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?
1, 2 and 3
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:
Let's define the variables:
Based on the definitions above, we can express the downstream and upstream speeds:
We know that Speed = Distance / Time. We are given the distance and time for both the downstream and upstream journeys.
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: \( v_m + v_s = 4.5 \)
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: \( v_m - v_s = 1.8 \)
We now have a system of two linear equations with two variables \( v_m \) and \( v_s \).
Let's evaluate each statement given in the question.
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.
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.
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.
Based on our calculations, all three statements are 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 (\( v_s \)) | \( \frac{\text{Downstream Speed} - \text{Upstream Speed}}{2} \) | \( \frac{4.5 - 1.8}{2} = \frac{2.7}{2} = 1.35 \text{ km/h} \) |
| 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. | \( v_m - v_s \) |
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:
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.
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