Solving Boat and Stream Problems: Calculating Time in Still Water

This problem involves the concepts of boat speed, stream speed, and their combined effects on speed when traveling downstream (with the current) and upstream (against the current). We are given information about the time taken for certain distances downstream and upstream, the speed of the stream, and we need to find the time taken to cover a specific distance in still water.

Understanding Boat and Stream Concepts

When a boat travels in water, its speed relative to the ground is affected by the speed of the water (the stream or current). Here's how the speeds are defined:

• Speed of boat in still water: This is the speed of the boat without any influence from a current. Let's denote this as \(B\).
• Speed of the stream: This is the speed of the water current. Let's denote this as \(S\).
• Speed downstream: When the boat travels with the stream, the speeds add up. Downstream Speed \(= B + S\).
• Speed upstream: When the boat travels against the stream, the stream's speed is subtracted from the boat's speed. Upstream Speed \(= B - S\).

We use the fundamental relationship between distance, speed, and time: \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\).

Setting Up the Problem

We are given:

• Distance downstream = 13 km
• Distance upstream = 7 km
• Time taken for 13 km downstream is the same as time taken for 7 km upstream.
• Speed of the stream (\(S\)) = 3 km/h.

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

Based on the given information, we can write the speeds:

• Downstream speed = \(B + S = B + 3\) km/h.
• Upstream speed = \(B - S = B - 3\) km/h.

The time taken for 13 km downstream is \(\frac{13}{B + 3}\) hours.

The time taken for 7 km upstream is \(\frac{7}{B - 3}\) hours.

Formulating and Solving the Equation

We are told that the time taken downstream is the same as the time taken upstream. So, we can set the two time expressions equal to each other:

\(\frac{13}{B + 3} = \frac{7}{B - 3}\)

Now, we solve this equation for \(B\):

1. Cross-multiply: \(13 \times (B - 3) = 7 \times (B + 3)\)
2. Distribute: \(13B - 39 = 7B + 21\)
3. Gather terms with \(B\) on one side and constants on the other: \(13B - 7B = 21 + 39\)
4. Simplify: \(6B = 60\)
5. Solve for \(B\): \(B = \frac{60}{6}\)
6. \(B = 10\) km/h

The speed of the boat in still water is 10 km/h.

Calculating Time in Still Water

We need to find the time taken to travel a distance of 44.8 km in still water. In still water, the speed of the boat is simply \(B\), which we found to be 10 km/h.

Distance = 44.8 km

Speed in still water = 10 km/h

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

Time = \(\frac{44.8}{10}\) hours

Time = 4.48 hours

Converting Decimal Time to Mixed Fraction

The answer options are given in mixed fraction form. We need to convert 4.48 hours into a mixed fraction:

4.48 hours = 4 whole hours + 0.48 hours

To convert 0.48 into a fraction, we can write it as \(\frac{48}{100}\). Now, simplify this fraction:

\(\frac{48}{100} = \frac{24}{50} = \frac{12}{25}\)

So, 0.48 hours is equal to \(\frac{12}{25}\) of an hour.

Therefore, 4.48 hours is equal to \(4 \frac{12}{25}\) hours.

The time taken to travel 44.8 km in still water is \(4 \frac{12}{25}\) hours.

Revision Table: Boat and Stream Speeds

Additional Information: Relative Speed Concept

Boat and stream problems are applications of the concept of relative speed. When objects move in the same direction, their relative speed is the difference between their speeds. When they move in opposite directions, their relative speed is the sum of their speeds.

• In the case of downstream travel, the boat and stream are moving in the "same direction" effectively adding their speeds relative to the river bed.
• In the case of upstream travel, the boat is moving against the stream, so their speeds are subtracted to find the boat's effective speed relative to the river bed.

Understanding relative speed helps in solving various motion-related problems, not just boat and stream scenarios but also problems involving trains, planes, or people moving relative to each other or a medium.