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?
16 hours
Solving Boat and Stream Problems: Calculating Total Time
This problem involves calculating the total time taken by a boat to travel a certain distance downstream and then a certain distance upstream. We are given the downstream speed of the boat and the speed of the stream. We need to find the speed of the boat in still water and the upstream speed first.
Understanding Boat and Stream Concepts
In boat and stream problems, the speed of the boat relative to the ground changes depending on whether it is moving with the stream (downstream) or against the stream (upstream).
- Downstream Speed: When the boat moves in the same direction as the stream, the speed of the stream adds to the speed of the boat in still water.
- Upstream Speed: When the boat moves in the opposite direction to the stream, the speed of the stream subtracts from the speed of the boat in still water.
- Speed of Boat in Still Water: This is the boat's speed without the influence of the stream.
- Speed of Stream: This is the speed of the water current.
Given Information
- Downstream speed of the boat ($S_{downstream}$) = 20 km/hr
- Speed of stream ($S_{stream}$) = 4 km/hr
- Downstream distance ($D_{downstream}$) = 160 km
- Upstream distance ($D_{upstream}$) = 96 km
Calculating Boat Speed in Still Water and Upstream Speed
Let the speed of the boat in still water be $S_{boat}$.
The relationship between downstream speed, boat speed in still water, and stream speed is:
\(S_{downstream} = S_{boat} + S_{stream}\)
We can use this to find the speed of the boat in still water:
\(20 \text{ km/hr} = S_{boat} + 4 \text{ km/hr}\)
\(S_{boat} = 20 - 4 = 16 \text{ km/hr}\)
Now that we have the speed of the boat in still water, we can calculate the upstream speed. The relationship between upstream speed, boat speed in still water, and stream speed is:
\(S_{upstream} = S_{boat} - S_{stream}\)
Substitute the values:
\(S_{upstream} = 16 \text{ km/hr} - 4 \text{ km/hr}\)
\(S_{upstream} = 12 \text{ km/hr}\)
Here is a summary of the calculated speeds:
| Speed Type | Value (km/hr) |
|---|---|
| Downstream Speed | 20 |
| Speed of Stream | 4 |
| Speed of Boat in Still Water | 16 |
| Upstream Speed | 12 |
Calculating Time Taken for Downstream Journey
The formula for time is:
\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)
For the downstream journey:
- Distance ($D_{downstream}$) = 160 km
- Speed ($S_{downstream}$) = 20 km/hr
Time taken for downstream journey ($T_{downstream}$):
\(T_{downstream} = \frac{D_{downstream}}{S_{downstream}} = \frac{160 \text{ km}}{20 \text{ km/hr}} = 8 \text{ hours}\)
Calculating Time Taken for Upstream Journey
For the upstream journey:
- Distance ($D_{upstream}$) = 96 km
- Speed ($S_{upstream}$) = 12 km/hr
Time taken for upstream journey ($T_{upstream}$):
\(T_{upstream} = \frac{D_{upstream}}{S_{upstream}} = \frac{96 \text{ km}}{12 \text{ km/hr}} = 8 \text{ hours}\)
Calculating Total Time Taken
The total time taken is the sum of the time taken for the downstream and upstream journeys.
Total Time = $T_{downstream} + T_{upstream}$
Total Time = 8 hours + 8 hours
Total Time = 16 hours
So, the total time taken by the boat to cover 160 km downstream and 96 km upstream is 16 hours.
Revision Table: Boat and Stream Formulas
| Concept | Formula | Notes |
|---|---|---|
| Downstream Speed ($S_d$) | \(S_d = S_b + S_s\) | \(S_b\): Boat speed in still water, \(S_s\): Stream speed |
| Upstream Speed ($S_u$) | \(S_u = S_b - S_s\) | \(S_b\): Boat speed in still water, \(S_s\): Stream speed |
| Boat Speed ($S_b$) | \(S_b = \frac{S_d + S_u}{2}\) | If both downstream and upstream speeds are known |
| Stream Speed ($S_s$) | \(S_s = \frac{S_d - S_u}{2}\) | If both downstream and upstream speeds are known |
| Time | \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\) | General formula for time, distance, and speed |
Additional Information on Boat and Stream Problems
Boat and stream problems are common in quantitative aptitude tests. They are based on the simple principle of relative speed. When objects move in the same direction (boat downstream), their speeds add up. When they move in opposite directions (boat upstream), the speed of the slower object (boat relative to stream) is subtracted from the speed of the faster object (stream relative to boat, which is equivalent to boat speed minus stream speed in terms of speed relative to ground). Always remember to distinguish between the boat's speed in still water and its effective speed relative to the ground (downstream or upstream speed).
Practicing different variations of these problems, such as finding distances, speeds, or time taken for round trips, helps in mastering this topic.
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 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.
A man covered a distance of 18 km in 3 hours, 7.2 km in 2 hours and 15 km in 5 hours. What is the average distance traveled by him per hour?
