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Question

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

This question was previously asked in
CDS I 2016 English Previous Year Paper (14-Feb-2016)
The correct answer is

4 km/hr

Solving Boat and Stream Speed Problems

This problem involves the concepts of speed, distance, and time, specifically in the context of a boat moving in water where there is a current. The current affects the boat's effective speed, making it faster when moving downstream (with the current) and slower when moving upstream (against the current).

Let's define the variables:

  • Let b be the speed of the boat in still water (km/hr).
  • Let c be the speed of the current (km/hr).

When the boat moves downstream, the speed of the current adds to the speed of the boat. The effective speed downstream is:

Speed downstream = Speed of boat in still water + Speed of current

Speed downstream \(= b + c\)

When the boat moves upstream, the speed of the current opposes the speed of the boat. The effective speed upstream is:

Speed upstream = Speed of boat in still water - Speed of current

Speed upstream \(= b - c\)

We know the relationship between speed, distance, and time is:

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

From the problem statement, we are given the following information:

  • Downstream distance = 20 km
  • Downstream time = 2 hours
  • Upstream distance = 4 km
  • Upstream time = 2 hours

Calculating Downstream and Upstream Speeds

Using the distance and time information, we can calculate the downstream and upstream speeds:

Downstream speed \(= \frac{\text{Downstream Distance}}{\text{Downstream Time}} = \frac{20 \text{ km}}{2 \text{ hours}} = 10 \text{ km/hr}\)

So, we have our first equation:

\(b + c = 10 \quad \text{(Equation 1)}\)

Upstream speed \(= \frac{\text{Upstream Distance}}{\text{Upstream Time}} = \frac{4 \text{ km}}{2 \text{ hours}} = 2 \text{ km/hr}\)

And our second equation:

\(b - c = 2 \quad \text{(Equation 2)}\)

Solving for the Speed of the Current

Now we have a system of two linear equations with two variables (\(b\) and \(c\)):

  1. \(b + c = 10\)
  2. \(b - c = 2\)

We want to find the speed of the current, which is represented by c. We can solve this system by adding the two equations together. This will eliminate the variable b:

Add Equation 1 and Equation 2:

\((b + c) + (b - c) = 10 + 2\)

\(b + c + b - c = 12\)

\(2b = 12\)

Divide by 2 to find b (the speed of the boat in still water):

\(b = \frac{12}{2} = 6 \text{ km/hr}\)

Now that we have the value of b, we can substitute it back into either Equation 1 or Equation 2 to find the value of c. Let's use Equation 1:

\(b + c = 10\)

Substitute \(b = 6\):

\(6 + c = 10\)

Subtract 6 from both sides to find c (the speed of the current):

\(c = 10 - 6 = 4 \text{ km/hr}\)

Alternatively, we could have subtracted Equation 2 from Equation 1 to directly solve for c:

Subtract Equation 2 from Equation 1:

\((b + c) - (b - c) = 10 - 2\)

\(b + c - b + c = 8\)

\(2c = 8\)

Divide by 2:

\(c = \frac{8}{2} = 4 \text{ km/hr}\)

Both methods yield the same result for the speed of the current.

Final Answer

The speed of the current is 4 km/hr.


Revision Table: Boat and Stream Concepts

Concept Description Formula
Speed of boat in still water (b) The speed at which the boat travels without the influence of a current. \(b = \frac{\text{Speed Downstream} + \text{Speed Upstream}}{2}\)
Speed of current (c) The speed at which the water flows. \(c = \frac{\text{Speed Downstream} - \text{Speed Upstream}}{2}\)
Speed downstream Boat speed when moving with the current. \(b + c = \frac{\text{Distance Downstream}}{\text{Time Downstream}}\)
Speed upstream Boat speed when moving against the current. \(b - c = \frac{\text{Distance Upstream}}{\text{Time Upstream}}\)

Additional Information: Understanding Relative Speed

Boat and stream problems are applications of the concept of relative speed. When two objects move in the same direction, their relative speed is the difference between their individual speeds. When they move in opposite directions (towards or away from each other), their relative speed is the sum of their individual speeds.

In boat and stream problems:

  • Moving downstream is like the boat and current moving in the same direction relative to the shore (current pushing the boat), so the effective speed (relative to shore) is the sum of their speeds (\(b+c\)).
  • Moving upstream is like the boat trying to move against the current, so the effective speed (relative to shore) is the difference between their speeds (\(b-c\)). The boat's speed must be greater than the current's speed (\(b > c\)) for it to move upstream.

The formulas used to find \(b\) and \(c\) from downstream and upstream speeds are direct results of solving the system of equations we used in the problem:

Given: Speed Downstream \(= S_d = b + c\) and Speed Upstream \(= S_u = b - c\)

Adding the two equations: \(S_d + S_u = (b + c) + (b - c) = 2b \implies b = \frac{S_d + S_u}{2}\)

Subtracting the second from the first: \(S_d - S_u = (b + c) - (b - c) = 2c \implies c = \frac{S_d - S_u}{2}\)

These formulas can be a quick way to find \(b\) and \(c\) once the downstream and upstream speeds are known.

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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. 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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