Calculating Combined Work: Sunil and Vijay

This problem involves calculating the time it takes for two individuals, Sunil and Vijay, to complete a work project together, given their relative work efficiencies and the difference in time they take individually.

Understanding Work Efficiency and Time

Work efficiency is inversely proportional to the time taken to complete a task. If one person is 'x' times as efficient as another, they will take $1/x$ times the duration to complete the same work.

• Sunil is $3\frac{1}{2}$ times as good a workman as Vijay.
• $3\frac{1}{2}$ can be written as $\frac{7}{2}$.
• Therefore, Sunil's efficiency is $\frac{7}{2}$ times Vijay's efficiency.
• This means Sunil takes $\frac{2}{7}$ times the days Vijay takes. Let $T_S$ be Sunil's days and $T_V$ be Vijay's days. So, $T_S = \frac{2}{7} T_V$.
• We are given that Sunil finishes the work in 45 days less than Vijay: $T_V - T_S = 45$ days.

Step-by-Step Calculation

1. Substitute the time relationship into the difference equation:

   Substitute $T_S = \frac{2}{7} T_V$ into $T_V - T_S = 45$:

   $T_V - \left(\frac{2}{7} T_V\right) = 45$

   $ \frac{7 T_V - 2 T_V}{7} = 45 $

   $ \frac{5 T_V}{7} = 45 $

2. Calculate Vijay's total time ($T_V$):

   $ T_V = \frac{45 \times 7}{5} $

   $ T_V = 9 \times 7 = 63 \text{ days} $

3. Calculate Sunil's total time ($T_S$):

   Using $T_V - T_S = 45$:

   $ T_S = 63 - 45 = 18 \text{ days} $

   (Alternatively, using $T_S = \frac{2}{7} T_V$):

   $ T_S = \frac{2}{7} \times 63 = 2 \times 9 = 18 \text{ days} $

4. Calculate the combined work rate:

   Sunil's rate = $\frac{1}{T_S} = \frac{1}{18}$ work per day.

   Vijay's rate = $\frac{1}{T_V} = \frac{1}{63}$ work per day.

   Combined rate = $\frac{1}{18} + \frac{1}{63}$.

5. Find the least common multiple (LCM) of 18 and 63:

   Prime factors of 18: $2 \times 3^2$.

   Prime factors of 63: $7 \times 3^2$.

   LCM$(18, 63) = 2 \times 7 \times 3^2 = 126$.

   Combined rate = $\frac{7}{126} + \frac{2}{126} = \frac{9}{126}$.

   Simplify the combined rate: $\frac{9}{126} = \frac{1}{14}$ work per day.

6. Calculate the total time taken working together:

   Time = $\frac{1}{\text{Combined rate}} = \frac{1}{1/14}$

   Time = $14$ days.

The calculation shows that Sunil and Vijay working together can finish the work in 14 days. Based on the provided options, the correct answer is indicated as Option A.