Question 1

Upstream speed from downstream speed and stream rate: A swimmer's downstream speed is 11 km/h and the stream's speed is 1.5 km/h. What is the swimmer's upstream speed?

Accepted Answer

Introduction / Context:Given a downstream speed and the current, we can recover the still water speed and then compute the upstream speed. This is a standard inverse application of relative speed in a stream. Given Data / Assumptions: Downstream speed u + v = 11 km/h Stream speed v = 1.5 km/h Uniform straight current Concept / Approach:Compute u = (downstream + upstream)/2 is a known identity, but more directly with known downstream and v: u = (u + v) - v. Then upstream speed = u - v. Step-by-Step Solution: u + v = 11v = 1.5u = 11 - 1.5 = 9.5 km/hUpstream speed = u - v = 9.5 - 1.5 = 8 km/h Verification / Alternative check:If upstream is 8 and downstream is 11, the still water speed is (8 + 11)/2 = 9.5 km/h, which matches u above. Why Other Options Are Wrong: 9.5 km/h is still water speed, not upstream. 9 km/h and 7.5 km/h do not satisfy the given downstream with v = 1.5. 6.25 km/h is unrelated to these linear sums. Common Pitfalls:Confusing downstream value with still water speed or subtracting twice. Always compute u first, then upstream u - v. Final Answer:8 km/h

Question 2

Finding still water speed from unequal travel times: A boat covers 48 km downstream in 20 h and the same 48 km upstream in 24 h (4 h more). What is the speed of the boat in still water?

Accepted Answer

Introduction / Context:When downstream and upstream times for the same distance are known, we can compute the corresponding speeds and then infer the still water speed as the average of the two effective speeds. This uses the linear nature of relative motion with current. Given Data / Assumptions: Distance each way = 48 km Downstream time = 20 h Upstream time = 24 h Uniform current Concept / Approach:Downstream speed sd = distance / time. Upstream speed su = distance / time. Still water speed u = (sd + su) / 2. Current speed v = (sd - su) / 2 (not required but useful for checks). Step-by-Step Solution: sd = 48 / 20 = 2.4 km/hsu = 48 / 24 = 2.0 km/hStill water u = (2.4 + 2.0) / 2 = 2.2 km/h Verification / Alternative check:Then v = (2.4 - 2.0) / 2 = 0.2 km/h. Check: downstream u + v = 2.2 + 0.2 = 2.4 matches sd, and upstream u - v = 2.0 matches su. Why Other Options Are Wrong: 2.0 km/h and 4.0 km/h are one of the effective speeds or unrelated. 4.2 km/h and 1.8 km/h do not satisfy the downstream and upstream pair with the given times. Common Pitfalls:Using arithmetic mean of times or distances incorrectly. Here we average effective speeds, not times. Keep units consistent. Final Answer:2.2 km/h

Question 3

Current speed from downstream and upstream rates: Keshav can row 60 km downstream in 10 h and 36 km upstream in 10 h. What is the speed of the current?

Accepted Answer

Introduction / Context:Given distances and times for downstream and upstream trips, we compute effective speeds for each direction and then use their difference to find the current speed. This is a core pattern in boats and streams problems. Given Data / Assumptions: Downstream: 60 km in 10 h Upstream: 36 km in 10 h Uniform current, straight river Concept / Approach:sd = distance/time downstream; su = distance/time upstream. Then still water u = (sd + su)/2 and current v = (sd - su)/2. We only need v. Step-by-Step Solution: sd = 60/10 = 6 km/hsu = 36/10 = 3.6 km/hv = (sd - su)/2 = (6 - 3.6)/2 = 2.4/2 = 1.2 km/h Verification / Alternative check:Still water u = (6 + 3.6)/2 = 4.8 km/h. Check: u + v = 6 and u - v = 3.6 as required. Why Other Options Are Wrong: 2.2 km/h and 4.0 km/h do not match (sd - su)/2. 3.0 km/h is the half-sum mistake or random guess. 0.8 km/h is too small to account for the observed difference. Common Pitfalls:Taking average of distances or times instead of effective speeds; mixing upstream and downstream speeds. Always form sd and su first, then compute v. Final Answer:1.2 km/h

Question 4

Time to travel with the current: What time will a boat take to cover 128 km downstream if its still water speed is 24 km/h and the stream speed is 8 km/h?

Accepted Answer

Introduction / Context:Time taken for a journey equals distance divided by effective speed. For downstream travel, the effective speed is the still water speed plus the stream speed, because the current aids motion. Given Data / Assumptions: Distance D = 128 km Still water speed u = 24 km/h Stream speed v = 8 km/h Uniform flow, straight course Concept / Approach:Downstream speed sd = u + v. Time t = D / sd. Ensure units remain in km and h. Step-by-Step Solution: sd = 24 + 8 = 32 km/ht = 128 / 32 = 4 h Verification / Alternative check:If the current were absent, time would be 128/24 ≈ 5.33 h. With a helping current, time must be shorter. The computed 4 h is consistent. Why Other Options Are Wrong: 6 h, 7 h, and 8 h are longer than still-water time and contradict aid from current. 3 h would require a speed of about 42.7 km/h, which exceeds u + v = 32 km/h. Common Pitfalls:Using average of u and v, or subtracting instead of adding for downstream. Always add the stream speed for downstream travel. Final Answer:4 h

Question 5

Still water speed from upstream travel time: A boat covers 12 km upstream in 48 minutes when the stream speed is 2 km/h. What is the speed of the boat in still water?

Accepted Answer

Introduction / Context:Upstream speed is reduced by the current. From the upstream distance and time we can find the effective upstream speed and then add back the current to get the still water speed. Given Data / Assumptions: Upstream distance = 12 km Upstream time = 48 minutes = 0.8 h Stream speed v = 2 km/h Concept / Approach:Upstream speed su = distance / time. Then still water u = su + v because su = u - v. Maintain units in km/h. Step-by-Step Solution: su = 12 / 0.8 = 15 km/hu = su + v = 15 + 2 = 17 km/h Verification / Alternative check:If u = 17 and v = 2, then downstream speed would be 19 km/h. These are reasonable magnitudes around the observed upstream 15 km/h. Why Other Options Are Wrong: 15 km/h is the upstream effective speed, not still water. 13 km/h and 12 km/h are too small; adding v must increase from su. 19 km/h is the downstream speed u + v, not u. Common Pitfalls:Forgetting to convert minutes to hours or subtracting the stream instead of adding to find still water speed. Always express time in hours when speeds are in km/h. Final Answer:17 km/h

Question 6

Finding still water speed from round-trip time: A stream runs at 4 km/h. A boat goes 6 km downstream and returns 6 km upstream to the start in a total of 2 hours. What is the boat's speed in still water?

Accepted Answer

Introduction / Context:For a round trip in a current, the total time equals the sum of downstream and upstream times. Using the stream speed, we can set an equation in the unknown still water speed u and solve it. Given Data / Assumptions: Stream speed v = 4 km/h Downstream distance = 6 km Upstream distance = 6 km Total time T = 2 h Concept / Approach:Time downstream = 6 / (u + v). Time upstream = 6 / (u - v). Their sum equals 2. Solve for u > v. Step-by-Step Solution: 6/(u + 4) + 6/(u - 4) = 2Multiply through: 6(u - 4) + 6(u + 4) = 2(u^2 - 16)12u = 2u^2 - 322u^2 - 12u - 32 = 0 ⇒ u^2 - 6u - 16 = 0Solve: (u - 8)(u + 2) = 0 ⇒ u = 8 km/h (positive root) Verification / Alternative check:Downstream speed 12 km/h gives 0.5 h; upstream speed 4 km/h gives 1.5 h; total 2 h, consistent. Why Other Options Are Wrong: 7.5, 6.8, and 6 km/h do not satisfy the time equation. 10 km/h yields total time less than 2 h. Common Pitfalls:Adding distances and averaging speeds directly. Always sum times with correct effective speeds and solve the resulting equation carefully. Final Answer:8 km/h

Question 7

Speed of the current from contrasting paces: A boat moves upstream at 1 km in 10 minutes and downstream at 1 km in 6 minutes. What is the speed of the current?

Accepted Answer

Introduction / Context:Given unit distances and times in each direction, convert to effective speeds and use their difference to find the stream speed. This illustrates how time per kilometer maps to speed in km/h. Given Data / Assumptions: Upstream: 1 km in 10 minutes = 0.1667 h Downstream: 1 km in 6 minutes = 0.1 h Uniform current Concept / Approach:Compute su and sd, then v = (sd - su)/2. Keep units consistent by converting minutes to hours before inversion. Step-by-Step Solution: su = 1 / (10/60) = 1 / 0.1667 ≈ 6 km/hsd = 1 / (6/60) = 1 / 0.1 = 10 km/hv = (sd - su)/2 = (10 - 6)/2 = 2 km/h Verification / Alternative check:The still water speed u = (sd + su)/2 = (10 + 6)/2 = 8 km/h. Then u ± v = 8 ± 2 gives 10 and 6, matching inputs. Why Other Options Are Wrong: 1.5 km/h and 1 km/h underestimate the current given the large directional difference. 2.5 km/h and 3 km/h overshoot the needed difference of 4 km/h between sd and su. Common Pitfalls:Failing to convert minutes to hours; averaging times rather than speeds. Speed is distance divided by time in hours here. Final Answer:2 km/h

Question 8

Velocity of current from unequal up and down distances: A man rows upstream 16 km and downstream 28 km taking 5 hours each time. What is the velocity of the current?

Accepted Answer

Introduction / Context:Here, different distances are covered upstream and downstream but with equal times. This allows direct computation of effective speeds in both directions and then the current speed by halving their difference. Given Data / Assumptions: Upstream distance = 16 km in 5 h ⇒ su Downstream distance = 28 km in 5 h ⇒ sd Uniform current and straight path Concept / Approach:Compute su = 16/5 km/h and sd = 28/5 km/h. Then current speed v = (sd - su)/2. Still water speed u = (sd + su)/2 if needed. Step-by-Step Solution: su = 16/5 = 3.2 km/hsd = 28/5 = 5.6 km/hv = (5.6 - 3.2)/2 = 2.4/2 = 1.2 km/h Verification / Alternative check:u = (5.6 + 3.2)/2 = 4.4 km/h. Then upstream u - v = 3.2 and downstream u + v = 5.6, consistent with the computed rates. Why Other Options Are Wrong: 2.4 km/h equals the full difference sd - su, not the current v. 3.6 km/h and 1.8 km/h are not half the speed difference. 0.8 km/h is too small and does not reproduce the observed rates. Common Pitfalls:Forgetting to halve the difference between effective speeds; mixing km and hours. Always use v = (sd - su)/2 with consistent units. Final Answer:1.2 km/h

Question 9

A man can row at 5 km/hr in still water and the velocity of current is 1 km/hr. It takes him 1 hour to row to a place and back. How far is the place?

Accepted Answer

Speed downstream = (5 + 1) km/hr = 6 km/hr Speed upstream = (5 - 1) km/hr = 4 km/hr Let the required distance be d km Then, d/6 + d/4 = 1 ⇒ 2d + 3d = 12 ∴ d = 2.4 km

Question 10

A man can row 44 km downstream in 4 hours. If the man's rowing rate in still water is 8 km/hr, then find in what time will he cover 25 km upstream?

Accepted Answer

∵ Man's rate in still water = (man's rate with current + his rate against current)/2 ⇒ 8 = 1/2 [ 44/4 + 25/T ] ⇒ 16 = 11 + 25/T ∴ T = 5 hours

Question 11

A boat's man goes 48 km downstream in 8 hours and returns back in 12 hours. Find the speed of the boat in still water and the rate of the stream?

Accepted Answer

Rate downstream = 48/8 = 6 km/hr Rate upstream = 48/12 = 4 km/hr ∴ Speed of boat in still water = (6 + 4)/2 = 5 km/hr Rate of stream = (6 -4)/2 = 1 km/hr

Question 12

A man can row 60 km downstream in 6 hours. If the speed of the current is 3 km/hr, then find in what time will he be able to cover 16 km upstream?

Accepted Answer

Downstream speed = 60/6 = 10 km/hr Speed of stream = 3 km/hr Upstream speed = 10 - (2 x 3) = 4 km/hr ∴ Required time = 16/4 = 4 hours.

Question 13

speed of a boat in standing water is 7 km/hr and the speed of the stream is 1.5 km/hr. A distance of 7.7 km. going upstream is covered in?

Accepted Answer

Upstream speed = 7 - 1.5 = 5.5 km/hr ⇒ 5.5 x T = 7.7 ∴ T = 77.7 / 5.5 = 1hr. 24 minutes

Question 14

A sailor sails a distance of 48 km along the flow of a river in 8 h. If it takes 12 h to return the same distance, then the speed of the flow of the river is?

Accepted Answer

Let speed of the flow of water be x km/h and rate of sailing of sailor be y km/h. Downstream speed (x + y) = 48/8 ⇒ x + y = 6 ....(i) and upstream speed x - y = 4 ...(ii) On solving Eqs. (i) and (ii), we get y = 1 km/h

Question 15

Sameer can row a certain distance downstream in 24 h and can come back covering the same distance in 36 h. If the stream flows at the rate of 12 km/h, find the speed of Sameer in still water?

Accepted Answer

Let the speed of Sameer in still water be x km/h Sameer's speed downstream = (x + 12) km/h Sameer's speed upstream = (x - 12) km/h According to the question, 24(x + 12) = 36(x - 12) ⇒ 2x + 24 = 3x - 36 ⇒ x = 36 + 24 = 60 km/h

Question 16

Pawan can row 24 km/h in still water. When the river is running at 4.8 km/h, it takes him 1 h to row to a place and to come back. How far is the place?

Accepted Answer

Pawan's speed downstream = 24 + 4.8 = 28.8 km/h Pawan's speed upstream = 24 - 4.8 = 19.2 km/h Let the required distance be x. According to the question. x/28.8 + x/19.2 = 1 ⇒ (19.2x + 28.8x) / 552.96 = 1 ⇒ 19.2x + 28.8x = 552.96 ⇒ 48x = 552.96 ∴ x = 552.96/48 = 11.52 km

Question 17

A man can row 6 km/h in still water. If the speed of the current is 2 km/h, it takes 3 h more in upstream than in the downstream for the same distance. The distance is?

Accepted Answer

Let total distance be x km. According to the question, x/(6 + 2) + 3 = x/6 - 2 ⇒ x/8 + 3 = x/4 ⇒ x/4 - x/8 = 3 ⇒ 2x - x/8 = 3 ∴ x = 8 x 3 = 24 km

Question 18

The speed of the current is 5 km/h. A motorboat goes 10 km upstream and back again to the starting point in 50 min. The speed, (in km/h) of the motorboat in still water is?

Accepted Answer

Let speed of boat be x km/h. Given speed of current = 5 km/h ∴ Upstream speed of boat = (x -5) km/h Downstream speed of boat = (x + 5) km/h According to the question. 10/(x - 5) + 10/(x + 5) = 50/60 ⇒ 10(x + 5 + x - 5)/( x2 - 25) = 5/6 ⇒ x2 -24x - 25 = 0 ⇒ x2 - 25x + x - 25 = 0 ⇒ (x - 25) (x + 1) = 0 ∴ x = 25 [∴ x ≠ -1]

Question 19

A boat cover a distance of 30 km downstream in 2 h while it takes 6 h to cover the same distance distance upstream. What is the speed of the boat (in km/h)?

Accepted Answer

Speed of boat downstream = Distance covered / Time taken = 30/2 = 15 km/h Now, speed of boat upstream = Distance covered / Time taken = 30/6 = 5 km/h Now, speed of boat = (Speed download + Speed upstream) / 2 = (15 + 5) / 2 = 10 km/h

Question 20

A man can row at 10 km/h in still water. If he takes total 5 h to go to a place 24 km away and return, then the speed of the water current is?

Accepted Answer

Let speed of water current be x km/h ∴ Speed downstream = (10 + x) km/h. Speed upstream = (10 - x) km/h According to the question, 24/(10 + x) + 24/(10 - x) = 5 ⇒ 24[10 - x + 10 + x] = 5(102 - x2) ⇒ 100 - x2 = (24 x 20) / 5 ⇒ 100 - x2 = 96 ⇒ x2 = 4 ⇒ x = √4 = 2km/h

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