Understanding Boat and Stream Problems

Boat and stream problems involve calculating speeds relative to the water flow. When a boat travels downstream, the speed of the stream adds to the boat's speed in still water. When the boat travels upstream, the speed of the stream subtracts from the boat's speed in still water.

• Downstream Speed: Speed of boat in still water + Speed of stream
• Upstream Speed: Speed of boat in still water - Speed of stream

The basic formula relating speed, distance, and time is:

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

Calculating Downstream and Upstream Speeds

We are given the following information:

• Distance = 35 km
• Time taken downstream = 2 hours
• Time taken upstream = 7 hours

Let \(B\) be the speed of the boat in still water (in km/h) and \(S\) be the speed of the stream (in km/h).

Calculating Downstream Speed

The downstream speed is the distance covered downstream divided by the time taken downstream.

\[ \text{Downstream Speed} = \frac{35 \text{ km}}{2 \text{ h}} = 17.5 \text{ km/h} \]

So, we have our first equation:

\[ B + S = 17.5 \quad \text{(Equation 1)} \]

Calculating Upstream Speed

The upstream speed is the distance covered upstream divided by the time taken upstream.

\[ \text{Upstream Speed} = \frac{35 \text{ km}}{7 \text{ h}} = 5 \text{ km/h} \]

So, we have our second equation:

\[ B - S = 5 \quad \text{(Equation 2)} \]

Solving for the Speed of the Boat in Still Water

We now have a system of two linear equations with two variables, \(B\) and \(S\):

\[ B + S = 17.5 \quad \text{(1)} \]

\[ B - S = 5 \quad \text{(2)} \]

To find the speed of the boat in still water (\(B\)), we can add Equation 1 and Equation 2. This will eliminate the variable \(S\) (speed of the stream).

Adding (1) and (2):

\[ (B + S) + (B - S) = 17.5 + 5 \]

\[ B + S + B - S = 22.5 \]

\[ 2B = 22.5 \]

Now, divide by 2 to find the value of \(B\):

\[ B = \frac{22.5}{2} \]

\[ B = 11.25 \]

The speed of the boat in still water is 11.25 km/h.

Summary of Calculation

By setting up and solving the equations based on downstream and upstream speeds, we found the boat's speed in still water.

The calculated speed of the boat in still water is 11.25 km/h.

Revision Table: Boat and Stream Formulas

Additional Information on Boat and Stream Problems

Boat and stream problems are a common type of question in time, speed, and distance topics. Understanding the effect of the stream's speed is crucial.

• If the boat travels in the same direction as the stream, the effective speed increases (downstream).
• If the boat travels against the direction of the stream, the effective speed decreases (upstream).
• The speed of the boat in still water is its inherent speed without any influence from water currents.
• The speed of the stream is the speed of the water flow itself.

Once you have the downstream speed and the upstream speed, you can always find both the speed of the boat in still water and the speed of the stream using the formulas listed in the revision table. For example, the speed of the stream in this problem would be \( (17.5 - 5) / 2 = 12.5 / 2 = 6.25 \) km/h.