Understanding River Speed and Swimming Concepts

This problem involves understanding how the speed of a river affects a swimmer's speed when moving upstream (against the current) and downstream (with the current).

Here's a breakdown of the key concepts:

• Speed in Still Water: This is the swimmer's own speed without any influence from the current. Let's call it \(u\).
• Speed of the Stream/Current: This is the speed at which the river water flows. Let's call it \(v\).
• Speed Downstream: When the swimmer goes with the current, the stream adds to their speed. Effective speed = Speed in still water + Speed of stream = \(u + v\).
• Speed Upstream: When the swimmer goes against the current, the stream opposes their movement. Effective speed = Speed in still water - Speed of stream = \(u - v\). Note that the speed in still water must be greater than the speed of the stream for the swimmer to make progress upstream.
• Time, Distance, and Speed Relationship: The fundamental formula connecting these is \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).

Analyzing the Given Information

We are given the following information about Arun's swim:

• Speed in still water (\(u\)) = 5 km/hr
• Speed of the stream (\(v\)) = 2 km/hr
• Let the distance between the two points be \(d\) km.
• Time taken upstream is 20 minutes more than time taken downstream.

Calculating Upstream and Downstream Speeds

Using the given speeds, we can calculate Arun's effective speed in each direction:

• Speed downstream = \(u + v = 5 + 2 = 7\) km/hr
• Speed upstream = \(u - v = 5 - 2 = 3\) km/hr

Notice that Arun's speed in still water (5 km/hr) is indeed greater than the stream's speed (2 km/hr), so he can swim upstream.

Setting Up the Time Equation

Let \(t_{down}\) be the time taken to swim downstream and \(t_{up}\) be the time taken to swim upstream.

Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

• \(t_{down} = \frac{d}{\text{Speed downstream}} = \frac{d}{7}\) hours
• \(t_{up} = \frac{d}{\text{Speed upstream}} = \frac{d}{3}\) hours

We are told that the time taken upstream is 20 minutes more than downstream. First, convert the time difference to hours:

\(20 \text{ minutes} = \frac{20}{60} \text{ hours} = \frac{1}{3} \text{ hours}\)

So, the relationship between the times is \(t_{up} - t_{down} = \frac{1}{3}\).

Substitute the expressions for \(t_{up}\) and \(t_{down}\):

\( \frac{d}{3} - \frac{d}{7} = \frac{1}{3} \)

Solving for the Distance

Now we need to solve the equation for \(d\). To combine the terms on the left side, find a common denominator, which is the least common multiple of 3 and 7, which is 21.

\( \frac{7d}{21} - \frac{3d}{21} = \frac{1}{3} \)

\( \frac{7d - 3d}{21} = \frac{1}{3} \)

\( \frac{4d}{21} = \frac{1}{3} \)

To isolate \(d\), multiply both sides of the equation by 21:

\( 4d = \frac{1}{3} \times 21 \)

\( 4d = 7 \)

Now, divide both sides by 4:

\( d = \frac{7}{4} \)

Converting the fraction to a decimal:

\( d = 1.75 \)

So, the distance between the two points is 1.75 km.

Verification

Let's quickly verify the answer:

• Distance \(d = 1.75\) km
• Speed downstream = 7 km/hr
• Speed upstream = 3 km/hr
• Time downstream = \( \frac{1.75}{7} = \frac{7/4}{7} = \frac{1}{4} \) hours
• Time upstream = \( \frac{1.75}{3} = \frac{7/4}{3} = \frac{7}{12} \) hours
• Time difference = \( \frac{7}{12} - \frac{1}{4} = \frac{7}{12} - \frac{3}{12} = \frac{4}{12} = \frac{1}{3} \) hours

\( \frac{1}{3} \) hours is equal to \( \frac{1}{3} \times 60 = 20 \) minutes. This matches the condition given in the problem.

Summary of Calculations

Conclusion on Distance Calculation

Based on the calculations using the given speeds and time difference, the distance between the two points is 1.75 km.

Revision Table: Speed, Distance, Time Concepts

Additional Information: Solving River Speed Problems

Problems involving boats or swimmers in rivers are common in quantitative aptitude. They test your understanding of relative speed. Here are a few additional points to keep in mind:

• Always identify the speed in still water and the speed of the stream clearly.
• Remember that the stream helps downstream motion and hinders upstream motion.
• If you are given the upstream and downstream speeds, you can find the speed in still water and the speed of the stream:
  • Speed in still water (\(u\)) = \( \frac{\text{Speed Downstream} + \text{Speed Upstream}}{2} \)
  • Speed of stream (\(v\)) = \( \frac{\text{Speed Downstream} - \text{Speed Upstream}}{2} \)
• Pay close attention to units (km/hr, m/s, minutes, hours) and convert them to be consistent before performing calculations.

Mastering these basic formulas and concepts will help you solve a wide variety of river-based speed, distance, and time problems.