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 ?
3 km/hr
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.
We are given:
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.
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:
\(\frac{42}{15 + V_s} = \frac{28}{15 - V_s}\)
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.
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.
| 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) | \(\frac{\text{Speed Downstream} - \text{Speed Upstream}}{2}\) |
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.
These problems often involve calculating speeds, distances, or times using the relationship \( \text{Distance} = \text{Speed} \times \text{Time} \).
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(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
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