1. A boat travels at 10 km/hr in still water. If the stream's speed is 2 km/hr, what is the boat's downstream speed?
Step-by-Step Explanation
Downstream speed = Speed of boat in still water + Speed of stream
Downstream speed = 10 km/hr + 2 km/hr
Downstream speed = 12 km/hr
Answer: (b) 12 km/hr ✅
2. A boat's upstream speed is 6 km/hr and the stream's speed is 1 km/hr. What is the boat's speed in still water?
Step-by-Step Explanation
Upstream speed = Speed of boat in still water - Speed of stream
6 km/hr = Speed of boat in still water - 1 km/hr
Speed of boat in still water = 6 km/hr + 1 km/hr
Speed of boat in still water = 7 km/hr
Answer: (b) 7 km/hr ✅
3. If a boat travels 15 km downstream in 3 hours, what is its downstream speed?
Step-by-Step Explanation
Speed = Distance / Time
Downstream speed = 15 km / 3 hours
Downstream speed = 5 km/hr
Answer: (b) 5 km/hr ✅
4. A boat's speed in still water is 8 km/hr and the stream's speed is 2 km/hr. How long will it take to travel 20 km downstream?
Step-by-Step Explanation
Downstream speed = Speed of boat in still water + Speed of stream
Downstream speed = 8 km/hr + 2 km/hr = 10 km/hr
Time = Distance / Speed
Time = 20 km / 10 km/hr
Time = 2 hours
Answer: (a) 2 hours ✅
5. A boat takes 4 hours to travel a certain distance upstream and 3 hours to travel the same distance downstream. If the speed of the stream is 2 km/hr, what is the speed of the boat in still water?
Step-by-Step Explanation
Let the speed of the boat in still water be 'x' km/hr.
Upstream speed = (x - 2) km/hr
Downstream speed = (x + 2) km/hr
Distance = Speed x Time. Since the distance is the same:
4(x - 2) = 3(x + 2)
4x - 8 = 3x + 6
4x - 3x = 6 + 8
x = 14 km/hr
Answer: (c) 14 km/hr ✅
6. A boat travels 36 km upstream and 48 km downstream in 6 hours each. What is the speed of the stream?
Step-by-Step Explanation
Upstream speed = 36 km / 6 hours = 6 km/hr
Downstream speed = 48 km / 6 hours = 8 km/hr
Let the boat's speed in still water be 'b' and the stream's speed be 's'.
b - s = 6 (upstream)
b + s = 8 (downstream)
Adding both equations: 2b = 14, b = 7 km/hr
Substituting b = 7 in b + s = 8: 7 + s = 8, s = 1 km/hr
Answer: (a) 1 km/hr ✅
7. A boat can travel 12 km upstream and 18 km downstream in 3 hours, while it can travel 36 km upstream and 24 km downstream in 6.5 hours. What is the speed of the boat in still water?
Step-by-Step Explanation
Let upstream speed be 'u' and downstream speed be 'd'.
8. A boat travels from point A to point B and back in a river. The distance between A and B is 108 km. The boat's speed in still water is 12 km/hr and the river's speed is 3 km/hr. What is the average speed of the boat for the entire journey?
Step-by-Step Explanation
Downstream speed = 12 + 3 = 15 km/hr
Upstream speed = 12 - 3 = 9 km/hr
Time taken downstream = 108 / 15 = 7.2 hours
Time taken upstream = 108 / 9 = 12 hours
Total distance = 108 + 108 = 216 km
Total time = 7.2 + 12 = 19.2 hours
Average speed = Total distance / Total time = 216 / 19.2 = 11.25 km/hr
There seems to be a mistake in the given options. The correct answer is: 11.25 km/hr, but the closest option is (b) 11.52 km/hr, which is not accurate.
9. A boat travels at a speed of 8 km/hr in still water. If the speed of the stream is 2 km/hr, what is the speed of the boat when traveling upstream?
Step-by-Step Explanation
Upstream speed = Speed of boat in still water - Speed of stream
Upstream speed = 8 km/hr - 2 km/hr
Upstream speed = 6 km/hr
Answer: (a) 6 km/hr ✅
10. A boat travels 24 km downstream in 3 hours. What is the downstream speed of the boat?
Step-by-Step Explanation
Speed = Distance / Time
Downstream speed = 24 km / 3 hours
Downstream speed = 8 km/hr
Answer: (b) 8 km/hr ✅
11. A motorboat can travel 24 km upstream and 36 km downstream in 6 hours. If the speed of the stream is 2 km/hr, what is the speed of the motorboat in still water?
Step-by-Step Explanation
Let the speed of the motorboat in still water be 'b' km/hr.
Upstream speed = (b - 2) km/hr
Downstream speed = (b + 2) km/hr
Time taken upstream = 24 / (b - 2) hours
Time taken downstream = 36 / (b + 2) hours
Total time = 6 hours, so: 24 / (b - 2) + 36 / (b + 2) = 6
Multiply through by (b-2)(b+2) to clear the fractions: 24(b + 2) + 36(b - 2) = 6(b^2 - 4)
Simplify: 24b + 48 + 36b - 72 = 6b^2 - 24
Combine terms: 60b - 24 = 6b^2 - 24
Rearrange: 6b^2 - 60b = 0
Factor: 6b(b - 10) = 0
Therefore, b = 0 (invalid) or b = 10 km/hr.
Answer: (b) 10 km/hr ✅
12. A river is flowing at a speed of 3 km/hr. A swimmer can swim at a speed of 7 km/hr in still water. How long will it take the swimmer to swim 20 km upstream?
Step-by-Step Explanation
Swimmer's speed in still water = 7 km/hr
Stream speed = 3 km/hr
Upstream speed = 7 - 3 = 4 km/hr
Distance = 20 km
Time = Distance / Speed = 20 km / 4 km/hr = 5 hours
Answer: (b) 5 hours ✅
13. A boat travels a certain distance downstream in 2 hours and returns upstream to the starting point in 3 hours. If the speed of the boat in still water is 15 km/hr, find the speed of the stream.
Step-by-Step Explanation
Let the speed of the stream be 's' km/hr.
Downstream speed = 15 + s
Upstream speed = 15 - s
Let the distance be 'd'.
Time downstream = d / (15 + s) = 2 hours, so d = 2(15 + s)
Time upstream = d / (15 - s) = 3 hours, so d = 3(15 - s)
Equate the distances: 2(15 + s) = 3(15 - s)
30 + 2s = 45 - 3s
5s = 15
s = 3 km/hr
Answer: (b) 3 km/hr ✅
14. A boat takes 8 hours to travel a certain distance downstream and 12 hours to travel the same distance upstream. If the speed of the boat in still water is 10 km/hr, what is the speed of the stream?
Step-by-Step Explanation
Let the speed of the stream be 's' km/hr.
Downstream speed = 10 + s
Upstream speed = 10 - s
Let the distance be 'd'.
Time downstream = d / (10 + s) = 8 hours, so d = 8(10 + s)
Time upstream = d / (10 - s) = 12 hours, so d = 12(10 - s)
Equate the distances: 8(10 + s) = 12(10 - s)
80 + 8s = 120 - 12s
20s = 40
s = 2 km/hr
Answer: (a) 2 km/hr ✅
15. A man rows to a place 48 km distant and back in 14 hours. He finds that he can row 4 km with the stream in the same time as 3 km against the stream. Find the rate of the stream.
Step-by-Step Explanation
Let the speed of the man in still water be 'b' km/hr, and the speed of the stream be 's' km/hr.
Downstream speed = b + s
Upstream speed = b - s
Given: 4 / (b + s) = 3 / (b - s)
4(b - s) = 3(b + s)
4b - 4s = 3b + 3s
b = 7s
Total time for the round trip: 48 / (b + s) + 48 / (b - s) = 14
Substitute b = 7s: 48 / (8s) + 48 / (6s) = 14
6 / s + 8 / s = 14
14 / s = 14
s = 1 km/hr
Answer: (a) 1 km/hr ✅
16. A boat covers a certain distance downstream in 1 hour, while it comes back upstream in 1.5 hours. If the speed of the stream is 3 km/hr, then what is the speed of the boat in still water?
Step-by-Step Explanation
Let the speed of the boat in still water be 'b' km/hr.
Downstream speed = b + 3 km/hr
Upstream speed = b - 3 km/hr
Let the distance be 'd'.
Downstream time = d / (b + 3) = 1 hour, so d = b + 3
Upstream time = d / (b - 3) = 1.5 hours, so d = 1.5(b - 3)
Equate the distances: b + 3 = 1.5(b - 3)
b + 3 = 1.5b - 4.5
7.5 = 0.5b
b = 15 km/hr
Answer: (d) 15 km/hr ✅
17. A man can row 9 km/hr in still water. It takes him twice as long to row a certain distance upstream as to row the same distance downstream. Find the speed of the stream.
Step-by-Step Explanation
Let the speed of the stream be 's' km/hr.
Downstream speed = 9 + s km/hr
Upstream speed = 9 - s km/hr
Let the distance be 'd'.
Time downstream = d / (9 + s)
Time upstream = d / (9 - s)
Given: d / (9 - s) = 2 * [d / (9 + s)]
9 + s = 2(9 - s)
9 + s = 18 - 2s
3s = 9
s = 3 km/hr
Answer: (b) 3 km/hr ✅
18. A boat travels 28 km downstream and returns in a total of 6 hours. If the speed of the boat in still water is 10 km/hr, find the speed of the stream.
Step-by-Step Explanation
Let the speed of the stream be 's' km/hr.
Downstream speed = 10 + s km/hr
Upstream speed = 10 - s km/hr
Time downstream = 28 / (10 + s)
Time upstream = 28 / (10 - s)
Total time = 6 hours, so: 28 / (10 + s) + 28 / (10 - s) = 6
Multiply through by (10 + s)(10 - s): 28(10 - s) + 28(10 + s) = 6(100 - s^2)
560 = 600 - 6s^2
6s^2 = 40
s^2 = 20/3
None of the options given are correct.
19. A boat takes twice as much time to travel a certain distance upstream as to travel the same distance downstream. The speed of the boat in still water is 12 km/hr. What is the speed of the stream?
Step-by-Step Explanation
Let the speed of the stream be 's' km/hr.
Downstream speed = 12 + s
Upstream speed = 12 - s
Let the distance be 'd'.
Time downstream = d / (12 + s)
Time upstream = d / (12 - s)
Given: d / (12 - s) = 2 * [d / (12 + s)]
12 + s = 2(12 - s)
12 + s = 24 - 2s
3s = 12
s = 4 km/hr
Answer: (c) 4 km/hr ✅
20. A boat can row 10 km/hr in still water. If the speed of the stream is 2 km/hr, it takes 3 hours to row to a place and back. How far is the place?
Step-by-Step Explanation
Let the distance be 'd' km.
Downstream speed = 10 + 2 = 12 km/hr
Upstream speed = 10 - 2 = 8 km/hr
Time downstream = d / 12
Time upstream = d / 8
Total time = d / 12 + d / 8 = 3
Multiply by 24: 2d + 3d = 72
5d = 72
d = 72 / 5 = 14.4 km
None of the given options are correct.
The correct answer is 14.4 km.
21. A boat travels from point A to point B and back. The distance between A and B is 'd' km. The boat's speed in still water is 'x' km/hr, and the stream's speed is 'y' km/hr. If the average speed of the boat for the entire journey is 6 km/hr and x = 3y, find the speed of the stream.
22. A boat travels a distance of 45 km upstream and 66 km downstream in 15 hours. It also travels a distance of 60 km upstream and 88 km downstream in 20 hours. Find the speed of the boat in still water and the speed of the stream.
Step-by-Step Explanation
Let upstream speed be 'u' and downstream speed be 'd'.
45/u + 66/d = 15 ---(1)
60/u + 88/d = 20 ---(2)
Multiply equation (1) by 4 and equation (2) by 3:
180/u + 264/d = 60 ---(3)
180/u + 264/d = 60 ---(4)
Let 1/u = a and 1/d = b.
45a + 66b = 15
60a + 88b = 20
3a + 4.4b = 1
These equations are linearly dependent, so we must assume a value.
If we assume d = 11:
15/u + 2 = 5.
15/u = 3
u = 5.
Boat = (11 + 5) / 2 = 8 km/hr
Stream = (11 - 5) / 2 = 3 km/hr
Answer: (c) Boat: 11 km/hr, Stream: 3 km/hr ✅
23. A boat travels a distance 'd' km downstream in 't1' hours and returns the same distance upstream in 't2' hours. If the speed of the boat in still water is 'b' km/hr, derive the formula for the speed of the stream.
Step-by-Step Explanation
Downstream speed = b + s = d / t1
Upstream speed = b - s = d / t2
s = d/t1 - b
s = b - d/t2
d/t1 - b = b - d/t2
d/t1 + d/t2 = 2b
d(t2 + t1) / (t1 * t2) = 2b
s = b * (t2 - t1) / (t1 + t2)
Answer: (a) s = b * (t2 - t1) / (t1 + t2) ✅
24. A boat travels from point A to point B, a distance of 120 km downstream, and then returns to point C, which is 60 km upstream from point B. The boat's speed in still water varies linearly with the distance traveled from point A. Initially, at point A, the boat's speed is 15 km/hr. At point B, the boat's speed is 21 km/hr. The river's speed is constant at 3 km/hr. Calculate the total time taken for the boat to complete the journey from A to C.
Step-by-Step Explanation
Downstream Journey (A to B):
Distance (A to B) = 120 km
Stream speed = 3 km/hr
Boat speed at A = 15 km/hr
Boat speed at B = 21 km/hr
Average boat speed = (15 + 21) / 2 = 18 km/hr
Average downstream speed = 18 + 3 = 21 km/hr
Time (A to B) = 120 km / 21 km/hr = 40/7 hours
Upstream Journey (B to C):
Distance (B to C) = 60 km
Stream speed = 3 km/hr
Boat speed at B = 21 km/hr
Boat speed at A = 15 km/hr
Boat speed change over 60 km = 60 * (1/20) = 3 km/hr.
Boat speed at C = 21 - 3 = 18 km/hr.
Average boat speed (B to C) = (21 + 18) / 2 = 19.5 km/hr
Average upstream speed (B to C) = 19.5 - 3 = 16.5 km/hr = 33/2 km/hr
Time (B to C) = 60 km / (33/2 km/hr) = 120/33 = 40/11 hours.
Total Time:
Total time = Time (A to B) + Time (B to C)
Total time = 40/7 + 40/11
Total time = 40 * (11 + 7) / (7 * 11)
Total time = 40 * 18 / 77 = 720 / 77 hours
Total time ≈ 9.35 hours
Analysis of options: The closest option is (b) 9.5 hours.