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?
z(x 2- y 2)/2x
Understanding Boat and Stream Problems
Boat and stream problems involve the concept of relative speed. When a boat moves in water, its speed is affected by the speed of the water (the stream or current).
- Speed in still water: This is the speed of the boat if there were no current. Let's call this \(x\) km/h, as given in the question.
- Speed of the stream (current): This is the speed at which the water is flowing. Let's call this \(y\) km/h, as given in the question.
Calculating Downstream and Upstream Speeds
When the boat moves with the flow of the stream, it's called downstream movement. When it moves against the flow, it's called upstream movement.
- Speed Downstream: When the boat moves in the same direction as the stream, the speed of the stream adds to the boat's speed in still water.
\(\text{Speed Downstream} = \text{Speed in still water} + \text{Speed of stream}\)
\(\text{Speed Downstream} = (x + y)\) km/h - Speed Upstream: When the boat moves against the direction of the stream, the speed of the stream subtracts from the boat's speed in still water.
\(\text{Speed Upstream} = \text{Speed in still water} - \text{Speed of stream}\)
\(\text{Speed Upstream} = (x - y)\) km/h
For upstream movement to be possible, the speed of the boat in still water (\(x\)) must be greater than the speed of the stream (\(y\)).
Setting Up the Equation for Total Time
The question states that the man rows to a place and back. This means he travels a certain distance downstream and then the same distance back upstream. The total time taken for this round trip is \(z\) hours.
We know the relationship between distance, speed, and time:
\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)
Let the distance between the two places be \(D\) km.
- Time taken to travel downstream (distance \(D\), speed \(x+y\)):
\(t_{\text{downstream}} = \frac{D}{x+y}\) hours - Time taken to travel upstream (distance \(D\), speed \(x-y\)):
\(t_{\text{upstream}} = \frac{D}{x-y}\) hours
The total time for the round trip is the sum of the downstream time and the upstream time:
\(t_{\text{total}} = t_{\text{downstream}} + t_{\text{upstream}}\)
We are given that the total time is \(z\) hours. So, the equation is:
\(\frac{D}{x+y} + \frac{D}{x-y} = z\)
Solving for the Distance (D)
Now, we need to solve the equation \(\frac{D}{x+y} + \frac{D}{x-y} = z\) for the variable \(D\).
First, find a common denominator for the fractions on the left side, which is \((x+y)(x-y)\):
\(\frac{D(x-y)}{(x+y)(x-y)} + \frac{D(x+y)}{(x-y)(x+y)} = z\)
\(\frac{D(x-y) + D(x+y)}{(x+y)(x-y)} = z\)
Expand the numerator:
\(\frac{Dx - Dy + Dx + Dy}{(x+y)(x-y)} = z\)
Combine like terms in the numerator (\(-Dy\) and \(+Dy\) cancel out):
\(\frac{2Dx}{(x+y)(x-y)} = z\)
Recall the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\). Apply this to the denominator:
\(\frac{2Dx}{x^2 - y^2} = z\)
Now, we want to isolate \(D\). Multiply both sides of the equation by \((x^2 - y^2)\):
\(2Dx = z(x^2 - y^2)\)
Finally, divide both sides by \(2x\) to find \(D\):
\(D = \frac{z(x^2 - y^2)}{2x}\)
This expression represents the distance between the two places.
Summary of the Solution
Given:
- Speed of man in still water = \(x\) km/h
- Speed of stream = \(y\) km/h
- Total time for round trip (to a place and back) = \(z\) hours
Calculations:
- Downstream speed = \((x+y)\) km/h
- Upstream speed = \((x-y)\) km/h
- Let distance = \(D\) km
- Time downstream = \(\frac{D}{x+y}\)
- Time upstream = \(\frac{D}{x-y}\)
- Total time equation: \(\frac{D}{x+y} + \frac{D}{x-y} = z\)
Solving the equation leads to:
\(D = \frac{z(x^2 - y^2)}{2x}\)
| Concept | Formula/Value |
|---|---|
| Speed in still water | \(x\) km/h |
| Speed of stream | \(y\) km/h |
| Downstream Speed | \((x+y)\) km/h |
| Upstream Speed | \((x-y)\) km/h |
| Total Time (Round Trip) | \(z\) hours |
| Distance (D) | \(\frac{z(x^2 - y^2)}{2x}\) km |
Revision Table: Boat and Stream Formulas
| Concept | Formula | Notes |
|---|---|---|
| Downstream Speed | \(S_d = S_w + S_s\) | \(S_w\) = speed in still water, \(S_s\) = speed of stream |
| Upstream Speed | \(S_u = S_w - S_s\) | \(S_w > S_s\) for movement against stream |
| Speed in Still Water | \(S_w = \frac{S_d + S_u}{2}\) | Calculated from downstream and upstream speeds |
| Speed of Stream | \(S_s = \frac{S_d - S_u}{2}\) | Calculated from downstream and upstream speeds |
| Time = Distance / Speed | \(T = \frac{D}{S}\) | Fundamental relationship |
Additional Information: Boat and Stream Concepts
Boat and stream problems are a common application of relative speed. Understanding how the stream affects the boat's speed is key.
- Relative Speed: The speed of an object with respect to another object. In this case, the boat's speed relative to the land is affected by the water's movement.
- Assumptions: These problems usually assume constant speeds for the boat in still water and for the stream throughout the journey.
- Round Trip Problems: When dealing with round trips (going to a place and coming back), the distance for both parts of the journey is the same, but the speeds (downstream and upstream) are different. The total time is the sum of the time taken for each leg of the journey.
- Solving for different variables: Depending on the problem, you might be asked to find the distance, the speed in still water, the speed of the stream, or the total time, given the other values. The core equations involving downstream speed, upstream speed, distance, and time remain the same.
Similar Questions
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?
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?
A person can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. What is the speed of the current?
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 ?
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
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?
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?
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?
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?
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.
