Q: A boat must travel upstream 20 km from point X to point Y and return from Y to X. The total time given is 41 minutes 40 seconds. Can the speed of the boat be determined? Choose the correct option.

Introduction / Context: The statement provides a very small total time (41 min 40 s ≈ 0.694 h) for a 40 km round trip against and with current, without specifying still-water speed or current. We must evaluate if a unique answer exists. Given Data / Assumptions: Upstream distance = 20 km; downstream distance = 20 km Total time T = 41 min 40 s = 41 + 40/60 min = 41.666... min ≈ 0.694 h Unknown still-water speed b and current c Concept / Approach: The time equation T = 20/(b − c) + 20/(b + c) has infinitely many (b, c) solutions for a fixed T unless additional constraints are given. Moreover, the given T seems unrealistically small for such a journey unless speeds are very high; still, with no b or c provided, there is no unique solution. Step-by-Step Reasoning: 0.694 ≈ 20/(b − c) + 20/(b + c)Multiple pairs (b, c) > 0 can satisfy this equation; without more information, the boat’s specific speed cannot be fixed. Verification / Alternative check: Trial values of b and c can be found to approximate the total, but uniqueness is absent. Also, the options list specific speeds without context; none can be uniquely justified. Why Other Options Are Wrong: 66, 72, or 48 km/h cannot be deduced from the data and would be arbitrary picks. Common Pitfalls: Assuming still water (c = 0) or guessing current; the problem provides neither, so the answer is indeterminate. Final Answer: Cannot be determined.

Question 1

A motorboat can travel at 10 km/h in still water. It goes 91 km downstream and returns 91 km upstream to the same place in a total of 20 hours. Find the rate of flow of the river (current speed in km/h).

Accepted Answer

Introduction / Context: With still-water speed and total time over symmetric downstream/upstream distances, the current can be solved from a single rational equation. Given Data / Assumptions: b = 10 km/h Distance each way = 91 km Total time = 20 h Let current be c Concept / Approach: Solve 91/(b + c) + 91/(b − c) = 20 for c (0 < c < b). Step-by-Step Solution: 91/(10 + c) + 91/(10 − c) = 2091 * ( (10 − c) + (10 + c) ) / (100 − c^2) = 20 ⇒ 91 * 20 / (100 − c^2) = 201820 / (100 − c^2) = 20 ⇒ 100 − c^2 = 91 ⇒ c^2 = 9 ⇒ c = 3 km/h Verification / Alternative check: Downstream 91/(13) = 7 h; upstream 91/7 = 13 h; total = 20 h, consistent. Why Other Options Are Wrong: 2, 4, 5 do not satisfy the sum-of-times equation with b = 10 km/h over 91 km each way. Common Pitfalls: Trying to average the two leg speeds directly; always sum times for unequal speeds over fixed distances. Final Answer: 3 km/h.

Question 2

A boat must travel upstream 20 km from point X to point Y and return from Y to X. The total time given is 41 minutes 40 seconds. Can the speed of the boat be determined? Choose the correct option.

Accepted Answer

Introduction / Context: The statement provides a very small total time (41 min 40 s ≈ 0.694 h) for a 40 km round trip against and with current, without specifying still-water speed or current. We must evaluate if a unique answer exists. Given Data / Assumptions: Upstream distance = 20 km; downstream distance = 20 km Total time T = 41 min 40 s = 41 + 40/60 min = 41.666... min ≈ 0.694 h Unknown still-water speed b and current c Concept / Approach: The time equation T = 20/(b − c) + 20/(b + c) has infinitely many (b, c) solutions for a fixed T unless additional constraints are given. Moreover, the given T seems unrealistically small for such a journey unless speeds are very high; still, with no b or c provided, there is no unique solution. Step-by-Step Reasoning: 0.694 ≈ 20/(b − c) + 20/(b + c)Multiple pairs (b, c) > 0 can satisfy this equation; without more information, the boat’s specific speed cannot be fixed. Verification / Alternative check: Trial values of b and c can be found to approximate the total, but uniqueness is absent. Also, the options list specific speeds without context; none can be uniquely justified. Why Other Options Are Wrong: 66, 72, or 48 km/h cannot be deduced from the data and would be arbitrary picks. Common Pitfalls: Assuming still water (c = 0) or guessing current; the problem provides neither, so the answer is indeterminate. Final Answer: Cannot be determined.

Question 3

A motorboat’s speed in still water is 45 km/h. It covers 80 km with the stream in 1 hour 20 minutes. Find the time it will take to cover the same 80 km against the stream.

Accepted Answer

Introduction / Context: From the downstream leg we can infer the current and then compute the upstream speed and time for the same distance. Given Data / Assumptions: b = 45 km/h Downstream distance = 80 km in 1 h 20 min = 4/3 h Concept / Approach: Downstream speed vd = distance/time = 80 / (4/3) = 60 km/h ⇒ current c = vd − b. Then upstream speed vu = b − c and time = 80 / vu. Step-by-Step Solution: vd = 60 km/h ⇒ c = 60 − 45 = 15 km/hvu = b − c = 45 − 15 = 30 km/hTime upstream = 80 / 30 = 8/3 h = 2 h 40 min Verification / Alternative check: Check dimensional consistency and reasonableness: upstream slower than b; time larger than 1 h 20 min, as expected. Why Other Options Are Wrong: Any time other than 2 h 40 min conflicts with the computed upstream speed 30 km/h over 80 km. Common Pitfalls: Using 45 km/h directly for upstream or not converting 1 h 20 min to hours correctly. Final Answer: 2 h 40 min.

Question 4

A boat’s speed in still water is 5 km/h and the river flows at 2 km/h. For the same distance upstream and downstream, the upstream time exceeds the downstream time by 2 hours. Find that distance (in km).

Accepted Answer

Introduction / Context: Time difference between upstream and downstream over the same distance allows solving for the distance using b ± c speeds. Given Data / Assumptions: b = 5 km/h; c = 2 km/h Upstream speed vu = 3 km/h Downstream speed vd = 7 km/h Let distance = d km Time difference = d/vu − d/vd = 2 h Concept / Approach: Form the linear equation in d and solve. Step-by-Step Solution: d/3 − d/7 = 2d * ( (7 − 3) / 21 ) = 2 ⇒ d * (4/21) = 2d = 2 * 21 / 4 = 10.5 km Verification / Alternative check: Up time = 10.5/3 = 3.5 h; down time = 10.5/7 = 1.5 h; difference = 2 h, as required. Why Other Options Are Wrong: Other distances do not yield exactly a 2-hour difference using 3 km/h and 7 km/h. Common Pitfalls: Taking the difference of speeds and dividing distance by it. Time difference is not distance divided by speed difference; use the proper fractional relation. Final Answer: 10.5 km.

Question 5

A boat takes 9 hours to travel a distance upstream and 3 hours to travel the same distance downstream. If the boat’s still-water speed is 4 km/h, find the speed of the stream (in km/h).

Accepted Answer

Introduction / Context: Equal distances with different times specify upstream and downstream effective speeds. Combine with the still-water speed identity to find the current. Given Data / Assumptions: Still-water speed b = 4 km/h Same distance d for upstream and downstream Upstream time = 9 h ⇒ vu = d/9 Downstream time = 3 h ⇒ vd = d/3 Concept / Approach: Use b = (vd + vu)/2 and c = (vd − vu)/2 because vd = b + c and vu = b − c. Step-by-Step Solution: b = (vd + vu)/2 = (d/3 + d/9)/2 = (4d/9)/2 = 2d/9Set 2d/9 = 4 ⇒ d = 18 kmc = (vd − vu)/2 = (d/3 − d/9)/2 = (2d/9)/2 = d/9 = 18/9 = 2 km/h Verification / Alternative check: vu = b − c = 2 km/h ⇒ time upstream = 18/2 = 9 h; vd = b + c = 6 km/h ⇒ time downstream = 18/6 = 3 h; consistent. Why Other Options Are Wrong: 3, 4, 6 contradict the still-water speed condition when reconstructed. Common Pitfalls: Trying to solve directly for speeds without relating them to distance; use the identities for sums and differences. Final Answer: 2 km/h.

Question 6

A boat running upstream covers 10 km in 30 minutes, and running downstream covers the same 10 km in 25 minutes. Find the speed of the water current (in km/h).

Accepted Answer

Introduction / Context: Equal distances allow conversion of times to effective speeds; the current is half the difference of downstream and upstream speeds. Given Data / Assumptions: Upstream: 10 km in 30 min = 0.5 h ⇒ vu = 20 km/h Downstream: 10 km in 25 min = 25/60 h ⇒ vd = 10 / (25/60) = 24 km/h Concept / Approach: Current c = (vd − vu)/2 and still-water speed b = (vd + vu)/2. Step-by-Step Solution: c = (24 − 20)/2 = 2 km/hb = (24 + 20)/2 = 22 km/h (for completeness) Verification / Alternative check: Check times: 10/24 ≈ 0.4167 h (25 min) and 10/20 = 0.5 h (30 min). Why Other Options Are Wrong: 3, 1, 0.5 do not reflect the half-difference of 24 and 20 which is exactly 2. Common Pitfalls: Using average time instead of converting each leg to a speed first, which is cleaner here. Final Answer: 2 km/h.

Question 7

A steamer goes from one port to another downstream in 4 hours and returns upstream in 5 hours. If the stream speed is 2 km/h, find the distance between the two ports (in km).

Accepted Answer

Introduction / Context: When upstream and downstream times are given for the same distance with a known current, we can deduce the still-water speed and then compute the distance. Given Data / Assumptions: Downstream time = 4 h, upstream time = 5 h Stream c = 2 km/h Let still-water speed be b; distance be D Concept / Approach: D = 4(b + 2) and D = 5(b − 2). Equate to solve for b, then compute D. Step-by-Step Solution: 4(b + 2) = 5(b − 2)4b + 8 = 5b − 10 ⇒ b = 18 km/hDistance D = 4(b + 2) = 4 * 20 = 80 km Verification / Alternative check: Upstream speed b − c = 16 km/h ⇒ time = 80/16 = 5 h; downstream speed b + c = 20 km/h ⇒ time = 80/20 = 4 h, as given. Why Other Options Are Wrong: 50, 60, 70 do not satisfy both time conditions with c = 2 km/h. Common Pitfalls: Using average of 4 and 5 hours or averaging speeds; equate distances via b ± c instead. Final Answer: 80 km.

Question 8

In a river, the ratio (speed of stream : speed of a boat in still water) for boat 1 is 2 : 5. For boat 2, the ratio (speed of stream : speed of that boat in still water) is 3 : 4. Find the ratio of the still-water speeds of boat 1 to boat 2.

Accepted Answer

Introduction / Context: The problem provides ratios of stream speed to each boat’s still-water speed. We are asked for the ratio of the boats’ still-water speeds independent of the actual stream magnitude. Given Data / Assumptions: For boat 1: v_stream / v_b1 = 2 / 5 ⇒ v_b1 = (5/2) v_stream For boat 2: v_stream / v_b2 = 3 / 4 ⇒ v_b2 = (4/3) v_stream Concept / Approach: Express both still-water speeds in terms of v_stream, then take the ratio v_b1 : v_b2 and simplify. Step-by-Step Solution: v_b1 : v_b2 = ( (5/2) v_s ) : ( (4/3) v_s )= (5/2) : (4/3) = (5/2) * (3/4) = 15/8 Verification / Alternative check: Choose any v_stream > 0 (e.g., 2). Then v_b1 = 5, v_b2 = 8/3 ⇒ ratio 15:8 holds. The ratio is independent of the actual stream value. Why Other Options Are Wrong: 10:7, 4:3, 5:4 do not equal 15:8 after simplification of the provided ratios. Common Pitfalls: Inverting the ratios or comparing stream-to-boat directly instead of converting to boat speeds first. Final Answer: 15 : 8.

Question 9

Rowing against and with the stream: A man rows 3/4 km upstream in 15 minutes and returns the same 3/4 km downstream in 10 minutes. What is the ratio of his speed in still water to the speed of the current?

Accepted Answer

Introduction / Context: Questions on boats and streams use effective speeds: upstream speed = (boat speed in still water - current speed) and downstream speed = (boat speed in still water + current speed). By comparing times for the same distance, we can solve for the ratio of speeds. Given Data / Assumptions: Upstream distance = 0.75 km covered in 15 minutes = 0.25 h. Downstream distance = 0.75 km covered in 10 minutes = 1/6 h. Let boat speed in still water = b km/h, current speed = c km/h. Concept / Approach: Upstream speed is b - c. Downstream speed is b + c. For a fixed distance, speed = distance / time. We compute the two effective speeds from the observed times, then solve for b and c and report the ratio b : c. Step-by-Step Solution: Upstream speed = 0.75 / 0.25 = 3 km/h Downstream speed = 0.75 / (1/6) = 4.5 km/h So b - c = 3 and b + c = 4.5 Add: 2b = 7.5 ⇒ b = 3.75 km/h Subtract: 2c = 1.5 ⇒ c = 0.75 km/h Ratio b : c = 3.75 : 0.75 = 5 : 1 Verification / Alternative check: Plug back: upstream = 3.75 - 0.75 = 3 km/h; downstream = 3.75 + 0.75 = 4.5 km/h — matches computed values; times are consistent. Why Other Options Are Wrong: 3 : 5 or 1 : 5 would imply the current is faster than the boat, which contradicts the data. 5 : 3 does not satisfy the two equations simultaneously. 'None of these' is invalid because 5 : 1 fits perfectly. Common Pitfalls: Using minutes directly without converting to hours; inverting ratios; assuming the ratio of times equals the ratio of speeds rather than the inverse for the same distance. Final Answer: 5 : 1

Question 10

A boat covers a fixed distance upstream in 528 minutes and the same distance downstream in 240 minutes. Find the ratio of the boat's speed in still water to the speed of the current.

Accepted Answer

Introduction / Context: This problem compares times to travel the same distance upstream and downstream to extract the speeds in still water and current via ratios. Given Data / Assumptions: Upstream time Tu = 528 minutes = 8.8 h. Downstream time Td = 240 minutes = 4 h. Let boat speed in still water = b, current speed = c (km/h), distance = D km. Concept / Approach: Upstream speed = b - c; downstream speed = b + c. Since D is the same, ratios of speeds equal inverse ratios of times: (b - c)/(b + c) = Td/Tu. Step-by-Step Solution: (b - c)/(b + c) = 4 / 8.8 = 5 / 11 11(b - c) = 5(b + c) 11b - 11c = 5b + 5c ⇒ 6b = 16c ⇒ b/c = 16/6 = 8/3 Verification / Alternative check: Any numbers with b : c = 8 : 3 (for example b = 8, c = 3) satisfy (b - c)/(b + c) = 5/11, which matches 4/8.8. Why Other Options Are Wrong: 2 : 1 and 3 : 2 do not produce 5/11 on substitution. 'Couldn't be determined' is wrong because the two times are sufficient to determine the ratio uniquely. Common Pitfalls: Using Tu/Td instead of Td/Tu; forgetting to convert minutes to hours; cancelling the distance incorrectly. Final Answer: 8 : 3

Question 11

A motorboat makes two trips at the same engine setting: (i) 25 km upstream and 39 km downstream in 8 hours, (ii) 35 km upstream and 52 km downstream in 11 hours. Find the speed of the stream.

Accepted Answer

Introduction / Context: When a boat travels upstream and downstream at a fixed engine setting, the still-water speed b and current speed c remain constant. Two mixed trips yield a solvable system for b and c. Given Data / Assumptions: Trip 1 time: 25 km upstream + 39 km downstream in 8 h. Trip 2 time: 35 km upstream + 52 km downstream in 11 h. Upstream speed = b - c; downstream speed = b + c. Concept / Approach: Use time = distance / speed on each leg and sum. This yields two equations in b and c. Step-by-Step Solution: 25/(b - c) + 39/(b + c) = 8 35/(b - c) + 52/(b + c) = 11 Solve simultaneously to obtain b = 9 km/h and c = 4 km/h. Verification / Alternative check: Check Trip 1: 25/5 + 39/13 = 5 + 3 = 8 h. Check Trip 2: 35/5 + 52/13 = 7 + 4 = 11 h. Perfect. Why Other Options Are Wrong: 2, 3, or 5 km/h do not satisfy both equations simultaneously; they break one of the total-time sums. Common Pitfalls: Adding numerators and denominators incorrectly; assuming average of speeds instead of harmonic structure; arithmetic slips when solving rational equations. Final Answer: 4 km/h

Question 12

A boat has speed 4 km/h in still water and the stream speed is 2 km/h. It takes 9 hours to go a certain distance upstream. How long will it take to come back the same distance downstream?

Accepted Answer

Introduction / Context: Given still-water and current speeds, we compute upstream and downstream speeds and transfer time information from one direction to the other for the same distance. Given Data / Assumptions: b = 4 km/h, c = 2 km/h. Upstream speed = b - c = 2 km/h; downstream speed = b + c = 6 km/h. Upstream time = 9 h over distance D. Concept / Approach: Distance D = speed * time. Then use D to find the downstream time with faster effective speed. Step-by-Step Solution: D = upstream speed * time = 2 * 9 = 18 km Downstream time = D / (b + c) = 18 / 6 = 3 h Verification / Alternative check: Round trip check: 9 h up + 3 h down = 12 h total; numbers are consistent and units align. Why Other Options Are Wrong: 2 or 2.5 h are too short for 18 km at a maximum of 6 km/h; 3.5 h is too long. Common Pitfalls: Using 4 km/h or 2 km/h directly for downstream; forgetting upstream is slower and downstream is faster; unit mismatches. Final Answer: 3 hrs

Question 13

Ramesh rows a fixed distance downstream in 6 hours and the same distance upstream in 9 hours. If his still-water speed is 12 km/h, find the speed of the stream.

Accepted Answer

Introduction / Context: Different times for the same distance downstream and upstream reflect different effective speeds. Knowing the still-water speed allows direct solution for the current speed. Given Data / Assumptions: Still-water speed b = 12 km/h. Downstream time Td = 6 h, upstream time Tu = 9 h for the same distance D. Downstream speed = b + c; upstream speed = b - c. Concept / Approach: Equate distances: (b + c) * Td = (b - c) * Tu. Solve for c. Step-by-Step Solution: (12 + c) * 6 = (12 - c) * 9 72 + 6c = 108 - 9c 15c = 36 ⇒ c = 36 / 15 = 2.4 km/h Verification / Alternative check: Downstream speed = 14.4 km/h; upstream speed = 9.6 km/h. For any D, D/14.4 divided by D/9.6 equals 2/3, matching 6 h vs 9 h. Why Other Options Are Wrong: 2 or 3 km/h do not satisfy the exact time ratio with b = 12 km/h; data is sufficient, so 'Data inadequate' is incorrect. Common Pitfalls: Adding times or averaging times; assuming average of speeds rather than solving the linear equation from equal distances. Final Answer: 2.4 km/hr

Question 14

A man rows to a place 48 km away and returns in a total of 14 hours. He finds that he can row 4 km with the stream in the same time as 3 km against the stream. Find the speed of the stream.

Accepted Answer

Introduction / Context: Relating equal times for unequal distances with and against the stream gives a ratio of effective speeds, which we can use with the total round-trip time to compute the current speed. Given Data / Assumptions: Distance one way = 48 km; total time = 14 h. Time for 4 km downstream equals time for 3 km upstream ⇒ vd : vu = 4 : 3, where vd = b + c and vu = b - c. Concept / Approach: Let vu = 3k and vd = 4k. Then the total time is 48/(3k) + 48/(4k) = 16/k + 12/k = 28/k. Set 28/k = 14 to get k, then compute c from (vd - vu)/2. Step-by-Step Solution: 28/k = 14 ⇒ k = 2 vu = 3k = 6 km/h; vd = 4k = 8 km/h c = (vd - vu)/2 = (8 - 6)/2 = 1 km/h Verification / Alternative check: Round-trip time = 48/6 + 48/8 = 8 + 6 = 14 h, as given. The 4 km vs 3 km equal-time condition also holds with speeds 8 and 6 km/h. Why Other Options Are Wrong: 2, 1.5, or 2.5 km/h do not produce both the correct round-trip sum and the 4:3 speed ratio simultaneously. Common Pitfalls: Assuming 4:3 is b:c instead of downstream:upstream; forgetting that times add over legs, not speeds; arithmetic slips in solving for k. Final Answer: 1 km/hr

Question 15

A boat sails 15 km upstream on a river in 5 hours. If the speed of the current is one-fourth of the speed of the boat in still water, how long will it take the boat to cover the same distance downstream?

Accepted Answer

Introduction / Context: Boat and stream problems are classic in quantitative aptitude, testing your understanding of relative speed in upstream and downstream motion. Here, the time taken upstream is given along with a relation between the speed of the current and the speed of the boat in still water. You need to determine the time taken to cover the same distance downstream. Given Data / Assumptions: Distance travelled upstream = 15 km. Time taken upstream = 5 hours. Speed of the current = one-fourth of the speed of the boat in still water. Let the speed of the boat in still water be b km/h and the speed of the current be c km/h. Upstream speed = (b - c), downstream speed = (b + c). Concept / Approach: First, we find the upstream speed from distance and time. Then we use the relation between b and c to express upstream speed in terms of b, solve for b, and then find downstream speed. Finally, time = distance / speed is used to compute the downstream time for the same 15 km. The key idea is that current reduces effective speed upstream and increases it downstream. Step-by-Step Solution: Step 1: Compute upstream speed. Upstream speed = distance / time = 15 / 5 = 3 km/h. Step 2: Use the relation between boat speed and current. Given c = (1/4)b, so upstream speed = b - c = b - b/4 = (3/4)b. But upstream speed is 3 km/h, so (3/4)b = 3. Step 3: Solve for b. b = 3 * (4/3) = 4 km/h. Then c = b/4 = 1 km/h. Step 4: Find downstream speed and time. Downstream speed = b + c = 4 + 1 = 5 km/h. Time downstream = distance / downstream speed = 15 / 5 = 3 hours. Verification / Alternative check: Check consistency: with b = 4 km/h and c = 1 km/h, upstream speed = 3 km/h and downstream speed = 5 km/h. Upstream time for 15 km: 15 / 3 = 5 hours (matches given). Downstream time: 15 / 5 = 3 hours, which agrees with our computed answer. Why Other Options Are Wrong: 1.8 h would correspond to a downstream speed of 15 / 1.8 ≈ 8.33 km/h, which is impossible given the still water speed is only 4 km/h. 4 h and 5 h imply downstream speeds of 3.75 km/h and 3 km/h respectively, both less than or equal to the upstream speed, which cannot be true when the current aids the boat downstream. Common Pitfalls: One common mistake is to ignore the relation c = b/4 or to misinterpret it. Another is to assume that upstream and downstream times are symmetric or to average speeds incorrectly. Always derive actual speeds using distance/time and the given relationships, then apply time = distance / speed carefully. Final Answer: The boat will take 3 h to cover the same 15 km distance downstream.

Question 16

A man can row a boat at 9 km/h in still water. For a given distance on a river, it takes him twice as long to row upstream as it takes to row downstream. What is the rate of flow of the stream (in km/h)?

Accepted Answer

Introduction / Context: This boat and stream problem focuses on the relationship between speeds upstream and downstream and the time taken to travel a fixed distance. When the time upstream is a simple multiple of the time downstream, you can use that ratio to find the speed of the current in terms of the speed of the boat in still water. Given Data / Assumptions: Speed of the boat in still water, b = 9 km/h. Let the speed of the stream be c km/h. For a fixed distance d, time upstream is twice the time downstream. Upstream speed = (b - c) km/h. Downstream speed = (b + c) km/h. Distance d is the same in both directions and cancels out in ratios. Concept / Approach: Time is equal to distance divided by speed. For the same distance, if the time upstream is twice the time downstream, then: d / (b - c) = 2 * [d / (b + c)]. The distance d cancels, leaving a relation involving b and c only. We then solve this equation for c using the known value of b = 9 km/h. Finally, we interpret c as the rate of the stream. Step-by-Step Solution: Step 1: Set up the time ratio equation. d / (b - c) = 2d / (b + c). Cancel d from both sides to get 1 / (b - c) = 2 / (b + c). Step 2: Cross-multiply to eliminate denominators. b + c = 2(b - c). b + c = 2b - 2c. Step 3: Group like terms. Bring terms together: 3c = b. Thus c = b / 3. Step 4: Substitute the value of b. c = 9 / 3 = 3 km/h. Verification / Alternative check: If c = 3 km/h, then downstream speed = 9 + 3 = 12 km/h and upstream speed = 9 - 3 = 6 km/h. For distance d, time downstream = d / 12, time upstream = d / 6 = 2 * (d / 12). This matches the condition that upstream time is twice downstream time, confirming that c = 3 km/h is correct. Why Other Options Are Wrong: 2 km/h or 1.5 km/h would give a time ratio that is not exactly 2:1 for upstream versus downstream for the same distance. 4 km/h or 5 km/h would make the current very strong relative to the boat, altering the times so that the upstream journey would take more than twice the downstream time or might even become too slow to be reasonable for b = 9 km/h. Common Pitfalls: Learners sometimes try to average speeds or assume that time differences translate directly into additive adjustments of speed. The correct approach is always to use time = distance / speed and set up equations using the given ratio of times. Also, ensure that you keep track of which speed is upstream (b - c) and which is downstream (b + c). Final Answer: The rate of flow of the stream is 3 km/h.

Question 17

A man travels 28 km downstream in a motor boat and immediately returns upstream to his starting point. The upstream journey takes twice as long as the downstream journey. If the speed of the river flow were doubled, the total time for going 28 km do…

Accepted Answer

Introduction / Context: This is a multi-step boats and streams problem involving both time ratios and a hypothetical change in the speed of the current. You must use relative speed concepts downstream and upstream, together with the given conditions, to determine both the speed of the boat in still water and the speed of the river flow. Given Data / Assumptions: Distance downstream = 28 km; same distance upstream back to start = 28 km. Time upstream is twice the time downstream. If the speed of the river flow were doubled, the total time for 28 km downstream and back would be 672 minutes = 11.2 hours. Let the speed of the boat in still water be b km/h. Let the speed of the river flow be c km/h. Downstream speed = b + c; upstream speed = b - c (with c < b so that upstream is possible). Concept / Approach: We first convert the time ratio into an algebraic relation using times for a fixed distance: time = distance / speed. This gives b in terms of c. Then, for the hypothetical scenario with doubled current speed (2c), we express the total round-trip time in terms of c only and set it equal to 11.2 hours. Solving this equation gives c, and substituting back yields b. Step-by-Step Solution: Step 1: Use the given time ratio condition. Downstream time t₁ = 28 / (b + c). Upstream time t₂ = 28 / (b - c). Given t₂ = 2t₁, so 28 / (b - c) = 2 * [28 / (b + c)]. Cancel 28: 1 / (b - c) = 2 / (b + c). Cross-multiply: b + c = 2(b - c) ⇒ b + c = 2b - 2c ⇒ 3c = b. So b = 3c. Step 2: Use the hypothetical doubled current scenario. New current speed = 2c, still-water speed remains b = 3c. New downstream speed = 3c + 2c = 5c. New upstream speed = 3c - 2c = c. Total time T = 28 / (5c) + 28 / c. Given T = 672 minutes = 11.2 hours. Compute T in terms of c: T = 28(1/(5c) + 1/c) = 28[(1 + 5)/(5c)] = 28 * 6/(5c) = 168/(5c). Set 168/(5c) = 11.2 ⇒ 168 = 11.2 * 5c = 56c ⇒ c = 3 km/h. Then b = 3c = 9 km/h. Verification / Alternative check: With b = 9 km/h and c = 3 km/h: original downstream speed = 12 km/h, upstream speed = 6 km/h. Downstream time = 28/12 ≈ 2.333 h, upstream time = 28/6 ≈ 4.667 h, and t₂ is twice t₁, as required. With doubled current 6 km/h, downstream speed = 15 km/h, upstream speed = 3 km/h. Total time = 28/15 + 28/3 ≈ 1.867 + 9.333 ≈ 11.2 h (672 minutes), confirming both conditions. Why Other Options Are Wrong: 12 km/h, 3 km/h and 8 km/h, 2 km/h do not satisfy both the original time ratio and the 672-minute condition simultaneously. 9 km/h, 6 km/h makes the current too fast; with b = 9 and c = 6, upstream speed would be only 3 km/h and the original time ratio would not match the given 2:1 relation correctly for 28 km each way. Common Pitfalls: Many students incorrectly add or average speeds instead of using the formula time = distance / speed in each phase. Another common mistake is forgetting to convert minutes to hours when inserting the 672-minute condition. Keeping symbols for b and c, solving the time-ratio equation first, and only then applying the second condition keeps the solution clear and manageable. Final Answer: The speed of the boat in still water is 9 km/h, and the speed of the river flow is 3 km/h.

Question 18

A boat takes a total of 19 hours to travel downstream from point A to point B and then return upstream to point C, where C is the midpoint of AB. If the speed of the stream is 4 km/h and the speed of the boat in still water is 14 km/h, what is the d…

Accepted Answer

Introduction / Context: This problem combines downstream and upstream travel over different portions of a route in a river with a known current. You are given the total time for a two-leg journey and the speeds in still water and in the stream, and you are asked to find the distance between two points on the river. Given Data / Assumptions: Let the distance from A to B be D km. Point C is the midpoint of AB, so AC = CB = D/2. The boat travels downstream from A to B, then upstream from B to C. Speed of boat in still water = 14 km/h. Speed of stream = 4 km/h. Downstream speed = 14 + 4 = 18 km/h. Upstream speed = 14 - 4 = 10 km/h. Total time for A → B (downstream) and B → C (upstream) = 19 hours. Concept / Approach: We compute the time for each leg using time = distance / speed. The downstream leg covers the full distance D, while the upstream leg covers only half the distance (from B to midpoint C, which is D/2). The sum of these two times is given as 19 hours. This gives a linear equation in D, which we can solve directly. Step-by-Step Solution: Step 1: Express the time taken downstream from A to B. Time downstream, t₁ = D / 18 hours. Step 2: Express the time taken upstream from B to C. Distance B to C = D/2. Upstream speed = 10 km/h. Time upstream, t₂ = (D/2) / 10 = D / 20 hours. Step 3: Set up the total time equation. t₁ + t₂ = 19. So D/18 + D/20 = 19. Step 4: Solve for D. Take LCM of 18 and 20, which is 180. (10D + 9D) / 180 = 19. 19D / 180 = 19 ⇒ D = 180 km. Verification / Alternative check: Check the times with D = 180 km: t₁ = 180 / 18 = 10 hours (downstream), t₂ = (180/2) / 10 = 90 / 10 = 9 hours (upstream). Total = 10 + 9 = 19 hours, which matches the given total time exactly, confirming that D = 180 km is correct. Why Other Options Are Wrong: For D = 160 km, total time would be 160/18 + 80/10 ≈ 8.89 + 8 = 16.89 h, not 19 h. For D = 140 km or 120 km, the total times would be even smaller, farther from the required 19 hours. Hence they cannot satisfy the given condition. Common Pitfalls: A common mistake is to assume the boat returns the entire distance from B back to A, instead of only to midpoint C, which changes the upstream distance from D/2 to D and gives the wrong equation. Another error is miscomputing the LCM or algebraic simplification when combining fractions. Keeping track of distances on each leg and carefully setting up the time equation avoids these problems. Final Answer: The distance between A and B is 180 km.

Question 19

A motor boat takes 12 hours to travel a certain distance downstream and 24 hours to travel the same distance upstream. For that distance, how much time would the boat take to travel in still water?

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Introduction / Context: This is a standard boat and stream question that uses downstream and upstream times for the same distance to infer the time required in still water. The focus is on understanding how speeds relate to times and how the still-water speed is the average of downstream and upstream speeds. Given Data / Assumptions: Let the one-way distance be d km. Time taken downstream = 12 hours. Time taken upstream = 24 hours. Downstream speed = d / 12 km/h. Upstream speed = d / 24 km/h. Speed of boat in still water = average of downstream and upstream speeds. We must find the time taken to cover distance d in still water. Concept / Approach: In boat-stream problems, the effective downstream speed is (b + c) and upstream speed is (b - c), where b is the boat's speed in still water and c is the current. Therefore, the boat's still-water speed is the average of these two: b = [(b + c) + (b - c)] / 2. Here, we compute these effective speeds from distance and time, then take their average to get b, and finally compute the time in still water as distance divided by b. Step-by-Step Solution: Step 1: Find downstream and upstream speeds. Downstream speed v_d = d / 12 km/h. Upstream speed v_u = d / 24 km/h. Step 2: Compute boat speed in still water. b = (v_d + v_u) / 2. b = (d/12 + d/24) / 2. Compute numerator: d/12 + d/24 = (2d + d) / 24 = 3d / 24 = d / 8. So b = (d/8) / 2 = d / 16 km/h. Step 3: Compute time in still water for the same distance d. Time in still water t = distance / speed = d / (d / 16) = 16 hours. Verification / Alternative check: We can loosely check consistency: the boat moves fastest downstream (taking 12 h) and slowest upstream (24 h). The still-water time must lie between these, and because the still-water speed is the average of the two speeds, its corresponding time is roughly in the middle of 12 and 24, which is 18, but when computed exactly for the same distance, we get 16 hours due to the harmonic-type relationship between speeds and times. The computed value 16 h satisfies the algebraic derivation and is consistent with the relationship between speeds and times. Why Other Options Are Wrong: 8 h is too small and would imply a speed faster than the downstream speed, which is impossible in this scenario. 15 h and 20 h do not match the exact algebraic result from combining speeds and would not create the correct 12 h and 24 h times when currents are reintroduced. Common Pitfalls: A frequent mistake is to average the times directly (giving 18 h) instead of working with speeds. Remember that the still-water speed is the average of downstream and upstream speeds, not of the times. Another pitfall is forgetting that time is inversely proportional to speed, so you must convert times to speeds before averaging and then convert back to time. Final Answer: The boat would take 16 h to travel the same distance in still water.

Question 20

The current of a stream flows at 1 km/h. A motor boat goes 35 km upstream from a point and then returns to the starting point, taking a total of 12 hours for the round trip. What is the speed of the motor boat in still water?

Accepted Answer

Introduction / Context: Here we are given the speed of the stream, the distance travelled upstream and downstream, and the total time taken for the round trip. The task is to determine the boat's speed in still water. This is a classic application of relative speed concepts in boats and streams. Given Data / Assumptions: Speed of the stream c = 1 km/h. Let the speed of the boat in still water be b km/h. Distance upstream from start to turning point = 35 km. Distance downstream back to start = 35 km. Total time for upstream and downstream = 12 hours. Upstream speed = (b - c) km/h. Downstream speed = (b + c) km/h. Concept / Approach: Time is equal to distance divided by speed. For the round trip, we sum the time taken upstream and downstream and set this equal to 12 hours. Since the stream speed is known, the only unknown is b. Solving the resulting equation gives the boat's still-water speed. Step-by-Step Solution: Step 1: Write expressions for times upstream and downstream. Upstream time t₁ = 35 / (b - 1). Downstream time t₂ = 35 / (b + 1). Total time t₁ + t₂ = 12 hours. Step 2: Set up the equation. 35 / (b - 1) + 35 / (b + 1) = 12. Divide both sides by 35: 1 / (b - 1) + 1 / (b + 1) = 12 / 35. Step 3: Combine the fractions on the left. (b + 1 + b - 1) / (b^2 - 1) = 12 / 35. 2b / (b^2 - 1) = 12 / 35. Step 4: Cross-multiply and solve for b. 2b * 35 = 12(b^2 - 1). 70b = 12b^2 - 12. Rearrange: 12b^2 - 70b - 12 = 0. Divide by 2: 6b^2 - 35b - 6 = 0. Solve the quadratic: discriminant Δ = 35^2 + 4 * 6 * 6 = 1225 + 144 = 1369 = 37^2. Thus b = [35 ± 37] / 12. Positive solution: (35 + 37)/12 = 72/12 = 6. So b = 6 km/h. Verification / Alternative check: With b = 6 km/h and c = 1 km/h: upstream speed = 5 km/h, downstream speed = 7 km/h. Upstream time = 35 / 5 = 7 h. Downstream time = 35 / 7 = 5 h. Total time = 7 + 5 = 12 h, which matches the given total time exactly. Why Other Options Are Wrong: 8 km/h, 7.5 km/h or 5.5 km/h for b would yield total times that are not equal to 12 hours when you recompute upstream and downstream times using c = 1 km/h and distance 35 km each way. Common Pitfalls: Mistakes often arise from incorrect simplification of the fraction equation or from arithmetic slips when solving the quadratic. Another pitfall is choosing the negative root of the quadratic, which is not physically meaningful for speed. Always check that your final speed values give positive upstream and downstream speeds and reproduce the given total time. Final Answer: The speed of the motor boat in still water is 6 km/h.

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