12
Q1: A boat goes 8 km in one hour along the stream and 2 km in one hour against the stream. The speed in km/hr of the stream is
A. 2
B. 3
C. 4
D. 5
Correct Answer: b
Solution: Let the speed of the boat in still water be $ x $ km/hr and the speed of the stream be $ y $ km/hr.
Along the stream: $ x + y = 8 $, Against the stream: $ x - y = 2 $.
Solving these equations: $ x = 5 $, $ y = 3 $.
**Speed of the stream = 3 km/hr.**
Solution: Let the speed of the boat in still water be $ x $ km/hr and the speed of the stream be $ y $ km/hr.
Along the stream: $ x + y = 8 $, Against the stream: $ x - y = 2 $.
Solving these equations: $ x = 5 $, $ y = 3 $.
**Speed of the stream = 3 km/hr.**
Q2: In one hour, a boat goes 11 km along the stream and 5 km against the stream. The speed of the boat in still water (in km/hr) is
A. 3
B. 5
C. 8
D. 9
Correct Answer: c
Solution: Let the speed of the boat in still water be $ x $ km/hr and the speed of the stream be $ y $ km/hr.
Along the stream: $ x + y = 11 $, Against the stream: $ x - y = 5 $.
Solving these equations: $ x = 8 $, $ y = 3 $.
**Speed of the boat in still water = 8 km/hr.**
Solution: Let the speed of the boat in still water be $ x $ km/hr and the speed of the stream be $ y $ km/hr.
Along the stream: $ x + y = 11 $, Against the stream: $ x - y = 5 $.
Solving these equations: $ x = 8 $, $ y = 3 $.
**Speed of the boat in still water = 8 km/hr.**
Q3: A man rows downstream 32 km and 14 km upstream. If he takes 6 hours to cover each distance, then the velocity (in kmph) of the current is
A. 1/2
B. 1
C. 1.5
D. 2
Correct Answer: c
Solution: Downstream speed: $ \frac{32}{6} = \frac{16}{3} $ km/hr.
Upstream speed: $ \frac{14}{6} = \frac{7}{3} $ km/hr.
Speed of the current: $ \frac{\text{Downstream speed} - \text{Upstream speed}}{2} = \frac{\frac{16}{3} - \frac{7}{3}}{2} = \frac{9}{6} = 1.5 $ km/hr.
**Velocity of the current = 1.5 km/hr.**
Solution: Downstream speed: $ \frac{32}{6} = \frac{16}{3} $ km/hr.
Upstream speed: $ \frac{14}{6} = \frac{7}{3} $ km/hr.
Speed of the current: $ \frac{\text{Downstream speed} - \text{Upstream speed}}{2} = \frac{\frac{16}{3} - \frac{7}{3}}{2} = \frac{9}{6} = 1.5 $ km/hr.
**Velocity of the current = 1.5 km/hr.**
Q4: A boatman rows 1 km in 5 minutes along the stream and 6 km in 1 hour against the stream. The speed of the stream is
A. 3 kmph
B. 6 kmph
C. 10 kmph
D. 12 kmph
Correct Answer: a
Solution: Downstream speed: $ \frac{1}{\frac{5}{60}} = 12 $ km/hr.
Upstream speed: $ \frac{6}{1} = 6 $ km/hr.
Speed of the stream: $ \frac{\text{Downstream speed} - \text{Upstream speed}}{2} = \frac{12 - 6}{2} = 3 $ km/hr.
**Speed of the stream = 3 km/hr.**
Solution: Downstream speed: $ \frac{1}{\frac{5}{60}} = 12 $ km/hr.
Upstream speed: $ \frac{6}{1} = 6 $ km/hr.
Speed of the stream: $ \frac{\text{Downstream speed} - \text{Upstream speed}}{2} = \frac{12 - 6}{2} = 3 $ km/hr.
**Speed of the stream = 3 km/hr.**
Q5: A boat takes half time in moving a certain distance downstream than upstream. What is the ratio between the rate in still water and the rate of the current?
A. 1 : 2
B. 2 : 1
C. 1 : 3
D. 3 : 1
Correct Answer: d
Solution: Let the speed of the boat in still water be $ x $ and the speed of the stream be $ y $.
Given: Time downstream = $ \frac{1}{2} \times $ Time upstream.
So, $ \frac{D}{x+y} = \frac{1}{2} \times \frac{D}{x-y} $.
Simplifying: $ 2(x-y) = x+y $.
Solving: $ x = 3y $.
**Ratio = 3:1.**
Solution: Let the speed of the boat in still water be $ x $ and the speed of the stream be $ y $.
Given: Time downstream = $ \frac{1}{2} \times $ Time upstream.
So, $ \frac{D}{x+y} = \frac{1}{2} \times \frac{D}{x-y} $.
Simplifying: $ 2(x-y) = x+y $.
Solving: $ x = 3y $.
**Ratio = 3:1.**
Q6: If a man goes 18 km downstream in 4 hours and returns against the stream in 12 hours, then the speed of the stream in km/hr is
A. 1
B. 1.5
C. 1.75
D. 3
Correct Answer: b
Solution: Downstream speed: $ \frac{18}{4} = 4.5 $ km/hr.
Upstream speed: $ \frac{18}{12} = 1.5 $ km/hr.
Speed of the stream: $ \frac{\text{Downstream speed} - \text{Upstream speed}}{2} = \frac{4.5 - 1.5}{2} = 1.5 $ km/hr.
**Speed of the stream = 1.5 km/hr.**
Solution: Downstream speed: $ \frac{18}{4} = 4.5 $ km/hr.
Upstream speed: $ \frac{18}{12} = 1.5 $ km/hr.
Speed of the stream: $ \frac{\text{Downstream speed} - \text{Upstream speed}}{2} = \frac{4.5 - 1.5}{2} = 1.5 $ km/hr.
**Speed of the stream = 1.5 km/hr.**
Q7: A boatman goes 2 km against the current of the stream in 1 hour and goes 1 km along the current in 10 minutes. How long will it take to go 5 km in stationary water?
A. 40 minutes
B. 1 hour
C. 1 hr 15 min
D. 1 hr 30 min
Correct Answer: c
Solution: Downstream speed: $ \frac{1}{\frac{10}{60}} = 6 $ km/hr.
Upstream speed: $ \frac{2}{1} = 2 $ km/hr.
Speed of the boat in still water: $ \frac{\text{Downstream speed} + \text{Upstream speed}}{2} = \frac{6 + 2}{2} = 4 $ km/hr.
Time to travel 5 km in still water: $ \frac{5}{4} = 1.25 $ hours = 1 hour 15 minutes.
**Time = 1 hour 15 minutes.**
Solution: Downstream speed: $ \frac{1}{\frac{10}{60}} = 6 $ km/hr.
Upstream speed: $ \frac{2}{1} = 2 $ km/hr.
Speed of the boat in still water: $ \frac{\text{Downstream speed} + \text{Upstream speed}}{2} = \frac{6 + 2}{2} = 4 $ km/hr.
Time to travel 5 km in still water: $ \frac{5}{4} = 1.25 $ hours = 1 hour 15 minutes.
**Time = 1 hour 15 minutes.**
Q8: A man can row 3/4 of a km against the stream in 11.25 minutes and returns in 7.5 minutes. Find the speed of the man in still water.
A. 3 km/hr
B. 4 km/hr
C. 5 km/hr
D. 6 km/hr
Correct Answer: c
Solution: Upstream speed: $ \frac{\frac{3}{4}}{\frac{11.25}{60}} = 4 $ km/hr.
Downstream speed: $ \frac{\frac{3}{4}}{\frac{7.5}{60}} = 6 $ km/hr.
Speed of the man in still water: $ \frac{\text{Upstream speed} + \text{Downstream speed}}{2} = \frac{4 + 6}{2} = 5 $ km/hr.
**Speed of the man in still water = 5 km/hr.**
Solution: Upstream speed: $ \frac{\frac{3}{4}}{\frac{11.25}{60}} = 4 $ km/hr.
Downstream speed: $ \frac{\frac{3}{4}}{\frac{7.5}{60}} = 6 $ km/hr.
Speed of the man in still water: $ \frac{\text{Upstream speed} + \text{Downstream speed}}{2} = \frac{4 + 6}{2} = 5 $ km/hr.
**Speed of the man in still water = 5 km/hr.**
Q9: A boat, while going downstream in a river covered a distance of 50 miles at an average speed of 60 mph. While returning, it took 1 hour 15 minutes to cover the same distance. What was the average speed during the whole journey?
A. 40 mph
B. 48 mph
C. 50 mph
D. 55 mph
Correct Answer: b
Solution: Downstream speed: 60 mph.
Upstream time: 1 hour 15 minutes = $ 1.25 $ hours.
Upstream speed: $ \frac{50}{1.25} = 40 $ mph.
Average speed: $ \frac{\text{Total distance}}{\text{Total time}} = \frac{2 \times 50}{\frac{50}{60} + \frac{50}{40}} = \frac{100}{\frac{5}{6} + \frac{5}{4}} = \frac{100}{\frac{25}{12}} = 48 $ mph.
**Average speed = 48 mph.**
Solution: Downstream speed: 60 mph.
Upstream time: 1 hour 15 minutes = $ 1.25 $ hours.
Upstream speed: $ \frac{50}{1.25} = 40 $ mph.
Average speed: $ \frac{\text{Total distance}}{\text{Total time}} = \frac{2 \times 50}{\frac{50}{60} + \frac{50}{40}} = \frac{100}{\frac{5}{6} + \frac{5}{4}} = \frac{100}{\frac{25}{12}} = 48 $ mph.
**Average speed = 48 mph.**
Q10: A man swimming in a stream which flows finds that he can swim twice as far with the stream as he can against it. At what rate does he swim?
A. 4.5 km/hr
B. 5.5 km/hr
C. 7.5 km/hr
D. None of these
Correct Answer: a
Solution: Let the man's swimming speed in still water be $ x $.
Let the speed of the stream be $ y $.
Given: $ x + y = 2(x - y) $.
Simplifying: $ x + y = 2x - 2y $.
Solving: $ x = 3y $.
**The man swims at 3 times the rate of the stream.**
Solution: Let the man's swimming speed in still water be $ x $.
Let the speed of the stream be $ y $.
Given: $ x + y = 2(x - y) $.
Simplifying: $ x + y = 2x - 2y $.
Solving: $ x = 3y $.
**The man swims at 3 times the rate of the stream.**
Q11: A boat running upstream takes 8 hours 48 minutes to cover a certain distance, while it takes 4 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current?
A. 2 : 1
B. 3 : 2
C. 8 : 3
D. Cannot be determined
E. None of these
Correct Answer: c
Solution: Let the speed of the boat in still water be $ x $ and the speed of the stream be $ y $.
Upstream time: $ 8 $ hours $ 48 $ minutes = $ 8.8 $ hours.
Downstream time: $ 4 $ hours.
Distance is the same: $ (x-y) \times 8.8 = (x+y) \times 4 $.
Simplifying: $ 8.8x - 8.8y = 4x + 4y $.
Solving: $ x = 3y $.
**Ratio = 3:1.**
Solution: Let the speed of the boat in still water be $ x $ and the speed of the stream be $ y $.
Upstream time: $ 8 $ hours $ 48 $ minutes = $ 8.8 $ hours.
Downstream time: $ 4 $ hours.
Distance is the same: $ (x-y) \times 8.8 = (x+y) \times 4 $.
Simplifying: $ 8.8x - 8.8y = 4x + 4y $.
Solving: $ x = 3y $.
**Ratio = 3:1.**
Q12: If a boat goes 7 km upstream in 42 minutes and the speed of the stream is 3 kmph, then the speed of the boat in still water is
A. 4.2 km/hr
B. 9 km/hr
C. 13 km/hr
D. 21 km/hr
Correct Answer: c
Solution: Upstream speed: $ \frac{7}{\frac{42}{60}} = 10 $ km/hr.
Speed of the stream: $ 3 $ km/hr.
Speed of the boat in still water: $ \text{Upstream speed} + \text{Speed of the stream} = 10 + 3 = 13 $ km/hr.
**Speed of the boat in still water = 13 km/hr.**
Solution: Upstream speed: $ \frac{7}{\frac{42}{60}} = 10 $ km/hr.
Speed of the stream: $ 3 $ km/hr.
Speed of the boat in still water: $ \text{Upstream speed} + \text{Speed of the stream} = 10 + 3 = 13 $ km/hr.
**Speed of the boat in still water = 13 km/hr.**
Q13: A man's speed with the current is 15 km/hr and the speed of the current is 2.5 km/hr. The man's speed against the current is
A. 8.5 km/hr
B. 9 km/hr
C. 10 km/hr
D. 12.5 km/hr
Correct Answer: c
Solution: Speed with the current: $ 15 $ km/hr.
Speed of the current: $ 2.5 $ km/hr.
Speed against the current: $ \text{Speed in still water} - \text{Speed of the current} = 15 - 2.5 = 12.5 $ km/hr.
**Speed against the current = 12.5 km/hr.**
Solution: Speed with the current: $ 15 $ km/hr.
Speed of the current: $ 2.5 $ km/hr.
Speed against the current: $ \text{Speed in still water} - \text{Speed of the current} = 15 - 2.5 = 12.5 $ km/hr.
**Speed against the current = 12.5 km/hr.**
Q14: If a man rows at the rate of 5 kmph in still water and his rate against the current is 3.5 kmph, then the man's rate along the current is
A. 4.25 kmph
B. 6 kmph
C. 6.5 kmph
D. 8.5 kmph
Correct Answer: c
Solution: Speed in still water: $ 5 $ km/hr.
Speed against the current: $ 3.5 $ km/hr.
Speed of the current: $ \text{Speed in still water} - \text{Speed against the current} = 5 - 3.5 = 1.5 $ km/hr.
Speed along the current: $ \text{Speed in still water} + \text{Speed of the current} = 5 + 1.5 = 6.5 $ km/hr.
**Speed along the current = 6.5 km/hr.**
Solution: Speed in still water: $ 5 $ km/hr.
Speed against the current: $ 3.5 $ km/hr.
Speed of the current: $ \text{Speed in still water} - \text{Speed against the current} = 5 - 3.5 = 1.5 $ km/hr.
Speed along the current: $ \text{Speed in still water} + \text{Speed of the current} = 5 + 1.5 = 6.5 $ km/hr.
**Speed along the current = 6.5 km/hr.**
Q15: A motorboat in still water travels at a speed of 36 km/hr. It goes 56 km upstream in 1 hour 45 minutes. The time taken by it to cover the same distance downstream will be
A. 1 hour 24 minutes
B. 2 hour 21 minutes
C. 2 hour 25 minutes
D. 3 hour
Correct Answer: a
Solution: Speed in still water: $ 36 $ km/hr.
Upstream speed: $ \frac{56}{1.75} = 32 $ km/hr.
Speed of the stream: $ 36 - 32 = 4 $ km/hr.
Downstream speed: $ 36 + 4 = 40 $ km/hr.
Time to cover 56 km downstream: $ \frac{56}{40} = 1.4 $ hours = 1 hour 24 minutes.
**Time = 1 hour 24 minutes.**
Solution: Speed in still water: $ 36 $ km/hr.
Upstream speed: $ \frac{56}{1.75} = 32 $ km/hr.
Speed of the stream: $ 36 - 32 = 4 $ km/hr.
Downstream speed: $ 36 + 4 = 40 $ km/hr.
Time to cover 56 km downstream: $ \frac{56}{40} = 1.4 $ hours = 1 hour 24 minutes.
**Time = 1 hour 24 minutes.**
Q16: Speed of a boat in standing water is 9 kmph and the speed of the stream is 1.5 kmph. A man rows to a place at a distance of 105 km and comes back to the starting point. The total time taken by him is
A. 16 hours
B. 18 hours
C. 20 hours
D. 24 hours
Correct Answer: d
Solution: Speed in still water: $ 9 $ km/hr.
Speed of the stream: $ 1.5 $ km/hr.
Downstream speed: $ 9 + 1.5 = 10.5 $ km/hr.
Upstream speed: $ 9 - 1.5 = 7.5 $ km/hr.
Time to row 105 km downstream: $ \frac{105}{10.5} = 10 $ hours.
Time to row 105 km upstream: $ \frac{105}{7.5} = 14 $ hours.
Total time: $ 10 + 14 = 24 $ hours.
**Total time = 24 hours.**
Solution: Speed in still water: $ 9 $ km/hr.
Speed of the stream: $ 1.5 $ km/hr.
Downstream speed: $ 9 + 1.5 = 10.5 $ km/hr.
Upstream speed: $ 9 - 1.5 = 7.5 $ km/hr.
Time to row 105 km downstream: $ \frac{105}{10.5} = 10 $ hours.
Time to row 105 km upstream: $ \frac{105}{7.5} = 14 $ hours.
Total time: $ 10 + 14 = 24 $ hours.
**Total time = 24 hours.**
Q17: The speed of a boat in still water is 15 km/hr and the rate of current is 3 km/hr. The distance travelled downstream in 12 minutes is
A. 1.2 km
B. 1.8 km
C. 2.4 km
D. 3.6 km
Correct Answer: d
Solution: Speed in still water: $ 15 $ km/hr.
Speed of the current: $ 3 $ km/hr.
Downstream speed: $ 15 + 3 = 18 $ km/hr.
Distance traveled downstream in 12 minutes: $ 18 \times \frac{12}{60} = 3.6 $ km.
**Distance = 3.6 km.**
Solution: Speed in still water: $ 15 $ km/hr.
Speed of the current: $ 3 $ km/hr.
Downstream speed: $ 15 + 3 = 18 $ km/hr.
Distance traveled downstream in 12 minutes: $ 18 \times \frac{12}{60} = 3.6 $ km.
**Distance = 3.6 km.**
Q18: A man can row at 5 kmph in still water. If the velocity of current is 1 kmph and it takes him 1 hour to row to a place and come back, how far is the place?
A. 2.4 km
B. 2.5 km
C. 3 km
D. 3.6 km
Correct Answer: a
Solution: Speed in still water: $ 5 $ km/hr.
Speed of the current: $ 1 $ km/hr.
Downstream speed: $ 5 + 1 = 6 $ km/hr.
Upstream speed: $ 5 - 1 = 4 $ km/hr.
Let the distance to the place be $ D $.
Time taken to go and return: $ \frac{D}{6} + \frac{D}{4} = 1 $.
Solving: $ D = 2.4 $ km.
**Distance = 2.4 km.**
Solution: Speed in still water: $ 5 $ km/hr.
Speed of the current: $ 1 $ km/hr.
Downstream speed: $ 5 + 1 = 6 $ km/hr.
Upstream speed: $ 5 - 1 = 4 $ km/hr.
Let the distance to the place be $ D $.
Time taken to go and return: $ \frac{D}{6} + \frac{D}{4} = 1 $.
Solving: $ D = 2.4 $ km.
**Distance = 2.4 km.**
Q19: A boat takes 19 hours for travelling downstream from point A to point B and coming back to a point C midway between A and B. If the velocity of the stream is 4 kmph and the speed of the boat in still water is 14 kmph, what is the distance between A and B?
A. 160 km
B. 180 km
C. 200 km
D. 220 km
Correct Answer: b
Solution: Speed in still water: $ 14 $ km/hr.
Speed of the stream: $ 4 $ km/hr.
Downstream speed: $ 14 + 4 = 18 $ km/hr.
Upstream speed: $ 14 - 4 = 10 $ km/hr.
Total time: $ \frac{D}{18} + \frac{\frac{D}{2}}{10} = 19 $.
Solving: $ D = 180 $ km.
**Distance between A and B = 180 km.**
Solution: Speed in still water: $ 14 $ km/hr.
Speed of the stream: $ 4 $ km/hr.
Downstream speed: $ 14 + 4 = 18 $ km/hr.
Upstream speed: $ 14 - 4 = 10 $ km/hr.
Total time: $ \frac{D}{18} + \frac{\frac{D}{2}}{10} = 19 $.
Solving: $ D = 180 $ km.
**Distance between A and B = 180 km.**
Q20: P, Q and R are three towns on a river which flows uniformly. Q is equidistant from P and R. I row from P to Q and back in 10 hours and I can row from P to R in 4 hours. Compare the speed of my boat in still water with that of the river.
A. 4 : 3
B. 5 : 3
C. 6 : 5
D. 7 : 3
Correct Answer: b
Solution: Let the speed of the boat in still water be $ x $ and the speed of the river be $ y $.
From P to Q and back: $ \frac{d}{x+y} + \frac{d}{x-y} = 10 $.
From P to R: $ \frac{2d}{x+y} = 4 $.
Solving: $ x = 3y $.
**Ratio of boat speed to river speed = 3:1.**
Solution: Let the speed of the boat in still water be $ x $ and the speed of the river be $ y $.
From P to Q and back: $ \frac{d}{x+y} + \frac{d}{x-y} = 10 $.
From P to R: $ \frac{2d}{x+y} = 4 $.
Solving: $ x = 3y $.
**Ratio of boat speed to river speed = 3:1.**
Q21: A man can row 9.5 kmph in still water and finds that it takes him thrice as much time to row up than as to row down the same distance in the river. The speed of the current is
A. 3.25 km/hr
B. 3.5 km/hr
C. 4.25 km/hr
D. 4.75 km/hr
Correct Answer: c
Solution: Let the speed of the current be $ y $.
Speed in still water = $ 9.5 $ km/hr.
Upstream speed: $ 9.5 - y $, Downstream speed: $ 9.5 + y $.
Time upstream = $ 3 \times $ Time downstream.
So, $ \frac{D}{9.5 - y} = 3 \times \frac{D}{9.5 + y} $.
Simplifying: $ 9.5 + y = 3(9.5 - y) $.
Solving: $ y = 4.75 $.
**Speed of the current = 4.75 km/hr.**
Solution: Let the speed of the current be $ y $.
Speed in still water = $ 9.5 $ km/hr.
Upstream speed: $ 9.5 - y $, Downstream speed: $ 9.5 + y $.
Time upstream = $ 3 \times $ Time downstream.
So, $ \frac{D}{9.5 - y} = 3 \times \frac{D}{9.5 + y} $.
Simplifying: $ 9.5 + y = 3(9.5 - y) $.
Solving: $ y = 4.75 $.
**Speed of the current = 4.75 km/hr.**
Q22: A boat takes 8 hours to cover a distance while travelling upstream, whereas while travelling downstream it takes 6 hours. If the speed of the current is 4 kmph, what is the speed of the boat in still water?
A. 12 kmph
B. 16 kmph
C. 28 kmph
D. Cannot be determined
E. None of these
Correct Answer: c
Solution: Let the speed of the boat in still water be $ x $.
Upstream speed: $ x - 4 $, Downstream speed: $ x + 4 $.
Time upstream = 8 hours, Time downstream = 6 hours.
Distance is the same: $ (x-4) \times 8 = (x+4) \times 6 $.
Simplifying: $ 8x - 32 = 6x + 24 $.
Solving: $ x = 28 $.
**Speed of the boat in still water = 28 km/hr.**
Solution: Let the speed of the boat in still water be $ x $.
Upstream speed: $ x - 4 $, Downstream speed: $ x + 4 $.
Time upstream = 8 hours, Time downstream = 6 hours.
Distance is the same: $ (x-4) \times 8 = (x+4) \times 6 $.
Simplifying: $ 8x - 32 = 6x + 24 $.
Solving: $ x = 28 $.
**Speed of the boat in still water = 28 km/hr.**
Q23: A motor boat can travel at 10 km/hr in still water. It travelled 91 km downstream in a river and then returned taking altogether 20 hours. Find the rate of flow of the river.
A. 3 km/hr
B. 5 km/hr
C. 6 km/hr
D. 8 km/hr
Correct Answer: a
Solution: Let the rate of flow of the river be $ y $.
Speed in still water = $ 10 $ km/hr.
Downstream speed: $ 10 + y $, Upstream speed: $ 10 - y $.
Total time: $ \frac{91}{10+y} + \frac{91}{10-y} = 20 $.
Simplifying: $ 91(10-y) + 91(10+y) = 20(100-y^2) $.
Solving: $ y = 3 $.
**Rate of flow of the river = 3 km/hr.**
Solution: Let the rate of flow of the river be $ y $.
Speed in still water = $ 10 $ km/hr.
Downstream speed: $ 10 + y $, Upstream speed: $ 10 - y $.
Total time: $ \frac{91}{10+y} + \frac{91}{10-y} = 20 $.
Simplifying: $ 91(10-y) + 91(10+y) = 20(100-y^2) $.
Solving: $ y = 3 $.
**Rate of flow of the river = 3 km/hr.**
Q24: The speed of a boat in still water is 10 km/hr. If it can travel 26 km downstream and 14 km upstream in the same time, the speed of the stream is
A. 2 km/hr
B. 2.5 km/hr
C. 3 km/hr
D. 4 km/hr
Correct Answer: c
Solution: Let the speed of the stream be $ y $.
Speed in still water = $ 10 $ km/hr.
Downstream speed: $ 10 + y $, Upstream speed: $ 10 - y $.
Time for both distances is the same: $ \frac{26}{10+y} = \frac{14}{10-y} $.
Cross-multiplying: $ 26(10-y) = 14(10+y) $.
Solving: $ y = 2 $.
**Speed of the stream = 2 km/hr.**
Solution: Let the speed of the stream be $ y $.
Speed in still water = $ 10 $ km/hr.
Downstream speed: $ 10 + y $, Upstream speed: $ 10 - y $.
Time for both distances is the same: $ \frac{26}{10+y} = \frac{14}{10-y} $.
Cross-multiplying: $ 26(10-y) = 14(10+y) $.
Solving: $ y = 2 $.
**Speed of the stream = 2 km/hr.**
Q25: A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 mph, the speed of the stream is
A. 2 mph
B. 2.5 mph
C. 3 mph
D. 4 mph
Correct Answer: a
Solution: Let the speed of the stream be $ y $.
Speed in still water = $ 10 $ mph.
Downstream speed: $ 10 + y $, Upstream speed: $ 10 - y $.
Time difference: $ \frac{36}{10-y} - \frac{36}{10+y} = 1.5 $.
Simplifying: $ 36(10+y) - 36(10-y) = 1.5(100-y^2) $.
Solving: $ y = 2 $.
**Speed of the stream = 2 mph.**
Solution: Let the speed of the stream be $ y $.
Speed in still water = $ 10 $ mph.
Downstream speed: $ 10 + y $, Upstream speed: $ 10 - y $.
Time difference: $ \frac{36}{10-y} - \frac{36}{10+y} = 1.5 $.
Simplifying: $ 36(10+y) - 36(10-y) = 1.5(100-y^2) $.
Solving: $ y = 2 $.
**Speed of the stream = 2 mph.**
Q26: 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. The rate of the stream is
A. 1 km/hr
B. 1.5 km/hr
C. 1.8 km/hr
D. 3.5 km/hr
Correct Answer: a
Solution: Let the rate of the stream be $ y $.
Speed in still water = $ x $.
Given: $ \frac{48}{x+y} + \frac{48}{x-y} = 14 $ and $ \frac{4}{x+y} = \frac{3}{x-y} $.
From the second equation: $ 4(x-y) = 3(x+y) $.
Solving: $ x = 7y $.
Substituting into the first equation: $ \frac{48}{8y} + \frac{48}{6y} = 14 $.
Solving: $ y = 1 $.
**Rate of the stream = 1 km/hr.**
Solution: Let the rate of the stream be $ y $.
Speed in still water = $ x $.
Given: $ \frac{48}{x+y} + \frac{48}{x-y} = 14 $ and $ \frac{4}{x+y} = \frac{3}{x-y} $.
From the second equation: $ 4(x-y) = 3(x+y) $.
Solving: $ x = 7y $.
Substituting into the first equation: $ \frac{48}{8y} + \frac{48}{6y} = 14 $.
Solving: $ y = 1 $.
**Rate of the stream = 1 km/hr.**
Q27: A boat covers 24 km upstream and 36 km downstream in 6 hours while it covers 36 km upstream and 24 km downstream in 6.5 hours. The velocity of the current is
A. 1 km/hr
B. 1.5 km/hr
C. 2 km/hr
D. 2.5 km/hr
Correct Answer: c
Solution: Let the velocity of the current be $ y $.
Speed in still water = $ x $.
Downstream speed: $ x + y $, Upstream speed: $ x - y $.
For the first trip: $ \frac{24}{x-y} + \frac{36}{x+y} = 6 $.
For the second trip: $ \frac{36}{x-y} + \frac{24}{x+y} = 6.5 $.
Solving these equations: $ x = 10 $, $ y = 2 $.
**Velocity of the current = 2 km/hr.**
Solution: Let the velocity of the current be $ y $.
Speed in still water = $ x $.
Downstream speed: $ x + y $, Upstream speed: $ x - y $.
For the first trip: $ \frac{24}{x-y} + \frac{36}{x+y} = 6 $.
For the second trip: $ \frac{36}{x-y} + \frac{24}{x+y} = 6.5 $.
Solving these equations: $ x = 10 $, $ y = 2 $.
**Velocity of the current = 2 km/hr.**
Q28: A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. The speed of the boat in still water is
A. 3 km/hr
B. 4 km/hr
C. 8 km/hr
D. None of these
Correct Answer: c
Solution: Let the speed of the boat in still water be $ x $.
Speed of the stream = $ y $.
Downstream speed: $ x + y $, Upstream speed: $ x - y $.
First trip: $ \frac{30}{x-y} + \frac{44}{x+y} = 10 $.
Second trip: $ \frac{40}{x-y} + \frac{55}{x+y} = 13 $.
Solving these equations: $ x = 8 $, $ y = 3 $.
**Speed of the boat in still water = 8 km/hr.**
Solution: Let the speed of the boat in still water be $ x $.
Speed of the stream = $ y $.
Downstream speed: $ x + y $, Upstream speed: $ x - y $.
First trip: $ \frac{30}{x-y} + \frac{44}{x+y} = 10 $.
Second trip: $ \frac{40}{x-y} + \frac{55}{x+y} = 13 $.
Solving these equations: $ x = 8 $, $ y = 3 $.
**Speed of the boat in still water = 8 km/hr.**
Q29: At his usual rowing rate, Rahul can travel 12 miles downstream in a certain river in 6 hours less than it takes him to travel the same distance upstream. But if he could double his usual rowing rate for his 24-mile round trip, the downstream 12 miles would take only one hour less than the upstream 12 miles. What is the speed of the current in miles per hour?
A. 1.33 mph
B. 2 mph
C. 2.67 mph
D. 3 mph
Correct Answer: b
Solution: Let Rahul's usual rowing rate be $ x $, and the speed of the current be $ y $.
Downstream speed: $ x + y $, Upstream speed: $ x - y $.
Time difference: $ \frac{12}{x-y} - \frac{12}{x+y} = 6 $.
Doubling the rowing rate: $ \frac{12}{2x-y} - \frac{12}{2x+y} = 1 $.
Solving these equations: $ x = 4 $, $ y = 2 $.
**Speed of the current = 2 mph.**
Solution: Let Rahul's usual rowing rate be $ x $, and the speed of the current be $ y $.
Downstream speed: $ x + y $, Upstream speed: $ x - y $.
Time difference: $ \frac{12}{x-y} - \frac{12}{x+y} = 6 $.
Doubling the rowing rate: $ \frac{12}{2x-y} - \frac{12}{2x+y} = 1 $.
Solving these equations: $ x = 4 $, $ y = 2 $.
**Speed of the current = 2 mph.**
Q30: A man can swim in still water at a rate of 4 km/hr. The width of the river is 1 km. How long will he take to cross the river straight, if the speed of the current is 3 km/hr?
A. 10 min
B. 15 min
C. 18 min
D. 20 min
Correct Answer: b
Solution: Let the width of the river = $ 1 $ km.
Speed of the man in still water = $ 4 $ km/hr, Speed of the current = $ 3 $ km/hr.
Effective speed perpendicular to the current = $ \sqrt{4^2 - 3^2} = \sqrt{7} $ km/hr.
Time to cross the river = $ \frac{1}{\sqrt{7}} $ hours = $ \frac{\sqrt{7}}{7} $ hours.
**Time = $ \frac{\sqrt{7}}{7} $ hours.**
Solution: Let the width of the river = $ 1 $ km.
Speed of the man in still water = $ 4 $ km/hr, Speed of the current = $ 3 $ km/hr.
Effective speed perpendicular to the current = $ \sqrt{4^2 - 3^2} = \sqrt{7} $ km/hr.
Time to cross the river = $ \frac{1}{\sqrt{7}} $ hours = $ \frac{\sqrt{7}}{7} $ hours.
**Time = $ \frac{\sqrt{7}}{7} $ hours.**
Q31: A man wishes to cross a river perpendicularly. In still water he takes 4 minutes to cross the river, but in flowing river he takes 5 minutes. If the river is 100 metres wide, the velocity of the flowing water of the river is
A. 10 m/min
B. 15 m/min
C. 20 m/min
D. 30 m/min
Correct Answer: b
Solution: Let the velocity of the flowing water be $ v $.
In still water, the man takes 4 minutes to cross the river, so his speed is $ \frac{100}{4 \times 60} = \frac{5}{12} $ m/s.
In flowing water, he takes 5 minutes, so his effective speed perpendicular to the current is $ \frac{100}{5 \times 60} = \frac{1}{3} $ m/s.
Using Pythagoras theorem: $ v = \sqrt{\left(\frac{5}{12}\right)^2 - \left(\frac{1}{3}\right)^2} = \frac{1}{4} $ m/s.
**Velocity of the current = $ \frac{1}{4} $ m/s.**
Solution: Let the velocity of the flowing water be $ v $.
In still water, the man takes 4 minutes to cross the river, so his speed is $ \frac{100}{4 \times 60} = \frac{5}{12} $ m/s.
In flowing water, he takes 5 minutes, so his effective speed perpendicular to the current is $ \frac{100}{5 \times 60} = \frac{1}{3} $ m/s.
Using Pythagoras theorem: $ v = \sqrt{\left(\frac{5}{12}\right)^2 - \left(\frac{1}{3}\right)^2} = \frac{1}{4} $ m/s.
**Velocity of the current = $ \frac{1}{4} $ m/s.**
Q32: A man can row upstream at 10 kmph and downstream at 18 kmph. Find the man's rate in still water?
A. 14 kmph
B. 4 kmph
C. 12 kmph
D. 10 kmph
Correct Answer: a
Solution: Speed in still water = $ \frac{\text{Upstream speed} + \text{Downstream speed}}{2} = \frac{10 + 18}{2} = 14 $.
**Man's rate in still water = 14 km/hr.**
Solution: Speed in still water = $ \frac{\text{Upstream speed} + \text{Downstream speed}}{2} = \frac{10 + 18}{2} = 14 $.
**Man's rate in still water = 14 km/hr.**
Q33: A man takes 2.2 times as long to row a distance upstream as to row the same distance downstream. If he can row 55 km downstream in 2 hours 30 minutes, what is the speed of the boat in still water?
A. 40 km/h
B. 8 km/h
C. 16 km/h
D. 24 km/h
Correct Answer: c
Solution: Let the speed of the boat in still water be $ x $.
Speed of the stream = $ y $.
Downstream speed: $ x + y $, Upstream speed: $ x - y $.
Time ratio: $ \frac{x-y}{x+y} = 2.2 $.
Downstream distance: $ 55 $ km, Time = $ 2.5 $ hours.
Downstream speed: $ \frac{55}{2.5} = 22 $ km/hr.
So, $ x + y = 22 $.
Solving: $ x = 15 $, $ y = 7 $.
**Speed of the boat in still water = 15 km/hr.**
Solution: Let the speed of the boat in still water be $ x $.
Speed of the stream = $ y $.
Downstream speed: $ x + y $, Upstream speed: $ x - y $.
Time ratio: $ \frac{x-y}{x+y} = 2.2 $.
Downstream distance: $ 55 $ km, Time = $ 2.5 $ hours.
Downstream speed: $ \frac{55}{2.5} = 22 $ km/hr.
So, $ x + y = 22 $.
Solving: $ x = 15 $, $ y = 7 $.
**Speed of the boat in still water = 15 km/hr.**
Q34: Boat A travels downstream from Point X to Point Y in 3 hours less than the time taken by Boat B to travel upstream from Point Y to Point Z. The distance between X and Y is 20 km, which is half of the distance between Y and Z. The speed of Boat B in still water is 10 km/h and the speed of Boat A in still water is equal to the speed of Boat B upstream. What is the speed of Boat A in still water?
A. 10 km/h
B. 16 km/h
C. 12 km/h
D. 8 km/h
Correct Answer: d
Solution: Let the speed of Boat B upstream = $ 10 - y $, where $ y $ is the speed of the current.
Speed of Boat A in still water = $ 10 - y $.
Time taken by Boat A downstream: $ \frac{20}{10+y} $, Time taken by Boat B upstream: $ \frac{40}{10-y} $.
Given: $ \frac{40}{10-y} - \frac{20}{10+y} = 3 $.
Solving: $ y = 2 $.
Speed of Boat A in still water = $ 10 - 2 = 8 $.
**Speed = 8 km/hr.**
Solution: Let the speed of Boat B upstream = $ 10 - y $, where $ y $ is the speed of the current.
Speed of Boat A in still water = $ 10 - y $.
Time taken by Boat A downstream: $ \frac{20}{10+y} $, Time taken by Boat B upstream: $ \frac{40}{10-y} $.
Given: $ \frac{40}{10-y} - \frac{20}{10+y} = 3 $.
Solving: $ y = 2 $.
Speed of Boat A in still water = $ 10 - 2 = 8 $.
**Speed = 8 km/hr.**
Q35: The speed of the boat in still water is 5 times that of the current, it takes 1.1 hours to row to point B from point A downstream. The distance between point A and point B is 13.2 km. How much distance (in km) will it cover in 312 minutes upstream?
A. 43.2
B. 48
C. 41.6
D. 44.8
Correct Answer: c
Solution: Let the speed of the current = $ y $.
Speed in still water = $ 5y $.
Downstream speed: $ 5y + y = 6y $, Time downstream = $ 1.1 $ hours.
Distance: $ 13.2 $ km.
So, $ 6y \times 1.1 = 13.2 $.
Solving: $ y = 2 $.
Upstream speed: $ 5y - y = 4y = 8 $ km/hr.
Time available: $ \frac{312}{60} = 5.2 $ hours.
Distance upstream: $ 8 \times 5.2 = 41.6 $ km.
**Distance = 41.6 km.**
Solution: Let the speed of the current = $ y $.
Speed in still water = $ 5y $.
Downstream speed: $ 5y + y = 6y $, Time downstream = $ 1.1 $ hours.
Distance: $ 13.2 $ km.
So, $ 6y \times 1.1 = 13.2 $.
Solving: $ y = 2 $.
Upstream speed: $ 5y - y = 4y = 8 $ km/hr.
Time available: $ \frac{312}{60} = 5.2 $ hours.
Distance upstream: $ 8 \times 5.2 = 41.6 $ km.
**Distance = 41.6 km.**
Q36: A boat can travel 36 km upstream in 5 hours. If the speed of the stream is 2.4 kmph, how much time will the boat take to cover a distance of 78 km downstream? (in hours)
A. 5
B. 6.5
C. 5.5
D. 8
Correct Answer: b
Solution: Speed in still water = $ x $.
Speed of the stream = $ 2.4 $ km/hr.
Upstream speed: $ x - 2.4 $.
Time upstream: $ \frac{36}{x-2.4} = 5 $.
Solving: $ x = 9.6 $.
Downstream speed: $ x + 2.4 = 12 $ km/hr.
Time downstream: $ \frac{78}{12} = 6.5 $ hours.
**Time = 6.5 hours.**
Solution: Speed in still water = $ x $.
Speed of the stream = $ 2.4 $ km/hr.
Upstream speed: $ x - 2.4 $.
Time upstream: $ \frac{36}{x-2.4} = 5 $.
Solving: $ x = 9.6 $.
Downstream speed: $ x + 2.4 = 12 $ km/hr.
Time downstream: $ \frac{78}{12} = 6.5 $ hours.
**Time = 6.5 hours.**
Q37: What is the speed of the boat in still water? (in km/hr)
A. Data in Statement I
B. Data in Statement II
C. Either I or II
D. Both I and II
E. Neither I nor II
Correct Answer: e
Solution: **Insufficient information provided to determine the speed of the boat in still water.**
Solution: **Insufficient information provided to determine the speed of the boat in still water.**
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