Exercise : Boat & Stream - General Questions
✔ Boat & Stream -
General Questions
11.
In a stream running at 2 km/h, a motorboat goes 10 km upstream and back again to the starting point in 55 minutes. Find the speed of the motorboat in still water.
View Answer
Answer: Option A
Explanation:
Explanation:
1. Let boat speed be \(b\). Total time = \(55/60 = 11/12\) hours.
2. Equation: \(\frac{10}{b-2} + \frac{10}{b+2} = \frac{11}{12}\).
3. \(10(b + 2 + b - 2) / (b^2 - 4) = \frac{11}{12}\).
4. \(\frac{20b}{b^2 - 4} = \frac{11}{12} \implies 240b = 11b^2 - 44\).
5. Solving the quadratic \(11b^2 - 240b - 44 = 0\), we get \(b = 22 \text{ km/h}\).
2. Equation: \(\frac{10}{b-2} + \frac{10}{b+2} = \frac{11}{12}\).
3. \(10(b + 2 + b - 2) / (b^2 - 4) = \frac{11}{12}\).
4. \(\frac{20b}{b^2 - 4} = \frac{11}{12} \implies 240b = 11b^2 - 44\).
5. Solving the quadratic \(11b^2 - 240b - 44 = 0\), we get \(b = 22 \text{ km/h}\).
12.
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.
View Answer
Answer: Option A
Explanation:
Explanation:
1. The ratio of Downstream Speed to Upstream Speed is \(4:3\). Let them be \(4x\) and \(3x\).
2. Total time: \(\frac{48}{4x} + \frac{48}{3x} = 14\).
3. \(\frac{12}{x} + \frac{16}{x} = 14 \implies \frac{28}{x} = 14 \implies x = 2\).
4. Downstream speed = 8 km/h, Upstream speed = 6 km/h.
5. Stream rate = \(\frac{8 - 6}{2} = 1 \text{ km/h}\).
2. Total time: \(\frac{48}{4x} + \frac{48}{3x} = 14\).
3. \(\frac{12}{x} + \frac{16}{x} = 14 \implies \frac{28}{x} = 14 \implies x = 2\).
4. Downstream speed = 8 km/h, Upstream speed = 6 km/h.
5. Stream rate = \(\frac{8 - 6}{2} = 1 \text{ km/h}\).
13.
A boat takes 19 hours for travelling downstream from point A to point B and coming back to point C which is at midway between A and B. If the speed of the stream is 4 km/h and speed of the boat in still water is 14 km/h, find the distance between A and B.
View Answer
Answer: Option A
Explanation:
Explanation:
1. Downstream Speed = \(14 + 4 = 18 \text{ km/h}\). Upstream Speed = \(14 - 4 = 10 \text{ km/h}\).
2. Let the distance \(AB = D\). Then \(BC = D/2\).
3. Total time = \(\frac{D}{18} + \frac{D/2}{10} = 19\).
4. \(\frac{D}{18} + \frac{D}{20} = 19 \implies \frac{10D + 9D}{180} = 19\).
5. \(\frac{19D}{180} = 19 \implies D = 180 \text{ km}\).
2. Let the distance \(AB = D\). Then \(BC = D/2\).
3. Total time = \(\frac{D}{18} + \frac{D/2}{10} = 19\).
4. \(\frac{D}{18} + \frac{D}{20} = 19 \implies \frac{10D + 9D}{180} = 19\).
5. \(\frac{19D}{180} = 19 \implies D = 180 \text{ km}\).
14.
The speed of a boat in still water is 15 km/h and the rate of current is 3 km/h. The distance travelled downstream in 12 minutes is?
View Answer
Answer: Option A
Explanation:
Explanation:
1. Downstream Speed = \(15 + 3 = 18 \text{ km/h}\).
2. Time = \(12 / 60 = 0.2 \text{ hours}\).
3. Distance = \(\text{Speed} \times \text{Time} = 18 \times 0.2 = 3.6 \text{ km}\).
2. Time = \(12 / 60 = 0.2 \text{ hours}\).
3. Distance = \(\text{Speed} \times \text{Time} = 18 \times 0.2 = 3.6 \text{ km}\).
15.
A man rows 750 m in 675 seconds against the stream and returns in 7.5 minutes. Find his rowing speed in still water.
View Answer
Answer: Option A
Explanation:
Explanation:
1. Upstream Speed = \(750 / 675 = 10/9 \text{ m/s}\).
2. Downstream Speed = \(750 / 450 \text{ seconds} = 5/3 \text{ m/s}\).
3. Speed in still water = \(\frac{10/9 + 15/9}{2} = \frac{25/18 \text{ m/s}}{2}\). Wait, the formula is \((U+D)/2\).
4. \(V_{still} = (10/9 + 5/3) / 2 = (25/9) / 2 = 25/18 \text{ m/s}\).
5. Convert to km/h: \(\frac{25}{18} \times 3.6 = 5 \text{ km/h}\).
2. Downstream Speed = \(750 / 450 \text{ seconds} = 5/3 \text{ m/s}\).
3. Speed in still water = \(\frac{10/9 + 15/9}{2} = \frac{25/18 \text{ m/s}}{2}\). Wait, the formula is \((U+D)/2\).
4. \(V_{still} = (10/9 + 5/3) / 2 = (25/9) / 2 = 25/18 \text{ m/s}\).
5. Convert to km/h: \(\frac{25}{18} \times 3.6 = 5 \text{ km/h}\).
16.
A boat goes 6 km an hour in still water, but takes thrice as much time in going the same distance against the current than going with the current. What is the speed of the current?
View Answer
Answer: Option A
Explanation:
Explanation:
1. Let current be \(c\). \(V_{down} = 6 + c\), \(V_{up} = 6 - c\).
2. Since Time Up = \(3 \times\) Time Down, then \(V_{down} = 3 \times V_{up}\).
3. \(6 + c = 3(6 - c) \implies 6 + c = 18 - 3c\).
4. \(4c = 12 \implies c = 3 \text{ km/h}\).
2. Since Time Up = \(3 \times\) Time Down, then \(V_{down} = 3 \times V_{up}\).
3. \(6 + c = 3(6 - c) \implies 6 + c = 18 - 3c\).
4. \(4c = 12 \implies c = 3 \text{ km/h}\).
17.
Two boats A and B start towards each other from two places 108 km apart. Speeds of A and B in still water are 12 km/h and 15 km/h respectively. A proceeds downstream and B upstream. In how many hours will they meet?
View Answer
Answer: Option A
Explanation:
Explanation:
1. Speed of A = \(12 + s\); Speed of B = \(15 - s\).
2. Relative Speed when moving toward each other = \((12 + s) + (15 - s) = 27 \text{ km/h}\).
3. Time to meet = \(\text{Distance} / \text{Relative Speed} = 108 / 27 = 4 \text{ hours}\).
2. Relative Speed when moving toward each other = \((12 + s) + (15 - s) = 27 \text{ km/h}\).
3. Time to meet = \(\text{Distance} / \text{Relative Speed} = 108 / 27 = 4 \text{ hours}\).
18.
A boat covers a certain distance downstream in 1 hour, while it comes back in 1.5 hours. If the speed of the stream is 3 km/h, find the speed of the boat in still water.
View Answer
Answer: Option A
Explanation:
Explanation:
1. Let boat speed be \(b\). Distance is constant, so \(\text{Speed} \times \text{Time}\) is equal for both ways.
2. \((b + 3) \times 1 = (b - 3) \times 1.5\).
3. \(b + 3 = 1.5b - 4.5 \implies 0.5b = 7.5\).
4. \(b = 15 \text{ km/h}\).
2. \((b + 3) \times 1 = (b - 3) \times 1.5\).
3. \(b + 3 = 1.5b - 4.5 \implies 0.5b = 7.5\).
4. \(b = 15 \text{ km/h}\).
19.
A man can row 30 km downstream and 20 km upstream in 10 hours. Also, he can row 40 km downstream and 25 km upstream in 13 hours. Find the speed of the stream.
View Answer
Answer: Option A
Explanation:
Explanation:
1. Let \(1/D = x\) and \(1/U = y\).
2. Equations: \(30x + 20y = 10\) and \(40x + 25y = 13\).
3. Solving these simultaneous equations leads to \(D = 10\) and \(U = 5\).
4. Stream Speed = \(\frac{D - U}{2} = \frac{10 - 5}{2} = 2.5 \text{ km/h}\).
2. Equations: \(30x + 20y = 10\) and \(40x + 25y = 13\).
3. Solving these simultaneous equations leads to \(D = 10\) and \(U = 5\).
4. Stream Speed = \(\frac{D - U}{2} = \frac{10 - 5}{2} = 2.5 \text{ km/h}\).
20.
A boat travels upstream from B to A and downstream from A to B in 3 hours. If the speed of the boat in still water is 9 km/h and the speed of the current is 3 km/h, what is the distance between A and B?
View Answer
Answer: Option A
Explanation:
Explanation:
1. Downstream speed = \(9 + 3 = 12 \text{ km/h}\); Upstream speed = \(9 - 3 = 6 \text{ km/h}\).
2. Let distance be \(D\). \(\frac{D}{12} + \frac{D}{6} = 3\).
3. \(\frac{D + 2D}{12} = 3 \implies \frac{3D}{12} = 3\).
4. \(D/4 = 1 \implies D = 12 \text{ km}\).
2. Let distance be \(D\). \(\frac{D}{12} + \frac{D}{6} = 3\).
3. \(\frac{D + 2D}{12} = 3 \implies \frac{3D}{12} = 3\).
4. \(D/4 = 1 \implies D = 12 \text{ km}\).