▶1. A boat goes 15 km upstream in 3 hours. If the stream speed is 2 km/h, find the boat's speed in still water.
Answer: 7 km/h
Explanation:
- Step 1: The boat travels 15 km upstream in 3 hours.
- Step 2: Upstream speed = Distance / Time = 15 km / 3 h = 5 km/h.
- Step 3: Let the boat's speed in still water be b km/h, and the stream speed be s = 2 km/h.
- Step 4: The formula for upstream speed is:
Upstream speed = Boat speed in still water - Stream speed
So, b - 2 = 5
- Step 5: Solving for b:
b = 5 + 2 = 7 km/h.
- Conclusion: The boat's speed in still water is 7 km/h.
▶2. A boat's downstream speed is 20 km/h, and upstream speed is 12 km/h. Find the stream speed.
Answer: 4 km/h
Explanation:
- Step 1: Downstream speed = 20 km/h, Upstream speed = 12 km/h.
- Step 2: Let the boat's speed in still water be b km/h, and the stream speed be s km/h.
- Step 3: The formulas are:
Downstream speed = b + s
Upstream speed = b - s
- Step 4: The stream speed is half the difference between downstream and upstream speeds:
s = (Downstream speed - Upstream speed) / 2
s = (20 - 12) / 2 = 8 / 2 = 4 km/h.
- Conclusion: The stream speed is 4 km/h.
▶3. In still water, a boat's speed is 10 km/h. The stream flows at 3 km/h. Find its downstream speed.
Answer: 13 km/h
Explanation:
- Step 1: Boat speed in still water = 10 km/h, Stream speed = 3 km/h.
- Step 2: The formula for downstream speed is:
Downstream speed = Boat speed in still water + Stream speed
Downstream speed = 10 + 3 = 13 km/h.
- Conclusion: The downstream speed is 13 km/h.
▶4. A man rows 12 km downstream in 2 hours. If the stream speed is 1 km/h, find his rowing speed in still water.
Answer: 5 km/h
Explanation:
- Step 1: Downstream distance = 12 km, Time = 2 hours.
- Step 2: Downstream speed = Distance / Time = 12 / 2 = 6 km/h.
- Step 3: Let the man's rowing speed in still water be b km/h, and stream speed s = 1 km/h.
- Step 4: Downstream speed = b + s → b + 1 = 6
- Step 5: Solving for b:
b = 6 - 1 = 5 km/h.
- Conclusion: The man's rowing speed in still water is 5 km/h.
▶5. A boat covers 24 km upstream in 4 hours. If the stream speed is 2 km/h, find the boat's speed in still water.
Answer: 8 km/h
Explanation:
- Step 1: Upstream distance = 24 km, Time = 4 hours.
- Step 2: Upstream speed = Distance / Time = 24 / 4 = 6 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s = 2 km/h.
- Step 4: Upstream speed = b - s → b - 2 = 6
- Step 5: Solving for b:
b = 6 + 2 = 8 km/h.
- Conclusion: The boat's speed in still water is 8 km/h.
▶6. The downstream speed is 18 km/h, and the stream speed is 3 km/h. Find the boat's speed in still water.
Answer: 15 km/h
Explanation:
- Step 1: Downstream speed = 18 km/h, Stream speed = 3 km/h.
- Step 2: Let the boat's speed in still water be b km/h.
- Step 3: Downstream speed = b + s → b + 3 = 18
- Step 4: Solving for b:
b = 18 - 3 = 15 km/h.
- Conclusion: The boat's speed in still water is 15 km/h.
▶7. A boat goes 30 km downstream in 3 hours and 15 km upstream in 5 hours. Find the stream speed.
Answer: 3.5 km/h
Explanation:
- Step 1: Downstream: 30 km in 3 hours → Downstream speed = 30 / 3 = 10 km/h.
- Step 2: Upstream: 15 km in 5 hours → Upstream speed = 15 / 5 = 3 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 10
Upstream speed = b - s = 3
- Step 5: To find stream speed, use:
s = (Downstream speed - Upstream speed) / 2
s = (10 - 3) / 2 = 7 / 2 = 3.5 km/h.
- Conclusion: The stream speed is 3.5 km/h.
▶8. A man rows 8 km downstream in 1 hour and 2 km upstream in 1 hour. Find the stream speed.
Answer: 3 km/h
Explanation:
- Step 1: Downstream speed = 8 km / 1 h = 8 km/h.
- Step 2: Upstream speed = 2 km / 1 h = 2 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 8
Upstream speed = b - s = 2
- Step 5: Stream speed is:
s = (Downstream speed - Upstream speed) / 2
s = (8 - 2) / 2 = 6 / 2 = 3 km/h.
- Conclusion: The stream speed is 3 km/h.
▶9. A boat takes twice as long to go upstream as to go downstream the same distance. If stream speed is 4 km/h, find the boat's speed in still water.
Answer: 12 km/h
Explanation:
- Step 1: Let the distance be d km.
- Step 2: Let the boat's speed in still water be b km/h, stream speed s = 4 km/h.
- Step 3: Upstream speed = b - s, Downstream speed = b + s.
- Step 4: Time upstream = d / (b - 4), Time downstream = d / (b + 4).
- Step 5: Given: Time upstream = 2 × Time downstream.
d / (b - 4) = 2 × [d / (b + 4)]
- Step 6: Cancel d (since d ≠ 0):
1 / (b - 4) = 2 / (b + 4)
- Step 7: Cross-multiply:
b + 4 = 2(b - 4)
b + 4 = 2b - 8
2b - b = 4 + 8 → b = 12 km/h.
- Conclusion: The boat's speed in still water is 12 km/h.
▶10. The speed of a boat in still water is 9 km/h. The stream flows at 1.5 km/h. Find the time to cover 21 km downstream.
Answer: 2 hours
Explanation:
- Step 1: Boat speed in still water = 9 km/h, Stream speed = 1.5 km/h.
- Step 2: Downstream speed = Boat speed + Stream speed = 9 + 1.5 = 10.5 km/h.
- Step 3: Time = Distance / Speed = 21 km / 10.5 km/h = 2 hours.
- Conclusion: It will take 2 hours to cover 21 km downstream.
▶11. A boat covers 40 km downstream in 5 hours and 20 km upstream in 4 hours. Find the boat's speed in still water.
Answer: 6.5 km/h
Explanation:
- Step 1: Downstream: 40 km in 5 hours → Downstream speed = 40 / 5 = 8 km/h.
- Step 2: Upstream: 20 km in 4 hours → Upstream speed = 20 / 4 = 5 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 8
Upstream speed = b - s = 5
- Step 5: The boat's speed in still water is the average of downstream and upstream speeds:
b = (Downstream speed + Upstream speed) / 2
b = (8 + 5) / 2 = 13 / 2 = 6.5 km/h.
- Conclusion: The boat's speed in still water is 6.5 km/h.
▶12. A boat travels downstream for 3 hours and covers 45 km. Upstream, it covers 24 km in 4 hours. Find the stream speed.
Answer: 4.5 km/h
Explanation:
- Step 1: Downstream: 45 km in 3 hours → Downstream speed = 45 / 3 = 15 km/h.
- Step 2: Upstream: 24 km in 4 hours → Upstream speed = 24 / 4 = 6 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 15
Upstream speed = b - s = 6
- Step 5: The stream speed is half the difference between downstream and upstream speeds:
s = (Downstream speed - Upstream speed) / 2
s = (15 - 6) / 2 = 9 / 2 = 4.5 km/h.
- Conclusion: The stream speed is 4.5 km/h.
▶13. In still water, a boat's speed is 12 km/h. The stream speed is x km/h. If downstream speed is 15 km/h, find x.
Answer: 3 km/h
Explanation:
- Step 1: Boat speed in still water = 12 km/h, Downstream speed = 15 km/h.
- Step 2: Let stream speed = x km/h.
- Step 3: Downstream speed = Boat speed + Stream speed → 12 + x = 15
- Step 4: Solving for x:
x = 15 - 12 = 3 km/h.
- Conclusion: The stream speed is 3 km/h.
▶14. A man can row 6 km/h in still water. The stream flows at 2 km/h. Find the time to row 16 km downstream and back.
Answer: 6 hours
Explanation:
- Step 1: Rowing speed in still water = 6 km/h, Stream speed = 2 km/h.
- Step 2: Downstream speed = 6 + 2 = 8 km/h.
- Step 3: Upstream speed = 6 - 2 = 4 km/h.
- Step 4: Time to row 16 km downstream = 16 / 8 = 2 hours.
- Step 5: Time to row 16 km upstream = 16 / 4 = 4 hours.
- Step 6: Total time = 2 + 4 = 6 hours.
- Conclusion: It will take 6 hours to row 16 km downstream and back.
▶15. The ratio of upstream to downstream speeds is 2:5. If the stream speed is 6 km/h, find the boat's speed in still water.
Answer: 14 km/h
Explanation:
- Step 1: Let upstream speed = 2x, downstream speed = 5x.
- Step 2: Let the boat's speed in still water be b km/h, stream speed s = 6 km/h.
- Step 3: Upstream speed = b - s = 2x
Downstream speed = b + s = 5x
- Step 4: Set up the equations:
b - 6 = 2x
b + 6 = 5x
- Step 5: Subtract the first from the second:
(b + 6) - (b - 6) = 5x - 2x → 12 = 3x → x = 4
- Step 6: Substitute x = 4 into b - 6 = 2x:
b - 6 = 8 → b = 14 km/h.
- Conclusion: The boat's speed in still water is 14 km/h.
▶16. A boat goes 30 km upstream and returns in 8 hours. If upstream speed is 5 km/h and downstream speed is 10 km/h, find the distance one way.
Answer: 26.67 km
Explanation:
- Step 1: Let the one-way distance be d km.
- Step 2: Upstream speed = 5 km/h, Downstream speed = 10 km/h.
- Step 3: Time upstream = d / 5, Time downstream = d / 10.
- Step 4: Total time = 8 hours:
d / 5 + d / 10 = 8
- Step 5: Find common denominator:
(2d + d) / 10 = 8 → 3d / 10 = 8
- Step 6: Solve for d:
3d = 80 → d = 80 / 3 ≈ 26.67 km.
- Conclusion: The one-way distance is approximately 26.67 km.
▶17. A boat covers 18 km downstream in 2 hours. The return journey takes 4.5 hours. Find the stream speed.
Answer: 2.5 km/h
Explanation:
- Step 1: Downstream: 18 km in 2 hours → Downstream speed = 18 / 2 = 9 km/h.
- Step 2: Upstream: 18 km in 4.5 hours → Upstream speed = 18 / 4.5 = 4 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 9
Upstream speed = b - s = 4
- Step 5: The stream speed is half the difference between downstream and upstream speeds:
s = (Downstream speed - Upstream speed) / 2
s = (9 - 4) / 2 = 5 / 2 = 2.5 km/h.
- Conclusion: The stream speed is 2.5 km/h.
▶18. The time to row 21 km downstream is 3 hours. The time to row the same distance upstream is 7 hours. Find the boat's speed in still water.
Answer: 5 km/h
Explanation:
- Step 1: Downstream: 21 km in 3 hours → Downstream speed = 21 / 3 = 7 km/h.
- Step 2: Upstream: 21 km in 7 hours → Upstream speed = 21 / 7 = 3 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 7
Upstream speed = b - s = 3
- Step 5: The boat's speed in still water is the average of downstream and upstream speeds:
b = (Downstream speed + Upstream speed) / 2
b = (7 + 3) / 2 = 10 / 2 = 5 km/h.
- Conclusion: The boat's speed in still water is 5 km/h.
▶19. A man rows to a place 15 km away and back in 8 hours. If his speed in still water is 5 km/h and the stream flows at 1 km/h, is this possible?
Answer: Yes
Explanation:
- Step 1: Distance one way = 15 km.
- Step 2: Speed in still water = 5 km/h, Stream speed = 1 km/h.
- Step 3: Downstream speed = 5 + 1 = 6 km/h.
- Step 4: Upstream speed = 5 - 1 = 4 km/h.
- Step 5: Time downstream = 15 / 6 = 2.5 hours.
- Step 6: Time upstream = 15 / 4 = 3.75 hours.
- Step 7: Total time = 2.5 + 3.75 = 6.25 hours.
- Step 8: Since 6.25 hours < 8 hours, it is possible.
- Conclusion: Yes, it is possible for the man to complete the journey in 8 hours.
▶20. A boat takes 5 hours to cover 50 km downstream and 4 hours for 24 km upstream. Find the stream speed.
Answer: 2 km/h
Explanation:
- Step 1: Downstream: 50 km in 5 hours → Downstream speed = 50 / 5 = 10 km/h.
- Step 2: Upstream: 24 km in 4 hours → Upstream speed = 24 / 4 = 6 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 10
Upstream speed = b - s = 6
- Step 5: The stream speed is half the difference between downstream and upstream speeds:
s = (Downstream speed - Upstream speed) / 2
s = (10 - 6) / 2 = 4 / 2 = 2 km/h.
- Conclusion: The stream speed is 2 km/h.
▶21. A boat travels 20 km downstream in 2 hours. The return journey takes 5 hours. Find the boat's speed in still water.
Answer: 7 km/h
Explanation:
- Step 1: Downstream: 20 km in 2 hours → Downstream speed = 20 / 2 = 10 km/h.
- Step 2: Upstream: 20 km in 5 hours → Upstream speed = 20 / 5 = 4 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 10
Upstream speed = b - s = 4
- Step 5: The boat's speed in still water is the average of downstream and upstream speeds:
b = (Downstream speed + Upstream speed) / 2
b = (10 + 4) / 2 = 14 / 2 = 7 km/h.
- Conclusion: The boat's speed in still water is 7 km/h.
▶22. A man rows 24 km upstream and 36 km downstream in 6 hours. He can row 36 km upstream and 24 km downstream in 6.5 hours. Find the stream speed.
Answer: 2 km/h
Explanation:
- Step 1: Let upstream speed = u, downstream speed = d.
- Step 2: First trip: 24/u + 36/d = 6 hours.
- Step 3: Second trip: 36/u + 24/d = 6.5 hours.
- Step 4: Solve these equations simultaneously (by elimination or substitution):
Multiply first by 3: 72/u + 108/d = 18
Multiply second by 2: 72/u + 48/d = 13
Subtract: (108 - 48)/d = 18 - 13 → 60/d = 5 → d = 12 km/h.
- Step 5: Substitute d = 12 into first: 24/u + 36/12 = 6 → 24/u + 3 = 6 → 24/u = 3 → u = 8 km/h.
- Step 6: Let boat speed in still water = b, stream speed = s.
d = b + s = 12, u = b - s = 8
So, b = (12 + 8)/2 = 10, s = (12 - 8)/2 = 2 km/h.
- Conclusion: The stream speed is 2 km/h.
▶23. A boat takes 3 hours to go 15 km upstream and 2 hours for 18 km downstream. Find the speed of the current.
Answer: 2 km/h
Explanation:
- Step 1: Upstream: 15 km in 3 hours → Upstream speed = 15 / 3 = 5 km/h.
- Step 2: Downstream: 18 km in 2 hours → Downstream speed = 18 / 2 = 9 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 9
Upstream speed = b - s = 5
- Step 5: The stream speed is half the difference between downstream and upstream speeds:
s = (Downstream speed - Upstream speed) / 2
s = (9 - 5) / 2 = 4 / 2 = 2 km/h.
- Conclusion: The speed of the current is 2 km/h.
▶24. A boat covers 32 km downstream in 4 hours. The same boat covers 12 km upstream in 3 hours. Find the boat's speed in still water.
Answer: 6 km/h
Explanation:
- Step 1: Downstream: 32 km in 4 hours → Downstream speed = 32 / 4 = 8 km/h.
- Step 2: Upstream: 12 km in 3 hours → Upstream speed = 12 / 3 = 4 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 8
Upstream speed = b - s = 4
- Step 5: The boat's speed in still water is the average of downstream and upstream speeds:
b = (Downstream speed + Upstream speed) / 2
b = (8 + 4) / 2 = 12 / 2 = 6 km/h.
- Conclusion: The boat's speed in still water is 6 km/h.
▶25. The difference between downstream and upstream speeds is 8 km/h. If the boat's speed in still water is 14 km/h, find the stream speed.
Answer: 4 km/h
Explanation:
- Step 1: Let boat speed in still water = b = 14 km/h, stream speed = s.
- Step 2: Downstream speed = b + s, Upstream speed = b - s.
- Step 3: The difference between downstream and upstream speeds is:
(b + s) - (b - s) = 2s
- Step 4: Set 2s = 8 → s = 4 km/h.
- Conclusion: The stream speed is 4 km/h.
▶26. A boat goes 40 km downstream and returns in 12 hours. If downstream speed is 10 km/h and upstream speed is 8 km/h, find the distance one way.
Answer: 53.33 km
Explanation:
- Step 1: Let the one-way distance be d km.
- Step 2: Downstream speed = 10 km/h, Upstream speed = 8 km/h.
- Step 3: Time downstream = d / 10, Time upstream = d / 8.
- Step 4: Total time = 12 hours:
d / 10 + d / 8 = 12
- Step 5: Find common denominator:
(4d + 5d) / 40 = 12 → 9d / 40 = 12
- Step 6: Solve for d:
9d = 480 → d = 480 / 9 ≈ 53.33 km.
- Conclusion: The one-way distance is approximately 53.33 km.
▶27. A man rows 15 km upstream and 22 km downstream in 5 hours. He can row 20 km upstream and 44 km downstream in 9 hours. Find the stream speed.
Answer: No valid solution (check problem data)
Explanation:
- Step 1: Let upstream speed = u, downstream speed = d.
- Step 2: First trip: 15/u + 22/d = 5 hours.
- Step 3: Second trip: 20/u + 44/d = 9 hours.
- Step 4: Try to solve these equations. The solution leads to a negative stream speed, which is not possible.
- Conclusion: The problem data may be inconsistent or incorrect.
▶28. A boat takes 2 hours to travel 16 km downstream and 1 hour for 5 km upstream. Find the stream speed.
Answer: 1.5 km/h
Explanation:
- Step 1: Downstream: 16 km in 2 hours → Downstream speed = 16 / 2 = 8 km/h.
- Step 2: Upstream: 5 km in 1 hour → Upstream speed = 5 / 1 = 5 km/h.
- Step 3: Let the boat's speed in still water be b km/h, stream speed s km/h.
- Step 4: The formulas:
Downstream speed = b + s = 8
Upstream speed = b - s = 5
- Step 5: The stream speed is half the difference between downstream and upstream speeds:
s = (Downstream speed - Upstream speed) / 2
s = (8 - 5) / 2 = 3 / 2 = 1.5 km/h.
- Conclusion: The stream speed is 1.5 km/h.
▶29. The ratio of boat speed to stream speed is 5:1. If the boat covers 72 km downstream in 6 hours, find the upstream speed.
Answer: 8 km/h
Explanation:
- Step 1: Downstream: 72 km in 6 hours → Downstream speed = 72 / 6 = 12 km/h.
- Step 2: Let boat speed in still water = 5k, stream speed = k.
- Step 3: Downstream speed = 5k + k = 6k = 12 → k = 2.
- Step 4: Upstream speed = 5k - k = 4k = 4 × 2 = 8 km/h.
- Conclusion: The upstream speed is 8 km/h.
▶30. A man can row 8 km/h in still water. He rows to a place 15 km away and back in 5 hours. Find the stream speed.
Answer: 4 km/h
Explanation:
- Step 1: Let stream speed = s km/h.
- Step 2: Time downstream = 15 / (8 + s), Time upstream = 15 / (8 - s).
- Step 3: Total time = 5 hours:
15 / (8 + s) + 15 / (8 - s) = 5
- Step 4: Multiply both sides by (8 + s)(8 - s):
15(8 - s) + 15(8 + s) = 5[(8 + s)(8 - s)]
15 × 8 - 15s + 15 × 8 + 15s = 5(64 - s^2)
120 = 320 - 5s^2
5s^2 = 320 - 120 = 200 → s^2 = 40 → s = 4 (since speed is positive).
- Conclusion: The stream speed is 4 km/h.
▶31. A boat takes 5 hours for 20 km upstream and 3 hours for 30 km downstream. Find the distance it can travel in still water in 4 hours.
Answer: 28 km
Explanation:
- Step 1: Upstream: 20 km in 5 hours → Upstream speed = 20 / 5 = 4 km/h.
- Step 2: Downstream: 30 km in 3 hours → Downstream speed = 30 / 3 = 10 km/h.
- Step 3: Boat speed in still water = (Upstream speed + Downstream speed) / 2 = (4 + 10) / 2 = 7 km/h.
- Step 4: In 4 hours, distance = 7 × 4 = 28 km.
- Conclusion: The boat can travel 28 km in still water in 4 hours.
▶32. The time for 30 km downstream is 2 hours. The time for 18 km upstream is 3 hours. Find the stream speed.
Answer: 4.5 km/h
Explanation:
- Step 1: Downstream: 30 km in 2 hours → Downstream speed = 30 / 2 = 15 km/h.
- Step 2: Upstream: 18 km in 3 hours → Upstream speed = 18 / 3 = 6 km/h.
- Step 3: Stream speed = (Downstream speed - Upstream speed) / 2 = (15 - 6) / 2 = 4.5 km/h.
- Conclusion: The stream speed is 4.5 km/h.
▶33. A boat covers 24 km upstream in 3 hours and 20 km downstream in 2 hours. Find the speed of the boat in still water.
Answer: 9 km/h
Explanation:
- Step 1: Upstream: 24 km in 3 hours → Upstream speed = 24 / 3 = 8 km/h.
- Step 2: Downstream: 20 km in 2 hours → Downstream speed = 20 / 2 = 10 km/h.
- Step 3: Boat speed in still water = (Upstream speed + Downstream speed) / 2 = (8 + 10) / 2 = 9 km/h.
- Conclusion: The boat's speed in still water is 9 km/h.
▶34. A man rows 12 km with the stream in 1.5 hours and 8 km against the stream in 2 hours. Find his rowing speed in still water.
Answer: 6 km/h
Explanation:
- Step 1: Downstream: 12 km in 1.5 hours → Downstream speed = 12 / 1.5 = 8 km/h.
- Step 2: Upstream: 8 km in 2 hours → Upstream speed = 8 / 2 = 4 km/h.
- Step 3: Rowing speed in still water = (Downstream speed + Upstream speed) / 2 = (8 + 4) / 2 = 6 km/h.
- Conclusion: The man's rowing speed in still water is 6 km/h.
▶35. A boat goes 60 km downstream in 4 hours and 20 km upstream in 5 hours. Find the stream speed.
Answer: 5.5 km/h
Explanation:
- Step 1: Downstream: 60 km in 4 hours → Downstream speed = 60 / 4 = 15 km/h.
- Step 2: Upstream: 20 km in 5 hours → Upstream speed = 20 / 5 = 4 km/h.
- Step 3: Stream speed = (Downstream speed - Upstream speed) / 2 = (15 - 4) / 2 = 5.5 km/h.
- Conclusion: The stream speed is 5.5 km/h.
▶36. The downstream speed is thrice the upstream speed. If the stream speed is 5 km/h, find the boat's speed in still water.
Answer: 10 km/h
Explanation:
- Step 1: Let boat speed in still water = b km/h, stream speed = s = 5 km/h.
- Step 2: Downstream speed = b + s, Upstream speed = b - s.
- Step 3: Given: Downstream speed = 3 × Upstream speed.
b + 5 = 3(b - 5)
- Step 4: Expand: b + 5 = 3b - 15 → 3b - b = 5 + 15 → 2b = 20 → b = 10 km/h.
- Conclusion: The boat's speed in still water is 10 km/h.
▶37. A boat takes 6 hours to cover 36 km downstream and 8 hours for 32 km upstream. Find the stream speed.
Answer: 1 km/h
Explanation:
- Step 1: Downstream: 36 km in 6 hours → Downstream speed = 36 / 6 = 6 km/h.
- Step 2: Upstream: 32 km in 8 hours → Upstream speed = 32 / 8 = 4 km/h.
- Step 3: Stream speed = (Downstream speed - Upstream speed) / 2 = (6 - 4) / 2 = 1 km/h.
- Conclusion: The stream speed is 1 km/h.
▶38. A man can row at 10 km/h in still water. If the stream flows at 2 km/h, how long will it take to row 48 km downstream?
Answer: 4 hours
Explanation:
- Step 1: Rowing speed in still water = 10 km/h, Stream speed = 2 km/h.
- Step 2: Downstream speed = 10 + 2 = 12 km/h.
- Step 3: Time = Distance / Speed = 48 / 12 = 4 hours.
- Conclusion: It will take 4 hours to row 48 km downstream.
▶39. A boat goes 30 km downstream and 20 km upstream in 7 hours. It goes 45 km downstream and 30 km upstream in 10.5 hours. Find the boat's speed in still water.
Answer: Insufficient data
Explanation:
- Step 1: Let upstream speed = u, downstream speed = d.
- Step 2: First trip: 30/d + 20/u = 7 hours.
- Step 3: Second trip: 45/d + 30/u = 10.5 hours.
- Step 4: These equations are dependent (not independent), so we cannot solve for unique values.
- Conclusion: More information is needed to solve for the boat's speed in still water.
▶40. The time to go 10 km upstream is equal to the time to go 20 km downstream. If stream speed is 3 km/h, find the boat's speed.
Answer: 9 km/h
Explanation:
- Step 1: Let boat speed in still water = b km/h, stream speed = s = 3 km/h.
- Step 2: Upstream speed = b - s, Downstream speed = b + s.
- Step 3: Time for 10 km upstream = 10 / (b - 3), Time for 20 km downstream = 20 / (b + 3).
- Step 4: Set times equal: 10 / (b - 3) = 20 / (b + 3).
- Step 5: Cross-multiply: 10(b + 3) = 20(b - 3) → 10b + 30 = 20b - 60 → 20b - 10b = 30 + 60 → 10b = 90 → b = 9 km/h.
- Conclusion: The boat's speed in still water is 9 km/h.
▶41. A boat takes 5 hours to go 50 km downstream and return to the start. If the boat's speed is 12 km/h in still water, find the stream speed.
Answer: 4√6 km/h
Explanation:
- Step 1: Let stream speed = s km/h.
- Step 2: Downstream speed = 12 + s, Upstream speed = 12 - s.
- Step 3: Time downstream = 50 / (12 + s), Time upstream = 50 / (12 - s).
- Step 4: Total time = 5 hours:
50 / (12 + s) + 50 / (12 - s) = 5
- Step 5: Multiply both sides by (12 + s)(12 - s):
50(12 - s) + 50(12 + s) = 5[(12 + s)(12 - s)]
600 - 50s + 600 + 50s = 5(144 - s^2)
1200 = 720 - 5s^2
5s^2 = 720 - 1200 = -480 (should be 1200 - 720 = 480)
5s^2 = 1200 - 720 = 480 → s^2 = 96 → s = 4√6 km/h.
- Conclusion: The stream speed is 4√6 km/h.
▶42. Two boats start from opposite banks of a river. The boats travel at 8 km/h and 10 km/h in still water. The river flows at 2 km/h. If the river is 3.6 km wide, when do they meet?
Answer: 12 minutes
Explanation:
- Step 1: Boat A speed in still water = 8 km/h, Boat B = 10 km/h, Stream speed = 2 km/h.
- Step 2: Boat A downstream speed = 8 + 2 = 10 km/h.
- Step 3: Boat B upstream speed = 10 - 2 = 8 km/h.
- Step 4: Relative speed = 10 + 8 = 18 km/h (since they move towards each other).
- Step 5: Time to meet = Distance / Relative speed = 3.6 / 18 = 0.2 hours.
- Step 6: 0.2 hours × 60 = 12 minutes.
- Conclusion: The boats meet after 12 minutes.
▶43. A boat travels 30 km upstream and 44 km downstream in 10 hours. Next day, it travels 40 km upstream and 55 km downstream in 13 hours. Find the stream speed.
Answer: 3 km/h
Explanation:
- Step 1: Let upstream speed = u, downstream speed = d.
- Step 2: First trip: 30/u + 44/d = 10 hours.
- Step 3: Second trip: 40/u + 55/d = 13 hours.
- Step 4: Solve these equations (by elimination or substitution):
Multiply first by 4: 120/u + 176/d = 40
Multiply second by 3: 120/u + 165/d = 39
Subtract: (176 - 165)/d = 40 - 39 → 11/d = 1 → d = 11 km/h.
- Step 5: Substitute d = 11 into first: 30/u + 44/11 = 10 → 30/u + 4 = 10 → 30/u = 6 → u = 5 km/h.
- Step 6: Let boat speed in still water = b, stream speed = s.
d = b + s = 11, u = b - s = 5
So, b = (11 + 5)/2 = 8, s = (11 - 5)/2 = 3 km/h.
- Conclusion: The stream speed is 3 km/h.
▶44. A man rows to a place 48 km away and back in 14 hours. He can row 12 km downstream and 8 km upstream in 2.5 hours. Find the stream speed.
Answer: 10 km/h
Explanation:
- Step 1: Let downstream speed = d, upstream speed = u.
- Step 2: 12/d + 8/u = 2.5 hours.
- Step 3: Let boat speed in still water = b, stream speed = s.
- Step 4: d = b + s, u = b - s.
- Step 5: Solve for d and u using the equation above and the total time for 48 km each way:
Time downstream = 48/d, Time upstream = 48/u, Total = 14 hours.
- Step 6: Use the two equations to solve for s. (Here, d = 24, u = 4, so s = (24 - 4)/2 = 10 km/h.)
- Conclusion: The stream speed is 10 km/h.
▶45. A boat takes 90 minutes to go 12 km downstream and return 4 km. If the boat's speed is 10 km/h, find the stream speed.
Answer: 2 km/h or 10/3 km/h
Explanation:
- Step 1: Let stream speed = s km/h.
- Step 2: Downstream speed = 10 + s, Upstream speed = 10 - s.
- Step 3: Time for 12 km downstream = 12 / (10 + s).
- Step 4: Time for 4 km upstream = 4 / (10 - s).
- Step 5: Total time = 90 minutes = 1.5 hours:
12 / (10 + s) + 4 / (10 - s) = 1.5
- Step 6: Solve this equation for s. (It gives two possible values: s = 2 km/h or s = 10/3 km/h.)
- Conclusion: The stream speed can be 2 km/h or 10/3 km/h.
▶46. A boat takes 2 hours less for 60 km downstream than for 40 km upstream. If the boat's speed is 15 km/h, find the stream speed.
Answer: 10√10 - 25 km/h
Explanation:
- Step 1: Let stream speed = s km/h.
- Step 2: Downstream speed = 15 + s, Upstream speed = 15 - s.
- Step 3: Time downstream = 60 / (15 + s), Time upstream = 40 / (15 - s).
- Step 4: Given: Time upstream - Time downstream = 2 hours.
40 / (15 - s) - 60 / (15 + s) = 2
- Step 5: Solve this equation for s. (The answer is s = 10√10 - 25 km/h.)
- Conclusion: The stream speed is 10√10 - 25 km/h.
▶47. Two boats start towards each other from opposite banks. After passing each other, they take 4 hours and 1 hour to reach their destinations. If the river flows at 3 km/h, find the boat speeds if they are in ratio 2:1.
Answer: No valid solution (problem data inconsistent)
Explanation:
- Step 1: Let boat speeds in still water be 2x and x, stream speed = 3 km/h.
- Step 2: Set up equations based on the time taken after meeting.
- Step 3: The system leads to an invalid (impossible) solution.
- Conclusion: The problem data is inconsistent or incorrect.
▶48. A boat sails 15 km upstream and 21 km downstream in 6 hours. It sails 20 km upstream and 28 km downstream in 8 hours. Find the boat's speed in still water.
Answer: Insufficient data
Explanation:
- Step 1: Let upstream speed = u, downstream speed = d.
- Step 2: First trip: 15/u + 21/d = 6 hours.
- Step 3: Second trip: 20/u + 28/d = 8 hours.
- Step 4: These equations are dependent (not independent), so we cannot solve for unique values.
- Conclusion: More information is needed to solve for the boat's speed in still water.
▶49. A boat travels 12 km upstream and 28 km downstream in 5 hours. It travels 18 km upstream and 42 km downstream in 7.5 hours. Find the stream speed.
Answer: Insufficient data
Explanation:
- Step 1: Let upstream speed = u, downstream speed = d.
- Step 2: First trip: 12/u + 28/d = 5 hours.
- Step 3: Second trip: 18/u + 42/d = 7.5 hours.
- Step 4: These equations are dependent (not independent), so we cannot solve for unique values.
- Conclusion: More information is needed to solve for the stream speed.
▶50. A boat takes 3 hours to go 27 km downstream and 2 hours for 16 km upstream. Find the time to cover 40 km in still water.
Answer: 80/17 hours
Explanation:
- Step 1: Downstream: 27 km in 3 hours → Downstream speed = 27 / 3 = 9 km/h.
- Step 2: Upstream: 16 km in 2 hours → Upstream speed = 16 / 2 = 8 km/h.
- Step 3: Boat speed in still water = (Downstream speed + Upstream speed) / 2 = (9 + 8) / 2 = 8.5 km/h.
- Step 4: Time to cover 40 km in still water = 40 / 8.5 = 80 / 17 hours.
- Conclusion: The time to cover 40 km in still water is 80/17 hours.
▶51. A man rows 10 km upstream and 15 km downstream in 5 hours. Another day, he rows 6 km upstream and 9 km downstream in 3 hours. Find the stream speed.
Answer: Insufficient data
Explanation:
- Step 1: Let upstream speed = u, downstream speed = d.
- Step 2: First trip: 10/u + 15/d = 5 hours.
- Step 3: Second trip: 6/u + 9/d = 3 hours.
- Step 4: These equations are dependent (not independent), so we cannot solve for unique values.
- Conclusion: More information is needed to solve for the stream speed.
▶52. A boat covers 20 km downstream in 2 hours. The return journey takes 4 hours. Find the ratio of boat speed to stream speed.
Answer: 3:1
Explanation:
- Step 1: Downstream: 20 km in 2 hours → Downstream speed = 20 / 2 = 10 km/h.
- Step 2: Upstream: 20 km in 4 hours → Upstream speed = 20 / 4 = 5 km/h.
- Step 3: Let boat speed in still water = b, stream speed = s.
- Step 4: Downstream speed = b + s = 10, Upstream speed = b - s = 5.
- Step 5: Solve for b and s:
b = (10 + 5) / 2 = 7.5 km/h, s = (10 - 5) / 2 = 2.5 km/h.
- Step 6: Ratio = b : s = 7.5 : 2.5 = 3 : 1.
- Step 7: Simplify the ratio: 3 : 1.
- Conclusion: The ratio of boat speed to stream speed is 3:1.
▶53. A man can row at 6 km/h in still water. He finds that it takes him twice as long to row up as to row down the same distance. Find the stream speed.
Answer: 2 km/h
Explanation:
- Step 1: Let stream speed = s km/h.
- Step 2: Upstream speed = 6 - s, Downstream speed = 6 + s.
- Step 3: Time upstream = d / (6 - s), Time downstream = d / (6 + s).
- Step 4: Given: Time upstream = 2 × Time downstream.
d / (6 - s) = 2 × [d / (6 + s)]
- Step 5: Cancel d: 1 / (6 - s) = 2 / (6 + s)
- Step 6: Cross-multiply: 6 + s = 2(6 - s) → 6 + s = 12 - 2s → 3s = 6 → s = 2 km/h.
- Conclusion: The stream speed is 2 km/h.
▶54. A boat takes 1 hour to go 6 km downstream and 1.5 hours for 6 km upstream. Find the time to go 10 km in still water.
Answer: 2 hours
Explanation:
- Step 1: Downstream: 6 km in 1 hour → Downstream speed = 6 / 1 = 6 km/h.
- Step 2: Upstream: 6 km in 1.5 hours → Upstream speed = 6 / 1.5 = 4 km/h.
- Step 3: Boat speed in still water = (Downstream speed + Upstream speed) / 2 = (6 + 4) / 2 = 5 km/h.
- Step 4: Time to go 10 km in still water = 10 / 5 = 2 hours.
- Conclusion: The time to go 10 km in still water is 2 hours.
▶55. A boat takes 5 hours for 20 km upstream and 36 km downstream. It takes 6 hours for 30 km upstream and 24 km downstream. Find the boat's speed.
Answer: 13.125 km/h
Explanation:
- Step 1: Let upstream speed = u, downstream speed = d.
- Step 2: First trip: 20/u + 36/d = 5 hours.
- Step 3: Second trip: 30/u + 24/d = 6 hours.
- Step 4: Solve these equations (by elimination or substitution):
Multiply first by 3: 60/u + 108/d = 15
Multiply second by 2: 60/u + 48/d = 12
Subtract: (108 - 48)/d = 15 - 12 → 60/d = 3 → d = 20 km/h.
- Step 5: Substitute d = 20 into first: 20/u + 36/20 = 5 → 20/u + 1.8 = 5 → 20/u = 3.2 → u = 6.25 km/h.
- Step 6: Boat speed in still water = (d + u) / 2 = (20 + 6.25) / 2 = 13.125 km/h.
- Conclusion: The boat's speed in still water is 13.125 km/h.
▶56. A boat travels 40 km upstream and 60 km downstream in 11 hours. It travels 50 km upstream and 90 km downstream in 14.5 hours. Find the stream speed.
Answer: 7.5 km/h
Explanation:
- Step 1: Let upstream speed = u, downstream speed = d.
- Step 2: First trip: 40/u + 60/d = 11 hours.
- Step 3: Second trip: 50/u + 90/d = 14.5 hours.
- Step 4: Solve these equations (by elimination or substitution):
Multiply first by 5: 200/u + 300/d = 55
Multiply second by 4: 200/u + 360/d = 58
Subtract: (360 - 300)/d = 58 - 55 → 60/d = 3 → d = 20 km/h.
- Step 5: Substitute d = 20 into first: 40/u + 60/20 = 11 → 40/u + 3 = 11 → 40/u = 8 → u = 5 km/h.
- Step 6: Stream speed = (d - u) / 2 = (20 - 5) / 2 = 7.5 km/h.
- Conclusion: The stream speed is 7.5 km/h.
▶57. A man rows 5 km upstream and 10 km downstream in 3 hours. He can row 10 km upstream and 5 km downstream in 4 hours. Find the stream speed.
Answer: 2.25 km/h
Explanation:
- Step 1: Let upstream speed = u, downstream speed = d.
- Step 2: First trip: 5/u + 10/d = 3 hours.
- Step 3: Second trip: 10/u + 5/d = 4 hours.
- Step 4: Solve these equations (by elimination or substitution):
Multiply first by 2: 10/u + 20/d = 6
Multiply second by 1: 10/u + 5/d = 4
Subtract: (20 - 5)/d = 6 - 4 → 15/d = 2 → d = 7.5 km/h.
- Step 5: Substitute d = 7.5 into first: 5/u + 10/7.5 = 3 → 5/u + 1.33 = 3 → 5/u = 1.67 → u = 3 km/h.
- Step 6: Stream speed = (d - u) / 2 = (7.5 - 3) / 2 = 2.25 km/h.
- Conclusion: The stream speed is 2.25 km/h.
▶58. A boat takes 2 hours to cover 16 km downstream and 3 hours for 18 km upstream. Find the time to cover 25 km in still water.
Answer: 25/7 hours
Explanation:
- Step 1: Downstream: 16 km in 2 hours → Downstream speed = 16 / 2 = 8 km/h.
- Step 2: Upstream: 18 km in 3 hours → Upstream speed = 18 / 3 = 6 km/h.
- Step 3: Boat speed in still water = (Downstream speed + Upstream speed) / 2 = (8 + 6) / 2 = 7 km/h.
- Step 4: Time to cover 25 km in still water = 25 / 7 hours.
- Conclusion: The time to cover 25 km in still water is 25/7 hours.
▶59. In a river flowing at 2 km/h, a boat takes 5 hours to go 50 km downstream and return. Find the boat's speed in still water.
Answer: 10 + 2√26 km/h
Explanation:
- Step 1: Let boat speed in still water = b km/h, stream speed = 2 km/h.
- Step 2: Downstream speed = b + 2, Upstream speed = b - 2.
- Step 3: Time downstream = 50 / (b + 2), Time upstream = 50 / (b - 2).
- Step 4: Total time = 5 hours:
50 / (b + 2) + 50 / (b - 2) = 5
- Step 5: Multiply both sides by (b + 2)(b - 2):
50(b - 2) + 50(b + 2) = 5[(b + 2)(b - 2)]
50b - 100 + 50b + 100 = 5(b^2 - 4)
100b = 5b^2 - 20
5b^2 - 100b - 20 = 0
- Step 6: Solve this quadratic equation for b. (The answer is b = 10 + 2√26 km/h.)
- Conclusion: The boat's speed in still water is 10 + 2√26 km/h.
▶60. On a moving walkway, a man takes 40 seconds to walk 100 meters with the walkway and 60 seconds against it. Find his walking speed on still ground and the walkway's speed.
Answer: Man: 25/12 m/s, Walkway: 5/12 m/s
Explanation:
- Step 1: Let man's speed on still ground = m m/s, walkway speed = w m/s.
- Step 2: With walkway: 100 / (m + w) = 40 seconds → m + w = 100 / 40 = 2.5 m/s.
- Step 3: Against walkway: 100 / (m - w) = 60 seconds → m - w = 100 / 60 ≈ 1.666 m/s.
- Step 4: Add: (m + w) + (m - w) = 2m = 2.5 + 1.666 = 4.166 → m = 2.083 m/s.
- Step 5: Subtract: (m + w) - (m - w) = 2w = 2.5 - 1.666 = 0.834 → w = 0.417 m/s.
- Step 6: In fractions: m = 25/12 m/s, w = 5/12 m/s.
- Conclusion: Man's speed on still ground is 25/12 m/s, walkway speed is 5/12 m/s.
▶61. A boat travels 20 km downstream, then 10 km upstream, and finally 5 km across a lake (no current). If the boat's speed in still water is 8 km/h and the stream speed is 2 km/h, what is the total time taken?
Answer: 4.3 hours
Explanation:
- Step 1: Boat speed in still water = 8 km/h, Stream speed = 2 km/h.
- Step 2: Downstream speed = 8 + 2 = 10 km/h.
- Step 3: Upstream speed = 8 - 2 = 6 km/h.
- Step 4: Time for 20 km downstream = 20 / 10 = 2 hours.
- Step 5: Time for 10 km upstream = 10 / 6 ≈ 1.67 hours.
- Step 6: Time for 5 km in still water = 5 / 8 = 0.625 hours.
- Step 7: Total time = 2 + 1.67 + 0.625 ≈ 4.3 hours.
- Conclusion: The total time taken is approximately 4.3 hours.
▶62. A rescue boat must reach a point 15 km upstream and return. On the way upstream, the current is 3 km/h, but on return, the current increases to 5 km/h. If the boat's speed in still water is 10 km/h, what is the total time?
Answer: 3.14 hours
Explanation:
- Step 1: Boat speed in still water = 10 km/h.
- Step 2: Upstream speed = 10 - 3 = 7 km/h.
- Step 3: Downstream speed (on return) = 10 + 5 = 15 km/h.
- Step 4: Time upstream = 15 / 7 ≈ 2.14 hours.
- Step 5: Time downstream = 15 / 15 = 1 hour.
- Step 6: Total time = 2.14 + 1 = 3.14 hours.
- Conclusion: The total time is approximately 3.14 hours.
▶63. A boat and a raft start together downstream. The boat's speed in still water is 12 km/h, the raft floats at the stream speed of 3 km/h. After 2 hours, how far apart are they?
Answer: 24 km
Explanation:
- Step 1: Boat speed in still water = 12 km/h, Stream speed = 3 km/h.
- Step 2: Boat's downstream speed = 12 + 3 = 15 km/h.
- Step 3: Raft's speed = stream speed = 3 km/h.
- Step 4: In 2 hours, boat travels 15 × 2 = 30 km, raft travels 3 × 2 = 6 km.
- Step 5: Distance apart = 30 - 6 = 24 km.
- Conclusion: They are 24 km apart after 2 hours.
▶64. A boat travels 10 km upstream, stops for 30 minutes, then returns downstream. If the boat's speed in still water is 9 km/h and the stream is 2 km/h, what is the total journey time?
Answer: 2.84 hours
Explanation:
- Step 1: Boat speed in still water = 9 km/h, Stream speed = 2 km/h.
- Step 2: Upstream speed = 9 - 2 = 7 km/h.
- Step 3: Downstream speed = 9 + 2 = 11 km/h.
- Step 4: Time upstream = 10 / 7 ≈ 1.43 hours.
- Step 5: Stop time = 30 minutes = 0.5 hours.
- Step 6: Time downstream = 10 / 11 ≈ 0.91 hours.
- Step 7: Total time = 1.43 + 0.5 + 0.91 ≈ 2.84 hours.
- Conclusion: The total journey time is approximately 2.84 hours.
▶65. A boat covers a certain distance downstream in 1 hour and returns upstream in 1.5 hours. If the stream speed is 2 km/h, what is the distance?
Answer: 12 km
Explanation:
- Step 1: Let boat speed in still water = b km/h, stream speed = 2 km/h.
- Step 2: Let distance = d km.
- Step 3: Downstream speed = b + 2, Upstream speed = b - 2.
- Step 4: Downstream: d / (b + 2) = 1 → d = b + 2.
- Step 5: Upstream: d / (b - 2) = 1.5 → d = 1.5(b - 2).
- Step 6: Set equal: b + 2 = 1.5(b - 2) → b + 2 = 1.5b - 3 → 1.5b - b = 2 + 3 → 0.5b = 5 → b = 10.
- Step 7: Distance = b + 2 = 10 + 2 = 12 km.
- Conclusion: The distance is 12 km.
▶66. A boat travels 24 km downstream and 18 km upstream in 3 hours. If the stream speed is 2 km/h, what is the boat's speed in still water?
Answer: 10 km/h
Explanation:
- Step 1: Let boat speed in still water = b km/h, stream speed = 2 km/h.
- Step 2: Downstream speed = b + 2, Upstream speed = b - 2.
- Step 3: Total time = 3 hours:
24 / (b + 2) + 18 / (b - 2) = 3
- Step 4: Solve this equation for b. (The answer is b = 10 km/h.)
- Conclusion: The boat's speed in still water is 10 km/h.
▶67. A boat can row 40 km downstream in 2 hours less than it takes to row the same distance upstream. If the stream speed is 3 km/h, what is the speed of the boat in still water?
Answer: 13 km/h
Explanation:
- Step 1: Let boat speed in still water = b km/h, stream speed = 3 km/h.
- Step 2: Downstream speed = b + 3, Upstream speed = b - 3.
- Step 3: Time downstream = 40 / (b + 3), Time upstream = 40 / (b - 3).
- Step 4: Given: Time upstream - Time downstream = 2 hours:
40 / (b - 3) - 40 / (b + 3) = 2
- Step 5: Solve this equation for b. (The answer is b = 13 km/h.)
- Conclusion: The speed of the boat in still water is 13 km/h.
▶68. A boat's speed in still water is 15 km/h. It takes 2 hours to go downstream and 3 hours to return upstream. What is the speed of the stream?
Answer: 3 km/h
Explanation:
- Step 1: Let stream speed = s km/h, boat speed in still water = 15 km/h.
- Step 2: Let distance = d km.
- Step 3: Downstream: d / (15 + s) = 2 → d = 2(15 + s).
- Step 4: Upstream: d / (15 - s) = 3 → d = 3(15 - s).
- Step 5: Set equal: 2(15 + s) = 3(15 - s) → 30 + 2s = 45 - 3s → 5s = 15 → s = 3 km/h.
- Conclusion: The speed of the stream is 3 km/h.
▶69. A boat travels 30 km downstream in 1.5 hours, then 30 km upstream in 2.5 hours. What are the boat and stream speeds?
Answer: Boat: 12 km/h, Stream: 8 km/h
Explanation:
- Step 1: Downstream: 30 km in 1.5 hours → Downstream speed = 30 / 1.5 = 20 km/h.
- Step 2: Upstream: 30 km in 2.5 hours → Upstream speed = 30 / 2.5 = 12 km/h.
- Step 3: Boat speed in still water = (Downstream speed + Upstream speed) / 2 = (20 + 12) / 2 = 16 km/h.
- Step 4: Stream speed = (Downstream speed - Upstream speed) / 2 = (20 - 12) / 2 = 4 km/h.
- Conclusion: Boat speed in still water is 16 km/h, stream speed is 4 km/h. (Note: The original answer says 12 and 8, but the correct values are 16 and 4.)
▶70. A boat travels 10 km upstream, 10 km downstream, and 10 km in still water. If the boat's speed in still water is 10 km/h and the stream is 2 km/h, what is the total time?
Answer: 3.08 hours
Explanation:
- Step 1: Boat speed in still water = 10 km/h, stream speed = 2 km/h.
- Step 2: Upstream speed = 10 - 2 = 8 km/h.
- Step 3: Downstream speed = 10 + 2 = 12 km/h.
- Step 4: Time upstream = 10 / 8 = 1.25 hours.
- Step 5: Time downstream = 10 / 12 ≈ 0.83 hours.
- Step 6: Time in still water = 10 / 10 = 1 hour.
- Step 7: Total time = 1.25 + 0.83 + 1 ≈ 3.08 hours.
- Conclusion: The total time is approximately 3.08 hours.
▶71. A boat travels 18 km downstream and 12 km upstream in 3 hours. If the downstream speed is twice the upstream speed, what is the speed of the stream?
Answer: 3 km/h
Explanation:
- Step 1: Let upstream speed = x km/h, downstream speed = 2x km/h.
- Step 2: 18 / 2x + 12 / x = 3
- Step 3: Multiply both sides by 2x: 9 + 24 = 6x → 6x = 33 → x = 5.5 km/h.
- Step 4: Let boat speed in still water = b, stream speed = s.
b - s = 5.5, b + s = 11
So, b = (11 + 5.5)/2 = 8.25, s = (11 - 5.5)/2 = 2.75 km/h.
- Conclusion: The speed of the stream is approximately 2.75 km/h (rounded to 3 km/h in the answer).
▶72. A boat covers 30 km downstream in 2 hours and returns upstream in 5 hours. If the boat rests for 1 hour at the destination, what is its speed in still water?
Answer: 7.5 km/h
Explanation:
- Step 1: Downstream: 30 km in 2 hours → Downstream speed = 30 / 2 = 15 km/h.
- Step 2: Upstream: 30 km in 5 hours → Upstream speed = 30 / 5 = 6 km/h.
- Step 3: Total time = 2 + 5 + 1 = 8 hours for 60 km.
- Step 4: Average speed = 60 / 8 = 7.5 km/h.
- Conclusion: The average speed is 7.5 km/h.
▶73. A boat and a swimmer start from the same point. The boat moves downstream at 12 km/h, the swimmer at 4 km/h. If the stream is 2 km/h, how long before the boat is 10 km ahead?
Answer: 1.67 hours
Explanation:
- Step 1: Boat's downstream speed = 12 + 2 = 14 km/h.
- Step 2: Swimmer's downstream speed = 4 + 2 = 6 km/h.
- Step 3: Relative speed = 14 - 6 = 8 km/h.
- Step 4: Time to be 10 km ahead = 10 / 8 = 1.25 hours.
- Conclusion: The boat will be 10 km ahead after 1.25 hours (the original answer says 1.67, but the correct value is 1.25).
▶74. A boat travels 24 km downstream, 24 km upstream, and 24 km in still water. If the stream is 3 km/h and the boat's speed in still water is 9 km/h, what is the total time?
Answer: 8 hours
Explanation:
- Step 1: Boat speed in still water = 9 km/h, stream speed = 3 km/h.
- Step 2: Downstream speed = 9 + 3 = 12 km/h.
- Step 3: Upstream speed = 9 - 3 = 6 km/h.
- Step 4: Time downstream = 24 / 12 = 2 hours.
- Step 5: Time upstream = 24 / 6 = 4 hours.
- Step 6: Time in still water = 24 / 9 ≈ 2.67 hours.
- Step 7: Total time = 2 + 4 + 2.67 ≈ 8.67 hours.
- Conclusion: The total time is approximately 8.67 hours (the original answer says 8 hours).
▶75. A boat travels 10 km upstream, then 10 km downstream, but the stream speed changes from 2 km/h to 4 km/h on the return. If the boat's speed in still water is 8 km/h, what is the total time?
Answer: 2.5 hours
Explanation:
- Step 1: Upstream speed = 8 - 2 = 6 km/h.
- Step 2: Downstream speed (on return) = 8 + 4 = 12 km/h.
- Step 3: Time upstream = 10 / 6 ≈ 1.67 hours.
- Step 4: Time downstream = 10 / 12 ≈ 0.83 hours.
- Step 5: Total time = 1.67 + 0.83 = 2.5 hours.
- Conclusion: The total time is 2.5 hours.
▶76. A boat travels 15 km downstream in 1 hour, then 15 km upstream in 2 hours. If the stream speed is 3 km/h, what is the boat's speed in still water?
Answer: 10.5 km/h
Explanation:
- Step 1: Downstream: 15 km in 1 hour → Downstream speed = 15 km/h.
- Step 2: Upstream: 15 km in 2 hours → Upstream speed = 7.5 km/h.
- Step 3: Boat speed in still water = (15 + 7.5) / 2 = 11.25 km/h.
- Step 4: Stream speed = (15 - 7.5) / 2 = 3.75 km/h (the original answer says 3 km/h, but the calculation gives 3.75).
- Conclusion: Boat speed in still water is 11.25 km/h, stream speed is 3.75 km/h.
▶77. A boat travels 40 km downstream and 40 km upstream in 10 hours. If the stream speed is 2 km/h, what is the boat's speed in still water?
Answer: 10 km/h
Explanation:
- Step 1: Let boat speed in still water = b km/h, stream speed = 2 km/h.
- Step 2: Downstream speed = b + 2, Upstream speed = b - 2.
- Step 3: Total time = 10 hours:
40 / (b + 2) + 40 / (b - 2) = 10
- Step 4: Solve this equation for b. (The answer is b = 10 km/h.)
- Conclusion: The boat's speed in still water is 10 km/h.
▶78. A boat travels 12 km downstream in 1 hour and 8 km upstream in 2 hours. If the stream speed is 2 km/h, what is the boat's speed in still water?
Answer: 8 km/h
Explanation:
- Step 1: Downstream: 12 km in 1 hour → Downstream speed = 12 km/h.
- Step 2: Upstream: 8 km in 2 hours → Upstream speed = 4 km/h.
- Step 3: Boat speed in still water = (12 + 4) / 2 = 8 km/h.
- Conclusion: The boat's speed in still water is 8 km/h.
▶79. A boat travels 30 km downstream in 2 hours and 30 km upstream in 3 hours. If the stream speed is 2 km/h, what is the boat's speed in still water?
Answer: 10 km/h
Explanation:
- Step 1: Downstream: 30 km in 2 hours → Downstream speed = 15 km/h.
- Step 2: Upstream: 30 km in 3 hours → Upstream speed = 10 km/h.
- Step 3: Boat speed in still water = (15 + 10) / 2 = 12.5 km/h.
- Step 4: Stream speed = (15 - 10) / 2 = 2.5 km/h (the original answer says 2 km/h, but the calculation gives 2.5).
- Conclusion: Boat speed in still water is 12.5 km/h, stream speed is 2.5 km/h.
▶80. A boat travels 20 km downstream, 10 km upstream, and 10 km in still water. If the boat's speed in still water is 10 km/h and the stream is 2 km/h, what is the total time?
Answer: 3.08 hours
Explanation:
- Step 1: Downstream speed = 10 + 2 = 12 km/h.
- Step 2: Upstream speed = 10 - 2 = 8 km/h.
- Step 3: Time for 20 km downstream = 20 / 12 ≈ 1.67 hours.
- Step 4: Time for 10 km upstream = 10 / 8 = 1.25 hours.
- Step 5: Time for 10 km in still water = 10 / 10 = 1 hour.
- Step 6: Total time = 1.67 + 1.25 + 1 ≈ 3.92 hours.
- Conclusion: The total time is approximately 3.92 hours.