Question 1

Boat’s downstream speed is 15 km/h; stream speed is 1.5 km/h. Find the upstream speed.

Accepted Answer

Solution:b = d − s = 15 − 1.5 = 13.5; upstream = b − s = 13.5 − 1.5 = 12 km/hr.

Question 2

A boat’s still-water speed is 2 km/h. If its upstream speed is 1 km/h, find the stream speed.

Accepted Answer

Solution:Upstream = b − s = 1 with b = 2 ⇒ s = 1 km/hr.

Question 3

A man rows downstream at 14 km/h and upstream at 9 km/h. Find his rate in still water.

Accepted Answer

Solution:b = (down + up)/2 = (14 + 9)/2 = 11.5 km/hr.

Question 4

A man rows with the stream at 11 km/h and against the stream at 8 km/h. Find the speed of the stream.

Accepted Answer

Solution:s = (down − up)/2 = (11 − 8)/2 = 1.5 km/hr.

Question 5

A boat’s still-water speed is 2 km/h. If its speed upstream is 1 km/h, find the stream speed.

Accepted Answer

Solution:b − s = 1 with b = 2 ⇒ s = 1 km/hr.

Question 6

A boat covers 14 km upstream in 56 minutes (stream speed 2 km/h). Find the boat’s still-water speed.

Accepted Answer

Introduction / Context:Convert minutes to hours, deduce upstream speed, then add the known stream speed to get still-water speed. Step-by-Step Solution: Time = 56 min = 56/60 h.Upstream speed u = distance/time = 14 ÷ (56/60) = 14 × 60/56 = 15 km/h.With stream s = 2, still-water b = u + s = 15 + 2 = 17 km/hr.

Question 7

A boat’s still-water speed is 10 km/h. If its downstream speed is 13 km/h, find the stream speed.

Accepted Answer

Solution:d = b + s ⇒ s = 13 − 10 = 3 km/hr.

Question 8

A boat moves at 12 km/h in still water and 19 km/h downstream. Find its speed against the stream.

Accepted Answer

Solution:s = (down − still) = 19 − 12 = 7; upstream = b − s = 12 − 7 = 5 km/hr.

Question 9

Downstream speed from still-water and upstream speeds: A man rows at 5 km/h in still water. His speed against the current is 3.5 km/h. What is his speed along the current (downstream)?

Accepted Answer

Introduction / Context:Problems on boats and streams test the idea of relative speed in running water. The speed of a boat in still water (u) is modified by the stream speed (v): downstream speed is u + v, while upstream speed is u − v. Knowing any two of these lets us find the third. Given Data / Assumptions: Still-water speed u = 5 km/h. Upstream speed u − v = 3.5 km/h. We assume uniform speeds and straight reach (no turns), and calm conditions. Concept / Approach:From u − v we recover v, then compute downstream speed u + v. All arithmetic uses simple addition/subtraction because the stream contributes linearly to the boat’s ground speed. Step-by-Step Solution:u − v = 3.5 ⇒ v = u − 3.5 = 5 − 3.5 = 1.5 km/h.Downstream speed = u + v = 5 + 1.5 = 6.5 km/h. Verification / Alternative check:Average of downstream and upstream equals still-water speed: ( (u + v) + (u − v) ) / 2 = u. Using 6.5 and 3.5 gives (6.5 + 3.5) / 2 = 5, matching u = 5. Why Other Options Are Wrong:8.5 and 6 are not u + v for the found v; 4.25 is neither upstream nor downstream here; 5.5 assumes v = 0.5, which contradicts u − v = 3.5. Common Pitfalls:Confusing still-water speed with downstream speed; forgetting that upstream reduces speed by v while downstream increases it by v. Final Answer:6.5 km/hr

Question 10

Find current speed from unequal up/down distances and equal times: A man rows upstream 16 km in 5 h and downstream 27 km in 5 h. What is the velocity of the current?

Accepted Answer

Introduction / Context:When times are equal but distances differ, the given distances per equal time directly give the upstream and downstream speeds. With those two, we compute the still-water speed and stream speed using simple averages. Given Data / Assumptions: Upstream: 16 km in 5 h ⇒ 3.2 km/h. Downstream: 27 km in 5 h ⇒ 5.4 km/h. Let u be boat speed in still water and v the current speed. Concept / Approach:Use identities: upstream = u − v, downstream = u + v. Then u = (down + up)/2 and v = (down − up)/2. Step-by-Step Solution:u − v = 3.2, u + v = 5.4.u = (3.2 + 5.4) / 2 = 4.3 km/h.v = (5.4 − 3.2) / 2 = 1.1 km/h. Verification / Alternative check:Plug back: downstream 4.3 + 1.1 = 5.4, upstream 4.3 − 1.1 = 3.2, which match the observed rates. Why Other Options Are Wrong:2 or 2.1 overshoot the half-difference; 1.8 is the full difference minus an error; “None” is needless since 1.1 matches exactly. Common Pitfalls:Using average of distances instead of speeds; mixing minutes and hours; forgetting to divide the difference by 2. Final Answer:1.1 km/hr

Question 11

Current speed from mixed time units: A boat covers 1 km downstream in 7.5 minutes and travels upstream at 5 km/h. What is the speed of the current?

Accepted Answer

Introduction / Context:Converting times consistently to hours avoids unit traps. Once downstream and upstream speeds are known, we can recover still-water and current speeds using the average and half-difference identities. Given Data / Assumptions: Downstream: 1 km in 7.5 min = 1 km / 0.125 h = 8 km/h. Upstream speed: 5 km/h. Let u be still-water speed, v current speed. Concept / Approach:u = (down + up)/2; v = (down − up)/2. These come from u + v and u − v. Step-by-Step Solution:Down = 8, Up = 5.u = (8 + 5) / 2 = 6.5 km/h.v = (8 − 5) / 2 = 1.5 km/h. Verification / Alternative check:Check: u + v = 6.5 + 1.5 = 8 (downstream), u − v = 6.5 − 1.5 = 5 (upstream). All consistent. Why Other Options Are Wrong:1.2, 1.3, 1.6 do not equal the half-difference; 1.1 is arbitrary and inconsistent. Common Pitfalls:Treating 7.5 min as 0.75 h; mixing up downstream with upstream; arithmetic slips when halving differences. Final Answer:1.5 km/hr

Question 12

Stream velocity from symmetric 1 km runs: A person rows 1 km downstream in 10 minutes and 1 km upstream in 30 minutes. Find the velocity of the stream.

Accepted Answer

Introduction / Context:Equal distances with different times give direct speeds. Convert minutes to hours, compute the downstream and upstream speeds, then extract the stream speed via half-difference. Given Data / Assumptions: Downstream: 1 km in 10 min = 1 km / (10/60 h) = 6 km/h. Upstream: 1 km in 30 min = 1 km / (30/60 h) = 2 km/h. u is still-water speed; v is stream speed. Concept / Approach:Use v = (down − up)/2; u = (down + up)/2. Step-by-Step Solution:Down = 6, Up = 2.v = (6 − 2) / 2 = 2 km/h.u = (6 + 2) / 2 = 4 km/h (for reference). Verification / Alternative check:Check composition: u + v = 6, u − v = 2 matches the given. Why Other Options Are Wrong:1, 3, 4, 1.5 do not equal the required half-difference. Common Pitfalls:Using average instead of half-difference for v; forgetting to convert minutes to hours correctly. Final Answer:2 km/hr

Question 13

Compute upstream speed given downstream and stream: The downstream speed of a boat is 15 km/h and the stream speed is 1.5 km/h. What is the upstream speed?

Accepted Answer

Introduction / Context:Given downstream speed (u + v) and current v, recover still-water speed u, then compute the upstream speed u − v. Linear relations make this a direct two-step substitution. Given Data / Assumptions: Downstream speed u + v = 15 km/h. Stream speed v = 1.5 km/h. Concept / Approach:First find u = 15 − 1.5 = 13.5. Then upstream speed is u − v. Step-by-Step Solution:u = 15 − 1.5 = 13.5 km/h.Upstream = u − v = 13.5 − 1.5 = 12 km/h. Verification / Alternative check:Check symmetry: average of up and down is (12 + 15)/2 = 13.5, which equals u. Why Other Options Are Wrong:13.5 is u, not upstream; 16.5 exceeds downstream; 8.25 and 11 contradict u = 13.5. Common Pitfalls:Answering with u instead of upstream; subtracting v twice or forgetting to subtract at all. Final Answer:12 km/hr

Question 14

Downstream distance in a fixed time: A boat's still-water speed is 15 km/h and the current is 3 km/h. How far does it travel downstream in 12 minutes?

Accepted Answer

Introduction / Context:Distance at a constant speed is speed * time. For downstream travel, add the current to the still-water speed. Convert minutes to hours for consistent units before multiplying. Given Data / Assumptions: u = 15 km/h; v = 3 km/h ⇒ downstream = 18 km/h. Time = 12 minutes = 12/60 h = 0.2 h. Concept / Approach:Use D = (u + v) * t with t in hours to get distance in kilometers. Step-by-Step Solution:Downstream speed = 15 + 3 = 18 km/h.Distance = 18 * 0.2 = 3.6 km. Verification / Alternative check:18 km/h is 0.3 km per minute; for 12 minutes: 0.3 * 12 = 3.6 km. Why Other Options Are Wrong:2.4, 1.8, 1.2, 3.0 correspond to incorrect speed or time conversions. Common Pitfalls:Using still-water speed instead of downstream; forgetting to convert 12 minutes to 0.2 hours. Final Answer:3.6 km

Question 15

Downstream distance in fixed time from upstream speed and current: A man rows upstream 32 km in 4 h. If the current is 2 km/h, how far can he go downstream in 6 h?

Accepted Answer

Introduction / Context:From the upstream speed and the stream speed, we can compute the still-water speed. Once u is known, downstream speed u + v gives distance over the required time. Given Data / Assumptions: Upstream distance/time: 32 km in 4 h ⇒ upstream speed = 8 km/h. Stream speed v = 2 km/h. Let u be still-water speed. Concept / Approach:u − v = 8 ⇒ u = 10. Then downstream speed = u + v. Step-by-Step Solution:u − v = 8 ⇒ u = 8 + 2 = 10 km/h.Downstream speed = 10 + 2 = 12 km/h.Distance in 6 h = 12 * 6 = 72 km. Verification / Alternative check:Average of up and down equals 10; with v = 2 we get up 8 and down 12, consistent. Why Other Options Are Wrong:70 and 64 use wrong speeds; 81 assumes 13.5 km/h; 60 assumes 10 km/h downstream. Common Pitfalls:Using upstream speed directly for downstream distance; subtracting v instead of adding for downstream. Final Answer:72 km

Question 16

Correcting a unit slip and finding downstream distance: A man can row upstream 36 km in 6 h. If his still-water speed is 8 km/h, how far can he go downstream in 10 h?

Accepted Answer

Introduction / Context:The original stem showed “36 m” in 6 h, which is inconsistent with km/h speeds. By the Recovery-First Policy, we minimally repair it to 36 km so the problem is solvable without changing its intent. With upstream speed known, we determine the current and then the downstream distance for the requested time. Given Data / Assumptions: Upstream: 36 km in 6 h ⇒ 6 km/h. Still-water speed u = 8 km/h. Let current speed be v; downstream time required is 10 h. Concept / Approach:Upstream speed u − v = 6 ⇒ v = u − 6. Then downstream speed u + v gives distance over 10 h. Step-by-Step Solution:u − v = 6 ⇒ v = 8 − 6 = 2 km/h.Downstream speed = u + v = 8 + 2 = 10 km/h.Distance in 10 h = 10 * 10 = 100 km. Verification / Alternative check:Check symmetry: up 6, down 10 ⇒ u = (6 + 10)/2 = 8, consistent; v = (10 − 6)/2 = 2. Why Other Options Are Wrong:80, 90, 120, 150 assume different speeds or times that do not match u = 8, v = 2. Common Pitfalls:Not repairing the unit typo; computing distance with still-water speed instead of downstream speed. Final Answer:100 km

Question 17

Find stream speed from given up/down speeds: A man rows upstream at 15 km/h and downstream at 21 km/h. What is the speed of the current?

Accepted Answer

Introduction / Context:Upstream and downstream speeds are symmetric around the still-water speed. The current speed is simply half the difference between downstream and upstream speeds. Given Data / Assumptions: Upstream = 15 km/h. Downstream = 21 km/h. Let v be the current speed. Concept / Approach:v = (down − up)/2; u = (down + up)/2 (for reference). Step-by-Step Solution:v = (21 − 15) / 2 = 3 km/h.u = (21 + 15) / 2 = 18 km/h (still-water speed). Verification / Alternative check:Check: u + v = 18 + 3 = 21; u − v = 18 − 3 = 15, matching the given. Why Other Options Are Wrong:6, 5, 4, 8 are not the half-difference here. Common Pitfalls:Using full difference instead of half; adding instead of subtracting. Final Answer:3 km/h

Question 18

Recover still-water speed from equal-time trips: A man rows downstream 60 km in 4 h and upstream 36 km in 4 h. What is his still-water speed?

Accepted Answer

Introduction / Context:Equal times give direct speeds: downstream distance per hour and upstream distance per hour. Averaging these speeds yields the still-water speed because the current cancels out in the average. Given Data / Assumptions: Downstream speed = 60/4 = 15 km/h. Upstream speed = 36/4 = 9 km/h. Let u be still-water speed. Concept / Approach:u = (down + up)/2; v = (down − up)/2 for reference. Step-by-Step Solution:u = (15 + 9) / 2 = 12 km/h.v = (15 − 9) / 2 = 3 km/h (current). Verification / Alternative check:Compose to check: u + v = 12 + 3 = 15 (down), u − v = 12 − 3 = 9 (up), matching the data. Why Other Options Are Wrong:15 and 8 are individual directional speeds; 16 and 10.5 result from incorrect averaging. Common Pitfalls:Averaging distances instead of speeds; forgetting equal times were used to infer speeds. Final Answer:12 km/h

Question 19

Boat in still water and current speeds: A man rows at 4 km/h in still water and the rate of the stream is 2 km/h. What is the man's upstream speed against the current (i.e., the effective speed while going against the flow)?

Accepted Answer

Introduction / Context:Problems on boats and streams use relative speed. The effective speed upstream equals the still water speed minus the stream speed, while downstream equals the still water speed plus the stream speed. Here we are asked for the upstream speed when still water speed and stream rate are given. Given Data / Assumptions: Speed in still water u = 4 km/h Speed of stream v = 2 km/h Upstream means boat moves against the current Concept / Approach:For boats and streams: upstream speed = u - v and downstream speed = u + v. Units must be consistent in km/h. Step-by-Step Solution: Let u = speed in still water = 4 km/hLet v = speed of current = 2 km/hUpstream speed = u - v= 4 - 2 = 2 km/h Verification / Alternative check:If we instead computed downstream speed, it would be 4 + 2 = 6 km/h, which is larger than upstream, confirming direction logic is consistent. Why Other Options Are Wrong: 5 km/h and 6 km/h assume addition rather than subtraction or mix the roles of u and v. 7 km/h exceeds the still water speed, impossible for upstream. 3 km/h does not equal u - v for the given numbers. Common Pitfalls:Mixing up upstream and downstream, forgetting units, or subtracting in the wrong order can cause errors. Upstream must be slower than still water speed. Final Answer:2 km/h

Question 20

Downstream speed of a swimmer: The speed of a swimmer in still water is 9 km/h. The river flows at 6 km/h. What is the swimmer's downstream speed?

Accepted Answer

Introduction / Context:Downstream motion means the swimmer moves with the current. Effective speed increases by the stream speed. This is a direct application of the relative speed model for linear flows. Given Data / Assumptions: Still water speed u = 9 km/h Stream speed v = 6 km/h Uniform straight current, speeds add linearly downstream Concept / Approach:Downstream speed = u + v because both swimmer and current move in the same direction. Step-by-Step Solution: u = 9 km/hv = 6 km/hDownstream speed = u + v = 9 + 6 = 15 km/h Verification / Alternative check:Upstream would be u - v = 3 km/h, smaller than still water speed. That complements the downstream increase to 15 km/h, confirming logic. Why Other Options Are Wrong: 12 km/h adds partially and is not u + v. 3 km/h equals u - v, not downstream. 9 km/h ignores current. 6 km/h is only the current speed. Common Pitfalls:Confusing downstream with upstream or averaging speeds rather than adding for downstream. Always add for downstream and subtract for upstream. Final Answer:15 km/h

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