All Exams Test series for 1 year @ ₹349 only
Question

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?

This question was previously asked in
CDS I 2017 General Knowledge Previous Year Paper (05-Feb-2017)
The correct answer is

1, 2 and 3

Understanding River Speed Problems

This question involves a classic concept in speed, time, and distance problems related to boats or men moving in water bodies where a current exists. The speed of the water (stream) affects the overall speed of the object moving in it.

There are two main scenarios:

  • Downstream: When the man rows in the same direction as the stream. The speed of the stream adds to the man's speed in still water.
  • Upstream: When the man rows against the direction of the stream. The speed of the stream is subtracted from the man's speed in still water.

Setting up Equations for Rowing Speeds

Let's define the variables:

  • Let \( v_m \) be the speed of the man in still water (in km/h).
  • Let \( v_s \) be the speed of the stream (in km/h).

Based on the definitions above, we can express the downstream and upstream speeds:

  • Speed downstream = \( v_m + v_s \)
  • Speed upstream = \( v_m - v_s \)

We know that Speed = Distance / Time. We are given the distance and time for both the downstream and upstream journeys.

Downstream Journey Analysis

Distance = 18 km

Time = 4 hours

Speed downstream = \(\text{Distance} / \text{Time} = 18 \text{ km} / 4 \text{ hours} = 4.5 \text{ km/h}\)

So, we get our first equation:

Equation 1: \( v_m + v_s = 4.5 \)

Upstream Journey Analysis

Distance = 18 km (returns the same distance)

Time = 10 hours

Speed upstream = \(\text{Distance} / \text{Time} = 18 \text{ km} / 10 \text{ hours} = 1.8 \text{ km/h}\)

So, we get our second equation:

Equation 2: \( v_m - v_s = 1.8 \)

Verifying the Statements

We now have a system of two linear equations with two variables \( v_m \) and \( v_s \).

  • \( v_m + v_s = 4.5 \) (Equation 1)
  • \( v_m - v_s = 1.8 \) (Equation 2)

Let's evaluate each statement given in the question.

Statement 1: The speed of the man against the stream is 1.8 km/h

The speed of the man against the stream is the upstream speed. From our analysis of the upstream journey, we calculated the speed against the stream to be 1.8 km/h.

Speed against the stream = \( v_m - v_s = 1.8 \) km/h.

Therefore, Statement (1) is correct.

Statement 2: The speed of the man in still water is 3.15 km/h

The speed of the man in still water is \( v_m \). We can find \( v_m \) by solving the system of equations. A common method is to add the two equations:

\((v_m + v_s) + (v_m - v_s) = 4.5 + 1.8\)

\(2v_m = 6.3\)

\(v_m = 6.3 / 2\)

\(v_m = 3.15 \text{ km/h}\)

Therefore, the speed of the man in still water is 3.15 km/h. Statement (2) is correct.

Statement 3: The speed of the stream is 1.35 km/h

The speed of the stream is \( v_s \). We can find \( v_s \) by substituting the value of \( v_m \) (3.15 km/h) into either Equation 1 or Equation 2.

Using Equation 1:

\(3.15 + v_s = 4.5\)

\(v_s = 4.5 - 3.15\)

\(v_s = 1.35 \text{ km/h}\)

Using Equation 2:

\(3.15 - v_s = 1.8\)

\(v_s = 3.15 - 1.8\)

\(v_s = 1.35 \text{ km/h}\)

Both equations give the same result. Therefore, the speed of the stream is 1.35 km/h. Statement (3) is correct.

Conclusion

Based on our calculations, all three statements are correct:

  • Statement (1): The speed of the man against the stream is 1.8 km/h. (Correct)
  • Statement (2): The speed of the man in still water is 3.15 km/h. (Correct)
  • Statement (3): The speed of the stream is 1.35 km/h. (Correct)

Thus, statements 1, 2, and 3 are all correct.

Quantity Formula/Value Calculation
Downstream Speed \( v_m + v_s \) \( 18 \text{ km} / 4 \text{ h} = 4.5 \text{ km/h} \)
Upstream Speed \( v_m - v_s \) \( 18 \text{ km} / 10 \text{ h} = 1.8 \text{ km/h} \)
Speed in Still Water (\( v_m \)) \( \frac{\text{Downstream Speed} + \text{Upstream Speed}}{2} \) \( \frac{4.5 + 1.8}{2} = \frac{6.3}{2} = 3.15 \text{ km/h} \)
Speed of Stream (\( v_s \)) \( \frac{\text{Downstream Speed} - \text{Upstream Speed}}{2} \) \( \frac{4.5 - 1.8}{2} = \frac{2.7}{2} = 1.35 \text{ km/h} \)

Revision Table: Key Speed Concepts

Concept Description Formula
Speed in Still Water (\( v_m \)) Speed of the object without any current effect. \( \frac{\text{Downstream Speed} + \text{Upstream Speed}}{2} \)
Speed of Stream (\( v_s \)) Speed of the water current. \( \frac{\text{Downstream Speed} - \text{Upstream Speed}}{2} \)
Downstream Speed Effective speed when moving with the stream. \( v_m + v_s \)
Upstream Speed (Speed Against Stream) Effective speed when moving against the stream. \( v_m - v_s \)

Additional Information on Boat and Stream Problems

Problems involving boats or swimmers in rivers are common in quantitative aptitude. They test your understanding of relative speed. The key is to correctly identify whether the motion is downstream (with the current) or upstream (against the current).

Remember these basic relationships:

  • When moving downstream, the stream adds to the object's speed.
  • When moving upstream, the stream subtracts from the object's speed.
  • If you know the downstream speed and upstream speed, you can easily find the speed in still water and the speed of the stream using the average concept shown in the Revision Table.

Always ensure units are consistent (e.g., km/h, m/s). In this problem, all units were given in km and hours, making the calculation straightforward in km/h.

Was this answer helpful?

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 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.

Need Expert Advice?
Test Series
CDS img
Defence
UPSC CDS 2026 Mock Test Series
536 Tests 4 Tests Free
1633 Attempts
4.3(174)
English, Hindi
More Questions from CDS

Start Your Preparation with Prepp Mobile App

Download the app from Google Play & App Store
Download the app from Google Play & App Store
Prepp Mobile App