Problem Statement:

A motor boat covers a certain distance downstream in a river in 5 hours and covers the same distance upstream in 5 and a half hours. The speed of the water is 1.5 km/h. Find the speed of the boat in still water.

Solution:

Let's assume the speed of the boat in still water is x km/h, and the speed of the current is given as 1.5 km/h.

1. Speed Downstream:
When the boat is moving downstream, it gets assistance from the current, which increases its effective speed. The effective speed can be calculated using the formula:

Effective Speed = Boat Speed + Current Speed

Therefore, the effective speed downstream is (x + 1.5) km/h.

The time taken to cover the distance downstream is given as 5 hours. We can use the formula:

Time = Distance / Speed

So, the distance downstream can be calculated as:

Distance = Time * Speed
Distance = 5 * (x + 1.5)
Distance = 5x + 7.5

2. Speed Upstream:
When the boat is moving upstream, it has to work against the current, which reduces its effective speed. The effective speed can be calculated using the formula:

Effective Speed = Boat Speed - Current Speed

Therefore, the effective speed upstream is (x - 1.5) km/h.

The time taken to cover the distance upstream is given as 5 and a half hours. We can convert this to hours by adding 0.5, which gives us 5.5 hours. Using the formula:

Time = Distance / Speed

The distance upstream can be calculated as:

Distance = Time * Speed
Distance = 5.5 * (x - 1.5)
Distance = 5.5x - 8.25

3. Equating the Distances:
Since the distance covered downstream and upstream is the same, we can equate the two distance equations:

5x + 7.5 = 5.5x - 8.25

4. Solving for x:
Let's simplify the equation:

5x - 5.5x = -8.25 - 7.5
-0.5x = -15.75
x = -15.75 / -0.5
x = 31.5

5. Conclusion:
The speed of the boat in still water is 31.5 km/h.