Solving Boat and Stream Distance Problems

This problem involves the concepts of boat and stream, specifically calculating the distance traveled based on the speed of the boat in still water, the speed of the river, and the total time taken for a round trip.

Understanding Boat and Stream Concepts

When a boat moves in a river, its speed relative to the ground changes depending on whether it is moving with the current (downstream) or against the current (upstream).

• Speed Downstream: The speed of the boat in still water plus the speed of the stream. The stream helps the boat move faster.
• Speed Upstream: The speed of the boat in still water minus the speed of the stream. The stream opposes the boat's movement, slowing it down.

Given Information

• Speed of the man in still water: 10 km/h
• Speed of the river (stream): 4.5 km/h
• Total time for the round trip (to the place and back): 2 hours

Calculating Speeds Downstream and Upstream

Using the given information, we can calculate the speed of the man when rowing downstream and upstream:

• Speed Downstream = Speed in still water + Speed of stream
• Speed Downstream = $10 \text{ km/h} + 4.5 \text{ km/h} = 14.5 \text{ km/h}$

• Speed Upstream = Speed in still water - Speed of stream
• Speed Upstream = $10 \text{ km/h} - 4.5 \text{ km/h} = 5.5 \text{ km/h}$

Setting up the Time Equation

Let the distance from the starting point to the place be $d$ km. The time taken to travel a certain distance is given by the formula:

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

The total time for the round trip is the sum of the time taken to row downstream (to the place) and the time taken to row upstream (back to the initial point).

Total Time = Time Downstream + Time Upstream

$2 \text{ hours} = \frac{d}{\text{Speed Downstream}} + \frac{d}{\text{Speed Upstream}}$

Substituting the calculated speeds:

$2 = \frac{d}{14.5} + \frac{d}{5.5}$

Solving for the Distance (d)

Now we solve the equation for $d$:

$2 = d \left( \frac{1}{14.5} + \frac{1}{5.5} \right)$

To add the fractions inside the parenthesis, find a common denominator or use the formula $\frac{1}{a} + \frac{1}{b} = \frac{b+a}{ab}$:

$2 = d \left( \frac{5.5 + 14.5}{14.5 \times 5.5} \right)$

$2 = d \left( \frac{20}{79.75} \right)$

Now, isolate $d$:

$d = 2 \times \frac{79.75}{20}$

$d = \frac{159.5}{20}$

$d = 7.975 \text{ km}$

Rounding the Distance

The question asks to round the distance off to two decimal places. The calculated distance is 7.975 km.

Rounding 7.975 to two decimal places, we look at the third decimal place, which is 5. Since it is 5 or greater, we round up the second decimal place.

$7.975 \approx 7.98 \text{ km}$

Conclusion

The distance to the place is approximately 7.98 km.

Revision Table: Boat and Stream Formulas

Additional Information: Solving Time and Distance Problems

Time, speed, and distance problems are common in quantitative aptitude. The key is to correctly identify the speeds in different conditions (like still water, downstream, upstream) and use the relationship between time, distance, and speed.

• For a fixed distance, time is inversely proportional to speed. If speed increases, time decreases.
• For a fixed time, distance is directly proportional to speed. If speed increases, distance increases.
• In round trip problems involving streams, the distance traveled upstream is equal to the distance traveled downstream.
• Always ensure that the units of speed, time, and distance are consistent before performing calculations. For example, if speed is in km/h and time is in hours, the distance will be in km.