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Question

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 ?  

This question was previously asked in
CDS I 2023 English Previous Year Paper (16-April-2023)
The correct answer is

3 km/hr 

Understanding Boat and Stream Speed Problems

This problem involves calculating the speed of a stream given the speed of a boat in still water and the distances traveled upstream and downstream in the same amount of time. These types of questions are common in quantitative aptitude tests and require understanding how the speed of the stream affects the boat's speed.

Key Concepts

  • Speed in Still Water (\(V_b\)): This is the speed of the boat without any influence from a current. Given as 15 km/hr.
  • Speed of the Stream (\(V_s\)): This is the speed of the water current. This is what we need to find.
  • Downstream Speed: When the boat travels in the same direction as the stream, the speed of the stream adds to the boat's speed. Downstream Speed \(= V_b + V_s\).
  • Upstream Speed: When the boat travels against the direction of the stream, the speed of the stream subtracts from the boat's speed. Upstream Speed \(= V_b - V_s\).

Setting up the Problem

We are given:

  • Speed of boat in still water, \(V_b = 15\) km/hr.
  • Distance traveled downstream, \(D_{downstream} = 42\) km.
  • Distance traveled upstream, \(D_{upstream} = 28\) km.
  • Time taken for downstream journey = Time taken for upstream journey.

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

The downstream speed will be \((15 + V_s)\) km/hr.

The upstream speed will be \((15 - V_s)\) km/hr.

Using the Time Formula

The formula connecting distance, speed, and time is: Time = Distance / Speed.

According to the problem, the time taken for both journeys is the same.

Time taken downstream \(= \frac{D_{downstream}}{\text{Downstream Speed}} = \frac{42}{15 + V_s}\)

Time taken upstream \(= \frac{D_{upstream}}{\text{Upstream Speed}} = \frac{28}{15 - V_s}\)

Since the times are equal, we can set up the equation:

\(\frac{42}{15 + V_s} = \frac{28}{15 - V_s}\)

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

Now, we need to solve this equation for \(V_s\).

Step 1: Cross-multiply the equation.

\(42 \times (15 - V_s) = 28 \times (15 + V_s)\)

\(630 - 42V_s = 420 + 28V_s\)

Step 2: Collect terms involving \(V_s\) on one side and constant terms on the other side.

\(630 - 420 = 28V_s + 42V_s\)

\(210 = 70V_s\)

Step 3: Solve for \(V_s\).

\(V_s = \frac{210}{70}\)

\(V_s = 3\)

So, the speed of the stream is 3 km/hr.

Verification (Optional but Recommended)

Let's check if the time taken is indeed the same with \(V_s = 3\) km/hr.

Downstream speed = \(15 + 3 = 18\) km/hr.

Time downstream = \(42 \text{ km} / 18 \text{ km/hr} = \frac{42}{18} \text{ hours} = \frac{7}{3}\) hours.

Upstream speed = \(15 - 3 = 12\) km/hr.

Time upstream = \(28 \text{ km} / 12 \text{ km/hr} = \frac{28}{12} \text{ hours} = \frac{7}{3}\) hours.

Since the time taken is the same (\(\frac{7}{3}\) hours) for both journeys, our calculated speed of the stream is correct.

The speed of the stream is 3 km/hr.

Revision Table: Boat and Stream Formulas

Concept Formula
Speed Downstream Speed of Boat in Still Water + Speed of Stream
Speed Upstream Speed of Boat in Still Water - Speed of Stream
Speed of Boat in Still Water (if Downstream/Upstream speeds are known) \(\frac{\text{Speed Downstream} + \text{Speed Upstream}}{2}\)
Speed of Stream (if Downstream/Upstream speeds are known) \(\frac{\text{Speed Downstream} - \text{Speed Upstream}}{2}\)

Additional Information: Solving Boat and Stream Problems

Boat and stream problems are a classic application of relative speed. When an object moves in a medium that is also moving, its effective speed (relative to a stationary observer) is the sum or difference of its speed in a still medium and the speed of the medium.

  • When moving with the medium (downstream), the speeds add up.
  • When moving against the medium (upstream), the speed of the medium subtracts from the object's speed. This is only possible if the speed of the object in still water is greater than the speed of the stream, otherwise, the object would be carried backward.

These problems often involve calculating speeds, distances, or times using the relationship \( \text{Distance} = \text{Speed} \times \text{Time} \).

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Similar Questions

  1. 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?

  2. 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?

  3. 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?

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

  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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