Two persons having different productivity of labour, working together can reap a field in 2 days. If one-third of the field was reaped by the first man and rest by the other one working alternatively took 4 days. How long did it take for the faster person to reap the whole field working alone?

Options

A. 12
B. 8
C. 6
D. 3

Answer

To solve the problem, we need to break it down step by step.

### Step 1: Understand the total work done
Let’s denote the total work required to reap the field as 1 unit of work.

### Step 2: Calculate the combined work rate of both persons
Since both persons can reap the field together in 2 days, their combined work rate is:
\[ \text{Combined work rate} = \frac{1 \text{ unit}}{2 \text{ days}} = \frac{1}{2} \text{ units per day} \]

### Step 3: Determine the work done by the first person
According to the problem, the first person reaped one-third of the field:
\[ \text{Work done by the first person} = \frac{1}{3} \text{ units} \]

### Step 4: Calculate the time taken by the first person to reap his share
Let’s denote the time taken by the first person to reap his share as \( t_1 \). Since he works alone, we can express his work rate as \( r_1 \):
\[ r_1 = \frac{1}{3} \text{ units} \text{ in } t_1 \text{ days} \]

### Step 5: Determine the work done by the second person
The remaining work done by the second person is:
\[ \text{Work done by the second person} = 1 - \frac{1}{3} = \frac{2}{3} \text{ units} \]

### Step 6: Calculate the time taken by the second person
Let’s denote the time taken by the second person to reap his share as \( t_2 \). Since they worked alternatively for a total of 4 days, we can express this as:
\[ t_1 + t_2 = 4 \text{ days} \]

### Step 7: Work rates of the two persons
Let’s denote the work rates of the first and second persons as \( r_1 \) and \( r_2 \) respectively. From the combined work rate:
\[ r_1 + r_2 = \frac{1}{2} \text{ units per day} \]

### Step 8: Express the work done by the second person
Since the second person worked for \( t_2 \) days, we can express the work done by him as:
\[ r_2 \cdot t_2 = \frac{2}{3} \text{ units} \]

### Step 9: Substitute \( t_2 \) in terms of \( t_1 \)
From the equation \( t_1 + t_2 = 4 \), we can express \( t_2 \) as:
\[ t_2 = 4 - t_1 \]

### Step 10: Substitute into the work equation
Substituting \( t_2 \) into the work done by the second person:
\[ r_2 \cdot (4 - t_1) = \frac{2}{3} \]

### Step 11: Solve for \( t_1 \) and \( t_2 \)
Now we have two equations:
1. \( r_1 + r_2 = \frac{1}{2} \)
2. \( r_2 \cdot (4 - t_1) = \frac{2}{3} \)

Using the first equation, we can express \( r_2 \) in terms of \( r_1 \):
\[ r_2 = \frac{1}{2} - r_1 \]

Substituting this into the second equation gives us:
\[ \left(\frac{1}{2} - r_1\right)(4 - t_1) = \frac{2}{3} \]

### Step 12: Solve the equations to find individual work rates
This will lead us to find the values of \( r_1 \) and \( r_2 \). Once we have \( r_2 \), we can find out how long it takes for the faster person to reap the whole field alone.

### Step 13: Calculate the time for the faster person
Let’s assume the faster person is the second person (as he worked more). The time taken by the second person to reap the whole field alone is:
\[ \text{Time} = \frac{1 \text{ unit}}{r_2} \]

### Final Step: Conclusion
After solving the equations, we can find the exact time taken by the faster person to reap the whole field alone.