Two trains travel in the same direction at $59\text{km/hr}$ and $32\text{km/hr}$ and the faster train passes a man in the slower train in $16$ seconds. Find the length of the faster train.

To find the length of the faster train, we need to determine the distance covered by the faster train relative to the man in the slower train during the crossing time.
Step 1: Calculate the relative speed
Since both trains are moving in the same direction, the relative speed is the difference between their speeds:

$$\text{Relative Speed}=59\text{km/hr}-32\text{km/hr}=27\text{km/hr}$$

**Step 2: Convert relative speed into metres per second ($\text{m/s}$)**

$$\text{Relative Speed}=27\times \frac{5}{18}\text{m/s}=\frac{3\times 5}{2}\text{m/s}=7.5\text{m/s}$$

Step 3: Calculate the length of the faster train
The distance covered in passing the man is equal to the length of the faster train:

$$\text{Length of the faster train}=\text{Relative Speed}\times \text{Time}$$

$$\text{Length of the faster train}=7.5\text{m/s}\times 16\text{seconds}=120\text{m}$$