Ahluwalia and Bimal together take 6 days to finish the work.'Bimal and Jalan together take 10. days to finish the Work What is the difference between number of day taken by Ahluwalia and Jalan when :they worked, alone to complete the whole work?

Options

A. 12 days
B. 16 days
C. 15 days
D. can’t be determined

Answer

To solve the problem, we need to determine the individual work rates of Ahluwalia, Bimal, and Jalan. Let's denote:

- A = number of days taken by Ahluwalia to complete the work alone.
- B = number of days taken by Bimal to complete the work alone.
- J = number of days taken by Jalan to complete the work alone.

### Step 1: Establish the equations based on the given information.

1. **Ahluwalia and Bimal together take 6 days to finish the work.**
This means their combined work rate is:
\[ \frac{1}{A} + \frac{1}{B} = \frac{1}{6} \]

2. **Bimal and Jalan together take 10 days to finish the work.**
This means their combined work rate is:
\[ \frac{1}{B} + \frac{1}{J} = \frac{1}{10} \]

### Step 2: Rearranging the equations.

From the first equation, we can express it as:
\[ \frac{1}{A} = \frac{1}{6} - \frac{1}{B} \quad \text{(1)} \]

From the second equation, we can express it as:
\[ \frac{1}{J} = \frac{1}{10} - \frac{1}{B} \quad \text{(2)} \]

### Step 3: Solve for B in terms of A and J.

From equation (1):
\[ \frac{1}{B} = \frac{1}{6} - \frac{1}{A} \]

From equation (2):
\[ \frac{1}{B} = \frac{1}{10} - \frac{1}{J} \]

### Step 4: Set the two expressions for \(\frac{1}{B}\) equal to each other.

\[ \frac{1}{6} - \frac{1}{A} = \frac{1}{10} - \frac{1}{J} \]

### Step 5: Cross-multiply to eliminate the fractions.

Cross-multiplying gives us:
\[ \left(\frac{1}{6} - \frac{1}{10}\right) = \left(\frac{1}{A} - \frac{1}{J}\right) \]

Calculating \(\frac{1}{6} - \frac{1}{10}\):
\[ \frac{5 - 3}{30} = \frac{2}{30} = \frac{1}{15} \]

So we have:
\[ \frac{1}{A} - \frac{1}{J} = \frac{1}{15} \]

### Step 6: Rearranging gives us the difference in days.

This can be expressed as:
\[ \frac{1}{A} = \frac{1}{J} + \frac{1}{15} \]

Taking the reciprocal gives:
\[ A = \frac{15J}{J + 15} \]

### Step 7: Find the difference between A and J.

To find the difference \( |A - J| \):
\[ |A - J| = \left| \frac{15J}{J + 15} - J \right| \]
\[ = \left| \frac{15J - J(J + 15)}{J + 15} \right| \]
\[ = \left| \frac{15J - J^2 - 15J}{J + 15} \right| \]
\[ = \left| \frac{-J^2}{J + 15} \right| \]

### Conclusion

The difference between the number of days taken by Ahluwalia and Jalan to complete the work alone cannot be determined without additional information about their individual efficiencies.