Abhishek can do a work in x hours. Bhavya who is 60% more efficient than Abhishek can complete a work 5 hours earlier than Abhimanyu. Abhimanyu worked for 5 hour and finds out that only 25% of work is completed and Remaining work is completed by Abhishek and Bhavya together. Find the total time in which work is completed. (in hours)

Options

A. `13(12)/(13)`
B. `15(12)/(13)`
C. `11(12)/(13)`
D. `17(12)/(13)`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define the efficiencies of Abhishek and Bhavya
Let the efficiency of Abhishek be \( E_A = 100 \) units of work per hour. Since Bhavya is 60% more efficient than Abhishek, her efficiency will be:
\[ E_B = E_A + 0.6 \times E_A = 100 + 60 = 160 \text{ units of work per hour} \]
Thus, the ratio of their efficiencies is:
\[ \text{Efficiency Ratio} = E_A : E_B = 100 : 160 = 5 : 8 \]

### Step 2: Define the time taken by each person
Let the time taken by Abhishek to complete the work be \( T_A = 8y \) hours and the time taken by Bhavya be \( T_B = 5y \) hours. Since Bhavya completes the work 5 hours earlier than Abhimanyu, we denote Abhimanyu's time as:
\[ T_C = 5y + 5 \text{ hours} \]

### Step 3: Work done by Abhimanyu
Abhimanyu worked for 5 hours and completed 25% of the work. Therefore, if the total work is \( W \), then:
\[ \text{Work done by Abhimanyu in 5 hours} = 0.25W \]
This implies that the total work \( W \) can be calculated as:
\[ W = 4 \times \text{Work done in 5 hours} = 4 \times 0.25W = 1W \]

### Step 4: Calculate total work based on time
From the time taken by Abhimanyu, we can express the total work in terms of his efficiency:
\[ \text{Total Work} = \text{Efficiency of Abhimanyu} \times \text{Time taken by Abhimanyu} \]
Let’s denote Abhimanyu's efficiency as \( E_C \). Therefore:
\[ W = E_C \times (5y + 5) \]

### Step 5: Equate the two expressions for total work
Since both expressions represent the same total work \( W \):
\[ E_A \times T_A = E_C \times (5y + 5) \]
Substituting the values:
\[ 100 \times 8y = E_C \times (5y + 5) \]

### Step 6: Find the value of \( y \)
From the work done by Abhimanyu, we know that:
\[ \text{Total Work} = 120 \text{ units} \quad \text{(from previous calculations)} \]
Thus, we can set up the equation:
\[ 120 = 100 \times 8y \]
This gives:
\[ y = \frac{120}{800} = \frac{3}{20} \text{ hours} \]

### Step 7: Calculate the time taken by each person
Now substituting \( y \) back into the time equations:
- Abhishek's time:
\[ T_A = 8y = 8 \times \frac{3}{20} = \frac{24}{20} = 1.2 \text{ hours} \]
- Bhavya's time:
\[ T_B = 5y = 5 \times \frac{3}{20} = \frac{15}{20} = 0.75 \text{ hours} \]
- Abhimanyu's time:
\[ T_C = 5y + 5 = 5 \times \frac{3}{20} + 5 = 0.75 + 5 = 5.75 \text{ hours} \]

### Step 8: Calculate the remaining work
After Abhimanyu worked for 5 hours and completed 25% of the work:
\[ \text{Remaining Work} = W - 0.25W = 0.75W = 90 \text{ units} \]

### Step 9: Calculate the time taken by Abhishek and Bhavya to complete the remaining work
The combined efficiency of Abhishek and Bhavya is:
\[ E_A + E_B = 100 + 160 = 260 \text{ units} \]
The time taken to complete the remaining work is:
\[ \text{Time} = \frac{\text{Remaining Work}}{\text{Combined Efficiency}} = \frac{90}{260} = \frac{9}{26} \text{ hours} \]

### Step 10: Calculate the total time taken
Total time taken to complete the work is:
\[ \text{Total Time} = 5 + \frac{9}{26} = \frac{130}{26} + \frac{9}{26} = \frac{139}{26} \text{ hours} \]

### Final Answer
The total time in which the work is completed is:
\[ \text{Total Time} = \frac{139}{26} \text{ hours} \]