} = x \times T_B = x \times 54 \]
Using A's efficiency:
\[ \text{Total Work} = 3x \times T_A = 3x \times 18 = 54x \]
Both expressions for total work are equal, confirming our calculations.

### Step 9: Calculate the combined work rate
When A and B work together, their combined efficiency is:
\[ \text{Combined Efficiency} = 3x + x = 4x \]

### Step 10: Calculate the time taken when working together
The time taken to complete the work together is:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} = \frac{54x}{4x} = \frac{54}{4} = 13.5 \text{ days} \]

### Final Answer
Thus, A and B will finish the job working together in **13.5 days**.