Tanush's firm can lay railway track between two stations in 16 days. Navansh's firm can do the same job in 12 days. With the help of Vridhi's firm, all can lay the track in 4 days. How many days will it take for Vridhi's firm alone to complete the work?

Options

A. `9(3//5)` days
B. `9(1//5)` days
C. `9(2//5)` days
D. 10 days

Answer

To solve the problem, we need to find out how many days it will take for Vridhi's firm alone to complete the work of laying the railway track. We will do this step by step.

### Step 1: Determine the work done by each firm in one day.
- Tanush's firm can complete the work in 16 days. Therefore, the work done by Tanush's firm in one day is:
\[ \text{Work by Tanush's firm in one day} = \frac{1}{16} \]

- Navansh's firm can complete the work in 12 days. Therefore, the work done by Navansh's firm in one day is:
\[ \text{Work by Navansh's firm in one day} = \frac{1}{12} \]

### Step 2: Determine the combined work done by all three firms in one day.
- Together, all three firms can complete the work in 4 days. Therefore, the work done by all three firms in one day is:
\[ \text{Combined work in one day} = \frac{1}{4} \]

### Step 3: Set up the equation for the combined work.
Let the work done by Vridhi's firm in one day be \( \frac{1}{x} \). Then, the equation for the combined work done by all three firms in one day is:
\[ \frac{1}{16} + \frac{1}{12} + \frac{1}{x} = \frac{1}{4} \]

### Step 4: Find a common denominator and solve for \( x \).
The least common multiple (LCM) of 16, 12, and 4 is 48. We can rewrite the equation with a common denominator:
\[ \frac{3}{48} + \frac{4}{48} + \frac{1}{x} = \frac{12}{48} \]

Combine the fractions:
\[ \frac{3 + 4}{48} + \frac{1}{x} = \frac{12}{48} \]
\[ \frac{7}{48} + \frac{1}{x} = \frac{12}{48} \]

### Step 5: Isolate \( \frac{1}{x} \).
Subtract \( \frac{7}{48} \) from both sides:
\[ \frac{1}{x} = \frac{12}{48} - \frac{7}{48} \]
\[ \frac{1}{x} = \frac{5}{48} \]

### Step 6: Solve for \( x \).
Taking the reciprocal gives:
\[ x = \frac{48}{5} = 9.6 \]

Thus, it will take Vridhi's firm alone **9.6 days** to complete the work.

### Final Answer:
Vridhi's firm alone will take **9.6 days** to complete the work.

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