To find the speed of the train, let's set up the problem using the given information. We will use the formula:

\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)

where the time taken for traveling both by train and taxi sums up to the total journey time.

Step 1: Set up equations for the two scenarios

Let's denote the speed of the train as \(x\) km/h, and the speed of the taxi as \(y\) km/h.

From the first journey: 300 km by train, 200 km by taxi, total time = 5 hours and 30 minutes = 5.5 hours.

Equation 1: \(\frac{300}{x} + \frac{200}{y} = 5.5\)

From the second journey: 260 km by train, 240 km by taxi, total time = 5 hours and 36 minutes = 5.6 hours.

Equation 2: \(\frac{260}{x} + \frac{240}{y} = 5.6\)

Step 2: Solve the equations

To simplify the problem, we eliminate one variable by solving one equation for it. Let's eliminate \(y\).

From Equation 1:

\(\frac{200}{y} = 5.5 - \frac{300}{x}\)

Therefore, \(y = \frac{200x}{5.5x - 300}\).

Substitute the expression for \(y\) into Equation 2:

\(\frac{260}{x} + \frac{240(5.5x - 300)}{200x} = 5.6\)

Simplify and solve for \(x\):

\(\frac{260}{x} + \frac{1320x - 72000}{200x} = 5.6\)

Multiply through by \(200x\) to clear the denominators:

\(52000 + 1320x - 72000 = 1120x\)

Simplify to:

x = 100 km/h.

Conclusion

The speed of the train is 100 km/hour.

This matches the correct answer given in the options.