A train, travelling @ $67.5\text{km/hr}$ passes a pole in $9\text{seconds}$ and a platform in $29\text{seconds}$. What is the length of the platform?

First, convert the speed of the train from $\text{km/hr}$ to $\text{m/s}$ by multiplying by $\frac{5}{18}$:
$$\text{Speed}=67.5\times \frac{5}{18}=\frac{337.5}{18}=18.75\text{m/s}$$

Step 1: Find the length of the train ($L_t$).
When the train passes a pole, it covers its own length in $9\text{seconds}$:

$$L_t=\text{Speed}\times \text{Time}=18.75\text{m/s}\times 9\text{s}=168.75\text{m}$$

Step 2: Find the length of the platform ($L_p$).
When the train passes a platform, it covers a total distance equal to its own length plus the platform length ($L_t+L_p$) in $29\text{seconds}$:

$$L_t+L_p=\text{Speed}\times \text{Time}$$

$$168.75+L_p=18.75\times 29$$

$$168.75+L_p=543.75$$

$$L_p=543.75-168.75=375\text{m}$$

Alternatively, the train takes $29-9=20\text{seconds}$ to cross the platform distance alone:

$$L_p=18.75\text{m/s}\times 20\text{s}=375\text{m}$$

Therefore, the length of the platform is $375\text{m}$.