Indian Railways employed $200$ workers to complete a certain work in $150$ days. If only one-fourth of the work had been done in $50$ days, then to complete the whole work in time, what is the number of additional workers to be employed?

Let the total work be $1$ unit.
Initial condition:

* Initial number of workers ($M_1$) = $200$
* Days worked ($D_1$) = $50\text{ days}$
* Work completed ($W_1$) = $\frac{1}{4}$

Remaining condition:

* Total targeted days = $150\text{ days}$
* Remaining days ($D_2$) = $150-50=100\text{ days}$
* Remaining work ($W_2$) = $1-\frac{1}{4}=\frac{3}{4}$
* Total workers required for the remaining work = $M_2$

Using the Chain Rule formula:

$$\frac{M_1\times D_1}{W_1}=\frac{M_2\times D_2}{W_2}$$

Substitute the values:

$$\frac{200\times 50}{\frac{1}{4}}=\frac{M_2\times 100}{\frac{3}{4}}$$

Since the denominators are equal, we can eliminate the $\frac{1}{4}$ factor from both sides:

$$200\times 50=\frac{M_2\times 100}{3}$$

$$10000=\frac{M_2\times 100}{3}$$

$$M_2=\frac{10000\times 3}{100}=300\text{ workers}$$

* Total workers needed for the remaining period = $300$
* Additional workers to be employed = $M_2-M_1=300-200=100$

Therefore, $100$ additional workers need to be employed.