Given below are two statements:

Statement I: A 60 m long train passes a 90 m platform in 10 seconds. The speed of the train is 54 km/hour.
Statement II: If the ratio of the speeds of A and B is a : b, then the ratio of the time taken by them to cover the same distance is a : b.
In light of the above statements, choose the correct answer from the options given below:

The correct option is: Statement I is correct but statement II is incorrect.

Let us analyze each statement step-by-step to understand why this option is correct.

Analysis of Statement I:

Statement I states: "A 60 m long train passes a 90 m platform in 10 seconds. The speed of the train is 54 km/hour."

When a train passes a platform completely, the total distance covered by the train is the sum of the length of the train and the length of the platform.

Total Distance ($d$) = Length of train + Length of platform
$d=60\text{m}+90\text{m}=150\text{m}$

The time taken ($t$) to cross this distance is given as 10 seconds.

Now, let us calculate the speed ($s$) of the train in meters per second (m/s) using the formula:
$s=\frac{d}{t}$

$s=\frac{150\text{m}}{10\text{s}}=15\text{m/s}$

To convert the speed from meters per second (m/s) to kilometers per hour (km/h), we multiply by $\frac{18}{5}$:
$\text{Speed in km/h}=15\times \frac{18}{5}$

$\text{Speed in km/h}=3\times 18=54\text{km/h}$

Since the calculated speed matches the speed given in the statement, Statement I is correct.

Analysis of Statement II:

Statement II states: "If the ratio of the speeds of A and B is a : b, then the ratio of the time taken by them to cover the same distance is a : b."

We know that speed is inversely proportional to time when the distance covered is constant:
$\mathrm{Speed}\propto \frac{1}{\mathrm{Time}}$

If the ratio of the speeds of A and B is $a:b$, then the ratio of their times taken to cover the same distance is the reciprocal of the speed ratio:
$\text{Ratio of time taken}=\frac{1}{a}:\frac{1}{b}=b:a$

Since the ratio of the times taken is $b:a$ (not $a:b$), Statement II is incorrect.

Therefore, Statement I is correct but Statement II is incorrect.