Question 1

A man rows 3⁄4 km upstream in 11¼ minutes and the same distance downstream in 7½ minutes. What is his speed in still water (km/h)?

Accepted Answer

Given data Distance each way = 0.75 km Upstream time = 11.25 min = 11.25/60 h Downstream time = 7.5 min = 7.5/60 h Concept / Approach Compute upstream speed (u) and downstream speed (d), then still-water speed = (u + d)/2. Step-by-step calculation u = 0.75 ÷ (11.25/60) = 0.75 ÷ 0.1875 = 4 km/hd = 0.75 ÷ (7.5/60) = 0.75 ÷ 0.125 = 6 km/hStill-water speed = (4 + 6)/2 = 5 km/h Verification Current speed = (d − u)/2 = (6 − 4)/2 = 1 km/h; times match given data. Final Answer 5 km/h

Question 2

A man’s speed with the current is 15 km/h and the current’s speed is 2.5 km/h. What is his speed against the current?

Accepted Answer

Concept / Approach With current: vb + vs; Against current: vb − vs. Step-by-step calculation vb + vs = 15, with vs = 2.5Boat speed in still water vb = 15 − 2.5 = 12.5Against current = 12.5 − 2.5 = 10 km/h Final Answer 10 km/h

Question 3

A man can row at 5 km/h in still water. The current is 1 km/h. If a to-and-fro trip takes 1 hour, how far is the place?

Accepted Answer

Given data Still-water speed = 5 km/h; current = 1 km/h Downstream speed = 5 + 1 = 6 km/h Upstream speed = 5 − 1 = 4 km/h Total time (down + up) = 1 h Concept / Approach Let one-way distance be d. Then d/6 + d/4 = 1. Step-by-step calculation d \/ 6 + d \/ 4 = 1 ⇒ d(1/6 + 1/4) = 1d(5/12) = 1 ⇒ d = 12/5 = 2.4 km Final Answer 2.4 km

Question 4

A boat runs 16 km downstream in 2 hours and the same distance upstream in 4 hours. What is the speed of the boat in still water?

Accepted Answer

Given data Downstream speed d = 16/2 = 8 km/h Upstream speed u = 16/4 = 4 km/h Concept / Approach Still-water speed v = (u + d)/2. Step-by-step calculation v = (8 + 4)/2 = 6 km/h Final Answer 6 km/h

Question 5

A motorboat's speed in still water is 15 km/h. It goes 30 km downstream and returns in 4 h 30 min. What is the stream speed?

Accepted Answer

Given data Still-water speed v = 15 km/h; stream speed = s Downstream speed = 15 + s; upstream speed = 15 − s Total time T = 4.5 h Concept / Approach Use time sum: 30/(15 + s) + 30/(15 − s) = 4.5. Step-by-step calculation 30 ( 1 / ( 15 + 𝑠 ) ) + ( 1 / ( 15 − 𝑠 ) ) (1/(15+s))+(1/(15−s)) = 4.530 × ( 30 ) / ( 225 − 𝑠 2 ) (30)/(225−s 2 ) = 4.5 ⇒ 900/(225 − s^2) = 4.5900 = 4.5(225 − s^2) ⇒ 900 = 1012.5 − 4.5s^24.5s^2 = 112.5 ⇒ s^2 = 25 ⇒ s = 5 km/h Verification Downstream 30/(20) = 1.5 h; Upstream 30/10 = 3 h; total = 4.5 h ✔ Final Answer 5 km/h

Question 6

A boat takes 90 minutes less to travel 36 miles downstream than upstream. If its still-water speed is 10 mph, what is the stream speed?

Accepted Answer

Given data Distance each way = 36 miles Boat speed in still water = 10 mph; stream speed = s mph Upstream time − downstream time = 90 min = 1.5 h Concept / Approach Use times: 36/(10 − s) − 36/(10 + s) = 1.5. Step-by-step calculation 36 1 10 − 𝑠 − 1 10 + 𝑠 10−s 1 − 10+s 1 = 1.536 × ( 10 + 𝑠 ) − ( 10 − 𝑠 ) 100 − 𝑠 2 100−s 2 (10+s)−(10−s) = 1.536 × 2 𝑠 100 − 𝑠 2 100−s 2 2s = 1.5 ⇒ \frac{72s}{100 − s^2} = 1.572s = 1.5(100 − s^2) ⇒ 1.5s^2 + 72s − 150 = 0Multiply by 2: 3s^2 + 144s − 300 = 0 ⇒ s^2 + 48s − 100 = 0Discriminant = 48^2 + 4×100 = 2704 = 52^2 ⇒ s = (−48 + 52)/2 = 2 Final Answer 2 mph

Question 7

A boat covers a distance downstream in 1 hour and upstream in 1½ hours. If the stream's speed is 3 km/h, what is the boat's speed in still water?

Accepted Answer

Given data Let the one-way distance be D km. Downstream time = 1 h ⇒ D = (v + 3) × 1 Upstream time = 1.5 h ⇒ D = (v − 3) × 1.5 Concept / Approach Equate the two expressions for D and solve for still-water speed v. Step-by-step calculation v + 3 = 1.5(v − 3)v + 3 = 1.5v − 4.5 ⇒ 7.5 = 0.5v ⇒ v = 15 km/h Verification Downstream speed 18 ⇒ D = 18 km; Upstream speed 12 ⇒ time = 18/12 = 1.5 h ✔ Final Answer 15 km/h

Question 8

A boat's speed in still water is 15 km/h and the current is 3 km/h. What distance is covered downstream in 12 minutes?

Accepted Answer

Given data Speed in still water, u = 15 km/h Speed of current, v = 3 km/h Time downstream, t = 12 minutes Concept / Approach Downstream speed = u + v (current aids motion). Distance = Speed × Time (time in hours). Step-by-step calculation Downstream speed = u + v = 15 + 3 = 18 km/hConvert time: 12 min = 12/60 = 0.2 hDistance = 18 × 0.2 = 3.6 km Verification In 1 hour at 18 km/h, distance = 18 km. In 0.2 h, 18 × 0.2 = 3.6 km. Common pitfalls Using u − v instead of u + v for downstream. Forgetting to convert minutes to hours. Final Answer 3.6 km

Question 9

A boatman rows 2 km upstream in 1 hour and 1 km downstream in 10 minutes. How long will he take to row 5 km in still water?

Accepted Answer

Given data Upstream: 2 km in 1 h ⇒ u − v = 2 km/h Downstream: 1 km in 10 min = 1/6 h ⇒ u + v = 6 km/h Need time for 5 km in still water (speed = u) Concept / Approach Let u be speed in still water; v be current speed. Solve the system for u, then compute time = distance / speed. Step-by-step calculation u − v = 2 … (1)u + v = 6 … (2)Adding (1) and (2): 2u = 8 ⇒ u = 4 km/hTime in still water for 5 km = 5 / 4 h = 1.25 h = 1 hour 15 minutes Verification v = (6 − 2)/2 = 2 km/h. Upstream speed = 2 km/h and downstream speed = 6 km/h match the data. Common pitfalls Using average of 2 and 6 as time; it is speed that averages (and specifically (up + down)/2 for u). Converting 10 minutes incorrectly. Final Answer 1 hour 15 minutes

Question 10

A man rows to a place 48 km away and returns in a total of 14 hours. He can row 4 km with the stream in the same time as 3 km against it. What is the stream's speed?

Accepted Answer

Given data Out-and-back distance: 48 km each way Total time = 14 h Time to row 4 km downstream equals time to row 3 km upstream ⇒ 4/(u + v) = 3/(u − v) Concept / Approach From 4/(u + v) = 3/(u − v), deduce relation between u and v. Use round-trip time formula: T = 48/(u + v) + 48/(u − v). Step-by-step calculation 4(u − v) = 3(u + v) ⇒ 4u − 4v = 3u + 3v ⇒ u = 7vTotal time T = 48/(u + v) + 48/(u − v) = 96u/(u^2 − v^2)With u = 7v: u^2 − v^2 = 49v^2 − v^2 = 48v^2T = 96·7v / (48v^2) = 14 / vSet T = 14 ⇒ 14/v = 14 ⇒ v = 1 km/h Verification u = 7 km/h; downstream = 8 km/h; upstream = 6 km/h; times: 48/8 + 48/6 = 6 + 8 = 14 h. Common pitfalls Assuming arithmetic average of speeds instead of using the given time ratio. Final Answer 1 km/h

Question 11

A boat's still-water speed is 9 km/h and the stream speed is 1.5 km/h. For a 105 km trip downstream and return upstream, what is the total time?

Accepted Answer

Given data u = 9 km/h, v = 1.5 km/h Distance each way = 105 km Concept / Approach Downstream speed = u + v; upstream speed = u − v. Total time = distance/downstream speed + distance/upstream speed. Step-by-step calculation Downstream speed = 9 + 1.5 = 10.5 km/h ⇒ time = 105/10.5 = 10 hUpstream speed = 9 − 1.5 = 7.5 km/h ⇒ time = 105/7.5 = 14 hTotal time = 10 + 14 = 24 hours Verification Check: 10.5 × 10 = 105; 7.5 × 14 = 105. Common pitfalls Using average of speeds for round trips; harmonic-like addition is required via time sum. Final Answer 24 hours

Question 12

In one hour the boat covers 11 km with the current and 5 km against it. What is the speed of the boat in still water (km/h)?

Accepted Answer

Given data Downstream speed = 11 km/h Upstream speed = 5 km/h Concept / Approach Let u be still-water speed and v be current speed. Then u + v = 11 and u − v = 5. Adding the two gives u. Step-by-step calculation u = ( (u + v) + (u − v) ) / 2 = (11 + 5)/2 = 8 km/h Verification v = (11 − 5)/2 = 3 km/h. Then downstream = 8 + 3 = 11, upstream = 8 − 3 = 5. Common pitfalls Taking a simple average of distances rather than speeds; here we directly average the given speeds to find u. Final Answer 8 km/h

Question 13

A man takes twice as long to row a distance against the stream as with the stream. What is the ratio of the boat's still-water speed to the stream's speed?

Accepted Answer

Given data For the same distance, time upstream is twice time downstream. Concept / Approach Time ratio equals inverse speed ratio for equal distances. If u is still-water speed and v is stream speed: (u + v) / (u − v) = 2. Step-by-step calculation (u + v)/(u − v) = 2 ⇒ u + v = 2u − 2vRearrange ⇒ u = 3vThus, u : v = 3 : 1 Verification Take v = 1, u = 3 ⇒ speeds: downstream 4, upstream 2 ⇒ time ratio upstream:downstream = 4:2 = 2:1. Common pitfalls Inverting the ratio leading to u = v, which contradicts the time relation. Final Answer 3 : 1

Question 14

A boat takes 8 hours 48 minutes upstream and 4 hours downstream for the same distance. Find the ratio of the boat's speed to the current's speed.

Accepted Answer

Given data Upstream time = 8 h 48 min = 8.8 h Downstream time = 4 h Concept / Approach For equal distances, speed ratio = inverse of time ratio. Let u be still-water speed, v be current speed. Then (u − v) : (u + v) = (1/8.8) : (1/4) = 4 : 8.8 = 5 : 11. Step-by-step calculation Assume u − v = 5k and u + v = 11kAdd: 2u = 16k ⇒ u = 8kSubtract: 2v = 6k ⇒ v = 3kTherefore, u : v = 8 : 3 Verification Pick k = 1 ⇒ speeds 8 and 3 ⇒ times are proportional to 1/(8 − 3) = 1/5 and 1/(8 + 3) = 1/11; ratio of times = 11:5 which matches 8.8:4 = 2.2:1 ⇒ 11:5 after scaling. Common pitfalls Confusing time ratio with speed ratio; they are reciprocals for equal distances. Final Answer 8 : 3

Question 15

A boat can travel at 13 km/h in still water. The stream flows at 4 km/h. Find the time taken to travel 68 km downstream.

Accepted Answer

Given data Still-water speed u = 13 km/h Stream speed v = 4 km/h Distance downstream = 68 km Concept / Approach Downstream speed = u + v. Time = Distance / Speed. Step-by-step calculation Downstream speed = 13 + 4 = 17 km/hTime = 68 / 17 = 4 hours Verification 17 × 4 = 68, consistent. Common pitfalls Using u − v (upstream) by mistake. Final Answer 4 hours

Question 16

The speed of boat upstream and speed of boat downstream are 7 km/h and 13 km/h, respectively. Then the speed of stream and the speed of boat in still water is?

Accepted Answer

Rate in still water=[(1/2)(u+v)] km/h Rate of current=[(1/2)(u-v)] km/h where u= speed of boat in downstream v=speed of boat in upstream Speed of boat in still water=[(1/2)(7+13)] =10 km/h Speed of the stream =[(1/2)(13-7)]=3 km/h

Question 17

The speed of a boat in still water is 10 km/h. If it can travel 26 km downstream and 14 km upstream in the same time , then the speed of stream is?

Accepted Answer

Let the speed of the stream = x km/h ∴Speed of boat in still water = 10 km/h Speed of boat downstream = ( x + 10) km/h Speed of boat upstream=(10 - x) km/h ∴ Time taken to travel 26 km downstream=[ 26 / ( 10 + x ) ] h Time taken to travel 14 km upstream=[ 14 /( 10 - x )] h ⇒ By condition [ 26 / ( 10 + x)]=[ 14 / ( 10 - x)] ⇒ 26(10-x)=14(10+x) ⇒ 260-26x =140 + 14x ⇒ 40x=120 ⇒ x = 3 Here, speed of stream = 3 km/h

Question 18

A boat goes 40 km. upstream in 8 hours and 36 km. downstream in 6 hours. The speed of the boat in standing water is?

Accepted Answer

Speed upstream = (40/8) km/hr = 5 km/hr Speed downstream = (36/6) km/hr = 6 km/hr Speed of boat in still water = (5 + 6)/2 km/hr = 5.5 km/hr.

Question 19

If a man's rate with the current is 12 km/hr and the rate of the current is 1.5 km/hr then man's rate against the current is?

Accepted Answer

Let the rate against the current be x km/hr. Then, (12 - x) / 2 = 1.5 ⇒ 12 - x = 3 ⇒ x = 9 km/hr

Question 20

If a man rows at 5 km/hr in still water and 3.5 km/hr against the current his rate along the current is?

Accepted Answer

Speed of current = 5 - 3.5 = 1.5 km/hr. So the speed of man along the current = 5 + 1.5 = 6.5 km/hr

Boats and Streams Questions