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Question

A motor boat has speed 30 km/hr in still water. It goes 60 km down stream and comes back in 9/2 hours. What is the speed of the stream?

This question was previously asked in
CDS II 2021 General Knowledge Previous Year Paper (14-Nov-2021)
The correct answer is

10 km/hr

Understanding Boat and Stream Problems

Boat and stream problems involve the concept of relative speed. When a boat travels in water, its speed is affected by the speed of the water flow (the stream). There are two main scenarios:

  • Downstream: The boat travels in the same direction as the stream. The speed of the stream adds to the boat's speed.
  • Upstream: The boat travels against the direction of the stream. The speed of the stream reduces the boat's speed.

Analyzing the Given Information

We are given the following details about the boat's journey:

  • Speed of the motor boat in still water (\(V_b\)): 30 km/hr
  • Distance traveled downstream (\(D\)): 60 km
  • Distance traveled upstream (\(D\)): 60 km (since it comes back)
  • Total time taken for the round trip (\(T_{total}\)): 9/2 hours or 4.5 hours

We need to find the speed of the stream (\(V_s\)).

Setting Up the Equations

Let the speed of the stream be \(V_s\) km/hr.

The speed of the boat when going downstream is the sum of its speed in still water and the speed of the stream:

Speed downstream (\(V_d\)) = \(V_b + V_s = (30 + V_s)\) km/hr

The speed of the boat when going upstream is the difference between its speed in still water and the speed of the stream:

Speed upstream (\(V_u\)) = \(V_b - V_s = (30 - V_s)\) km/hr

The time taken to travel a certain distance is given by the formula: Time = Distance / Speed.

Time taken to go downstream (\(T_d\)) = \(\frac{\text{Distance}}{\text{Speed downstream}} = \frac{60}{30 + V_s}\) hours

Time taken to come back upstream (\(T_u\)) = \(\frac{\text{Distance}}{\text{Speed upstream}} = \frac{60}{30 - V_s}\) hours

The total time for the round trip is the sum of the time taken for the downstream and upstream journeys:

\(T_{total} = T_d + T_u\)

We are given that the total time is 9/2 hours.

So, the equation is:

\(\frac{9}{2} = \frac{60}{30 + V_s} + \frac{60}{30 - V_s}\)

Solving for the Speed of the Stream (\(V_s\))

Now, we solve the equation for \(V_s\):

\(\frac{9}{2} = 60 \left( \frac{1}{30 + V_s} + \frac{1}{30 - V_s} \right)\)

\(\frac{9}{2} = 60 \left( \frac{(30 - V_s) + (30 + V_s)}{(30 + V_s)(30 - V_s)} \right)\)

\(\frac{9}{2} = 60 \left( \frac{60}{30^2 - V_s^2} \right)\)

\(\frac{9}{2} = \frac{3600}{900 - V_s^2}\)

Now, cross-multiply:

\(9 (900 - V_s^2) = 2 \times 3600\)

\(8100 - 9V_s^2 = 7200\)

Rearrange the terms to solve for \(V_s^2\):

\(8100 - 7200 = 9V_s^2\)

\(900 = 9V_s^2\)

\(V_s^2 = \frac{900}{9}\)

\(V_s^2 = 100\)

Taking the square root of both sides:

\(V_s = \sqrt{100}\)

\(V_s = 10\) (Since speed must be a positive value)

So, the speed of the stream is 10 km/hr.

Verifying the Solution

Let's check if a stream speed of 10 km/hr gives the correct total time.

If \(V_s = 10\) km/hr:

  • Speed downstream (\(V_d\)) = \(30 + 10 = 40\) km/hr
  • Speed upstream (\(V_u\)) = \(30 - 10 = 20\) km/hr
  • Time downstream (\(T_d\)) = \(\frac{60}{40} = 1.5\) hours
  • Time upstream (\(T_u\)) = \(\frac{60}{20} = 3\) hours
  • Total time (\(T_{total}\)) = \(T_d + T_u = 1.5 + 3 = 4.5\) hours

4.5 hours is equal to 9/2 hours, which matches the given total time. The solution is correct.

The speed of the stream is 10 km/hr.

Concept Formula Value (with \(V_s = 10\))
Speed in Still Water (\(V_b\)) Given 30 km/hr
Speed of Stream (\(V_s\)) To find (x) 10 km/hr
Speed Downstream (\(V_d\)) \(V_b + V_s\) \(30 + 10 = 40\) km/hr
Speed Upstream (\(V_u\)) \(V_b - V_s\) \(30 - 10 = 20\) km/hr
Time Downstream (\(T_d\)) Distance / \(V_d\) \(60 / 40 = 1.5\) hours
Time Upstream (\(T_u\)) Distance / \(V_u\) \(60 / 20 = 3\) hours
Total Time (\(T_{total}\)) \(T_d + T_u\) \(1.5 + 3 = 4.5\) hours or 9/2 hours

Revision Table: Boat and Stream Formulas

Concept Formula Notes
Speed Downstream (\(V_d\)) \(V_b + V_s\) Boat speed + Stream speed
Speed Upstream (\(V_u\)) \(V_b - V_s\) Boat speed - Stream speed
Speed in Still Water (\(V_b\)) \(\frac{V_d + V_u}{2}\) Average of downstream and upstream speeds
Speed of Stream (\(V_s\)) \(\frac{V_d - V_u}{2}\) Half the difference between downstream and upstream speeds

Additional Information: Relative Speed Concepts

The concept of relative speed is key in boat and stream problems. When moving with the stream, the stream assists the boat, increasing its effective speed relative to the ground. When moving against the stream, the stream opposes the boat, decreasing its effective speed relative to the ground.

It is important to note that the speed of the boat in still water (\(V_b\)) is its intrinsic speed without any influence from the water flow. The speed of the stream (\(V_s\)) is the speed of the water itself.

If \(V_s > V_b\), the boat cannot move upstream against the current; it will be carried downstream.

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Similar Questions

  1. A man rows down a river 18 km in 4 hours with the stream and returns in 10 hours. Consider the following statements:

    (1) The speed of the man against the stream is 1.8 km/h

    (2) The speed of the man in still water is 3.15 km/h

    (3) The speed of the stream is 1.35 km/h

    Which of the above statements are correct?

  2. A man can row at a speed of x km/h in still water. If in a stream which is flowing at a speed of y km/h it takes him z hours to row to a place and back, then what is the distance between the two places?

  3. A person can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. What is the speed of the current?

  4. The speed of a boat in still water is 15 km/hr. If it can travel 42 km downstream and 28 km upstream in the same time, then what is the speed of the stream ?  

  5. A boatman can row to a place (Y) at a distance of 24 km from the starting point (X) and back in 14 hours. If he can row 4 km with the stream in the same time as he can row 3 km against it, what is the speed of the stream?


Important Questions from Boat and River

  1. A boat sails 15 km of a river towards upstream in 5 hours. How long (in hours) will it take to cover the same distance downstream, if the speed of river is one-fourth the speed of the boat in still water?

  2. The speed of boat upstream is 5 kmph. Its speed in still water is 10 kmph. How many minutes will it take to row 25 km downstream?

  3. Sudha can travel a certain distance downstream in 6 hours by boat and return to the starting point in 9 hours. If the stream flows at a speed of 3 km/h, how long (in hours) will it take to cover a distance of 67.5 km in still water?

  4. The downstream speed of a boat is 20 km/hr and the speed of stream is 4 km/hr. What will be the total time taken by the boat to cover 160 km downstream and 96 km upstream?

  5. The ratio of the speeds of a boat in still water and the speed of the river is 3 ∶ 1. The boat takes 45 minutes for a round trip journey. Find the distance of the whole trip if the speed of the stream is 4 km/hr.

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