A is thrice as good a workman B and therefore is able to finish a job in 40 days less titan B. Working together, they can do it in

Options

A. 14 days
B. 13 days
C. 20 days
D. 15 days

Answer

To solve the problem step by step, we need to find out how long it takes for A and B to complete the work together.

### Step 1: Define the work rates of A and B
Let the time taken by A to complete the work be \( x \) days. Since A is thrice as efficient as B, the time taken by B to complete the work will be \( x + 40 \) days.

### Step 2: Set up the equation based on work done
The amount of work done can be expressed in terms of their efficiencies. The work done by A in \( x \) days is equal to the work done by B in \( x + 40 \) days. We can express this as:
\[ \text{Work done by A} = \text{Work done by B} \]
This can be formulated as:
\[ \frac{1}{x} = \frac{1}{x + 40} \]
Since A is thrice as efficient as B, we can express this as:
\[ 3 \cdot x = (x + 40) \]

### Step 3: Solve for \( x \)
Rearranging the equation gives:
\[ 3x = x + 40 \]
Subtract \( x \) from both sides:
\[ 3x - x = 40 \]
\[ 2x = 40 \]
Now, divide both sides by 2:
\[ x = 20 \]

### Step 4: Find the time taken by B
Now that we have \( x \), we can find the time taken by B:
\[ \text{Time taken by B} = x + 40 = 20 + 40 = 60 \text{ days} \]

### Step 5: Calculate the total work
The total work can be calculated using A's efficiency:
\[ \text{Total work} = \text{Efficiency of A} \times \text{Time taken by A} = 3 \times 20 = 60 \text{ units of work} \]

### Step 6: Calculate the combined work rate of A and B
Now, we need to find out how long it will take for A and B to complete the work together. The combined work rate of A and B is:
\[ \text{Efficiency of A} + \text{Efficiency of B} = \frac{1}{20} + \frac{1}{60} \]
To add these fractions, we need a common denominator, which is 60:
\[ \frac{3}{60} + \frac{1}{60} = \frac{4}{60} = \frac{1}{15} \]

### Step 7: Calculate the time taken when working together
If their combined work rate is \( \frac{1}{15} \), then the time taken to complete the work together is:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Work Rate}} = \frac{60}{\frac{1}{15}} = 60 \times 15 = 900 \text{ days} \]

### Final Answer
Thus, working together, A and B can complete the job in **15 days**.

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