Understanding boat and stream problems involves knowing how the speed of the boat is affected by the current of the stream. When a boat travels with the stream, its speed increases (downstream), and when it travels against the stream, its speed decreases (upstream).

Boat and Stream Fundamentals

Let's define the key terms used in these types of problems:

• Speed of the boat in still water (\(v_b\)): This is the speed of the boat when there is no current, meaning the water is not moving.
• Speed of the stream or current (\(v_s\)): This is the speed at which the water is flowing.
• Downstream speed: When the boat travels in the same direction as the stream, its speed is the sum of its speed in still water and the speed of the stream.
  Downstream Speed \( = v_b + v_s\)
• Upstream speed: When the boat travels against the direction of the stream, its speed is the difference between its speed in still water and the speed of the stream.
  Upstream Speed \( = v_b - v_s\)

Analyzing the Given Information

We are provided with the following information about Sudha's boat journey:

• Time taken to travel a certain distance downstream (\(t_d\)) = 6 hours
• Time taken to return to the starting point (upstream, \(t_u\)) = 9 hours
• Speed of the stream (\(v_s\)) = 3 km/h

The distance traveled downstream is the same as the distance traveled upstream (since she returns to the starting point).

Calculating Boat Speed in Still Water

Let \(D\) be the distance Sudha traveled one way. We can express the distance using the speed and time for both downstream and upstream journeys.

For the downstream journey:

Distance \(D = \text{Downstream Speed} \times \text{Downstream Time}\)

$D = (v_b + v_s) \times t_d$

$D = (v_b + 3) \times 6$ --- (Equation 1)

For the upstream journey:

Distance \(D = \text{Upstream Speed} \times \text{Upstream Time}\)

$D = (v_b - v_s) \times t_u$

$D = (v_b - 3) \times 9$ --- (Equation 2)

Since the distance \(D\) is the same in both cases, we can equate Equation 1 and Equation 2 to find the speed of the boat in still water (\(v_b\)).

$(v_b + 3) \times 6 = (v_b - 3) \times 9$

Now, let's solve this equation step-by-step:

$6v_b + 18 = 9v_b - 27$

To isolate \(v_b\), we will move all terms involving \(v_b\) to one side and constants to the other side:

$18 + 27 = 9v_b - 6v_b$

$45 = 3v_b$

Now, divide both sides by 3 to find \(v_b\):

$v_b = \frac{45}{3}$

$v_b = 15 \text{ km/h}$

So, the speed of the boat in still water is 15 km/h.

Time to Cover Distance in Still Water

The question asks for the time it will take to cover a distance of 67.5 km in still water. We have just calculated the boat's speed in still water.

Distance to cover = 67.5 km

Speed in still water (\(v_b\)) = 15 km/h

The formula for time is:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Time \( = \frac{67.5}{15}\) hours

Let's perform the division:

$\frac{67.5}{15} = \frac{675}{150}$

We can simplify this fraction. Both numbers are divisible by 5:

$\frac{675 \div 5}{150 \div 5} = \frac{135}{30}$

Again, both are divisible by 5:

$\frac{135 \div 5}{30 \div 5} = \frac{27}{6}$

Now, both are divisible by 3:

$\frac{27 \div 3}{6 \div 3} = \frac{9}{2}$

Converting the fraction to a decimal:

$\frac{9}{2} = 4.5$ hours

Therefore, it will take 4.5 hours to cover a distance of 67.5 km in still water.

Summary of Steps

Here’s a quick recap of the process: