Question 1

A man can row at 9 km/h in still water. It takes him twice as long to row a certain distance upstream as it takes to row the same distance downstream. What is the speed of the stream?

Accepted Answer

Introduction / Context:This is a classic boats and streams question involving a comparison of the time taken to travel upstream and downstream. The man's speed in still water is given, and you know that it takes him twice as long to row upstream as downstream over the same distance. From this, you must deduce the speed of the stream. Such problems help you practice equations involving relative speed and time. Given Data / Assumptions: Speed of man (boat) in still water b = 9 km/h. Let speed of stream be c km/h. Downstream speed = (b + c) km/h. Upstream speed = (b − c) km/h. Time upstream is twice the time downstream for the same distance. Concept / Approach:If the distance in each direction is D, then: Time downstream = D / (b + c). Time upstream = D / (b − c). The condition given is that time upstream is twice time downstream, so D / (b − c) = 2 * D / (b + c). We cancel D and solve the resulting equation for c. Once c is known, we have the speed of the stream directly. Step-by-Step Solution:Step 1: Write the time condition: D / (b − c) = 2 * D / (b + c).Step 2: Cancel D from both sides, giving 1 / (b − c) = 2 / (b + c).Step 3: Cross multiply: b + c = 2(b − c).Step 4: Expand right side: b + c = 2b − 2c.Step 5: Bring like terms together: b + c − 2b + 2c = 0 ⇒ −b + 3c = 0.Step 6: So 3c = b ⇒ c = b / 3.Step 7: Since b = 9 km/h, c = 9 / 3 = 3 km/h.Step 8: Therefore, the speed of the stream is 3 km/h. Verification / Alternative check:With b = 9 km/h and c = 3 km/h, downstream speed = 9 + 3 = 12 km/h and upstream speed = 9 − 3 = 6 km/h. For any distance D, time downstream is D/12 and time upstream is D/6. The ratio of times is (D/6) : (D/12) = 2 : 1, which matches the condition that upstream time is twice downstream time. Why Other Options Are Wrong:If c were 2, 2.5, 3.5 or 4 km/h, substituting into upstream and downstream speeds would not produce the required 2:1 ratio of times. For example, if c = 2, downstream speed = 11 and upstream speed = 7, and the time ratio is not exactly 2. Only c = 3 yields upstream and downstream speeds (6 and 12) with the correct time relationship. Common Pitfalls: Reversing the relationship and taking downstream time as twice upstream time. Miswriting speeds as (b − 2c) or (b + 2c) by mistake. Forgetting to cancel the distance D when setting up the time ratio. Attempting to plug in options directly without forming the equation, which can be slower and error-prone. Final Answer:The rate of the stream is 3 km/h.

Question 2

A boat takes 19 hours to travel downstream from point A to point B and then return upstream to point C, where point 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 distan…

Accepted Answer

Introduction / Context:This boats and streams question involves a trip where a boat goes downstream from one point to another and then returns only halfway upstream. You must carefully interpret the distances and use upstream and downstream speeds to form an equation based on the total time. The challenge is to correctly express the total journey time in terms of the unknown distance between the two points. Given Data / Assumptions: Let distance between A and B be D km. Point C is the midpoint of AB, so AC = CB = D/2. The boat travels from A to B downstream and then from B to C upstream. Total time for A → B (downstream) and B → C (upstream) is 19 hours. Speed of boat in still water b = 14 km/h. Speed of stream c = 4 km/h. Downstream speed = b + c = 18 km/h. Upstream speed = b − c = 10 km/h. Concept / Approach:We express the total time as sum of downstream and upstream times: Time downstream A → B = distance / downstream speed = D / 18. Time upstream B → C = distance / upstream speed = (D/2) / 10 = D / 20. We are told this total time equals 19 hours, so we form an equation D/18 + D/20 = 19 and solve for D. Step-by-Step Solution:Step 1: Time from A to B downstream = D / 18 hours.Step 2: Time from B to C upstream = (D/2) / 10 = D / 20 hours.Step 3: Total time = D / 18 + D / 20 = 19.Step 4: Factor D: D(1/18 + 1/20) = 19.Step 5: Compute 1/18 + 1/20 = (20 + 18) / 360 = 38 / 360 = 19 / 180.Step 6: So D * (19 / 180) = 19.Step 7: Divide both sides by 19: D / 180 = 1 ⇒ D = 180.Step 8: Therefore, the distance between A and B is 180 km. Verification / Alternative check:Using D = 180, time downstream from A to B is 180 / 18 = 10 hours. Time upstream from B to C (90 km at 10 km/h) is 90 / 10 = 9 hours. Total time = 10 + 9 = 19 hours, which matches the given condition, confirming that D = 180 km is correct. Why Other Options Are Wrong:Distances like 160 km or 140 km would give totals of time that are not equal to 19 hours when substituted into the equation D / 18 + D / 20. For example, with D = 160, the total time is less than 19 hours. Distances like 120 or 200 km similarly fail to satisfy the given condition. Common Pitfalls: Misinterpreting the journey and assuming the boat returns all the way from B back to A rather than just to midpoint C. Using wrong upstream or downstream speeds (for example, adding instead of subtracting for upstream). Forgetting that the upstream distance on the return leg is only D/2, not D. Making arithmetic errors when adding fractions 1/18 and 1/20. Final Answer:The distance between A and B is 180 km.

Question 3

A man travels 28 km downstream in a motor boat and immediately returns the same 28 km upstream. The upstream trip takes twice as long as the downstream trip. If the speed of the river flow were doubled, the total time for the downstream and upstream…

Accepted Answer

Introduction / Context:This is a more challenging boats and streams problem that involves two scenarios. In the first scenario, you know that the upstream journey takes twice as long as the downstream journey. In the second scenario, the speed of the stream is doubled, and the combined time for going downstream and upstream is given. From this information, you must determine both the speed of the boat in still water and the speed of the stream. This type of problem tests multi-step reasoning and algebraic skill. Given Data / Assumptions: Distance downstream from start to destination = 28 km. Distance upstream back to starting point = 28 km. In the first situation, time upstream is twice time downstream. Let speed of boat in still water be b km/h. Let speed of river flow be s km/h. Downstream speed = b + s; upstream speed = b − s. In the second situation, the speed of the stream becomes 2s. Total time downstream and upstream together in the second situation = 672 minutes = 11.2 hours. Concept / Approach:The problem has two stages: First, use the time ratio condition (upstream time is twice downstream time) to relate b and s. Second, use the changed stream speed and the total time in the second scenario to solve for the numerical values of b and s. We will derive equations from each scenario and solve them step by step. Step-by-Step Solution:Step 1: In the first scenario, downstream time td = 28 / (b + s) and upstream time tu = 28 / (b − s).Step 2: Condition: tu = 2 td ⇒ 28 / (b − s) = 2 * 28 / (b + s).Step 3: Cancel 28: 1 / (b − s) = 2 / (b + s).Step 4: Cross multiply: b + s = 2(b − s) ⇒ b + s = 2b − 2s ⇒ −b + 3s = 0 ⇒ b = 3s.Step 5: In the second scenario, stream speed is doubled to 2s. Then downstream speed = b + 2s and upstream speed = b − 2s.Step 6: Using b = 3s, downstream speed = 3s + 2s = 5s and upstream speed = 3s − 2s = s.Step 7: Total time for 28 km downstream and 28 km upstream is 11.2 hours: 28/(5s) + 28/s = 11.2.Step 8: Factor 28: 28(1/(5s) + 1/s) = 11.2.Step 9: Inside bracket, 1/(5s) + 1/s = 1/(5s) + 5/(5s) = 6/(5s).Step 10: So total time = 28 * 6 / (5s) = 168 / (5s) = 11.2.Step 11: Solve 168 / (5s) = 11.2 ⇒ 168 = 11.2 * 5s = 56s ⇒ s = 168 / 56 = 3 km/h.Step 12: From b = 3s, b = 3 * 3 = 9 km/h. Verification / Alternative check:Check the first scenario: with b = 9 and s = 3, downstream speed = 12 km/h and upstream speed = 6 km/h. Downstream time = 28/12 hours; upstream time = 28/6 hours, which is exactly twice as long. For the second scenario, new downstream speed = 9 + 2*3 = 15 km/h and new upstream speed = 9 − 2*3 = 3 km/h. Total time = 28/15 + 28/3 = 28/15 + 140/15 = 168/15 = 11.2 hours or 672 minutes, exactly as given. This confirms the correctness of the values. Why Other Options Are Wrong:Pairs like 12 km/h & 3 km/h or 8 km/h & 2 km/h do not satisfy both the time ratio and the second-scenario timing when checked carefully. The pair 9 km/h & 6 km/h gives impossible upstream or downstream speeds in the doubled-stream scenario. Only 9 km/h, 3 km/h fulfills all the given conditions. Common Pitfalls: Forgetting to convert 672 minutes into hours (11.2 hours). Not correctly using the fact that the upstream time is twice the downstream time. Making algebraic mistakes when simplifying 28/(5s) + 28/s. Confusing the first scenario's speeds with the second scenario's doubled-stream speeds. 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 4

A motor boat takes 12 hours to travel a certain distance downstream and 24 hours to travel the same distance upstream. How many hours would it take to cover this distance in still water?

Accepted Answer

Introduction / Context:This question compares travel times downstream and upstream for the same distance, and asks for the time in still water. It is a standard boats and streams problem where the key is to relate the downstream and upstream speeds to the still-water speed and the stream speed. Once you find the effective distance-speed relationship, you can determine how long the trip would take in still water. Given Data / Assumptions: Time taken downstream for a certain distance D = 12 hours. Time taken upstream for the same distance D = 24 hours. Let speed of boat in still water be b km/h. Let speed of stream be s km/h. Downstream speed = b + s. Upstream speed = b − s. We are asked for the time taken to cover the distance D in still water at speed b. Concept / Approach:First we use time = distance / speed to express downstream and upstream speeds in terms of D. Then we relate those speeds back to b and s. This gives a simple algebraic relation between b and s. From that relation, we express D in terms of b and finally compute the time in still water as D / b. Step-by-Step Solution:Step 1: Downstream speed = distance / time = D / 12.Step 2: Upstream speed = distance / time = D / 24.Step 3: We know downstream speed = b + s and upstream speed = b − s.Step 4: So b + s = D / 12 and b − s = D / 24.Step 5: Add the two equations: (b + s) + (b − s) = D / 12 + D / 24 ⇒ 2b = D(1/12 + 1/24) = D(3/24) = D/8.Step 6: Therefore b = D / 16.Step 7: Time in still water to cover distance D = D / b = D / (D/16) = 16 hours. Verification / Alternative check:We can also express D in terms of b and s. From downstream, D = 12(b + s); from upstream, D = 24(b − s). Equating gives 12(b + s) = 24(b − s) ⇒ b + s = 2(b − s) ⇒ b = 3s. Using this, we find D = 12(b + s) = 12(3s + s) = 48s and b = 3s. Then time in still water D / b = 48s / (3s) = 16 hours, which confirms our previous result. Why Other Options Are Wrong:Times like 15 hours or 20 hours do not match the derived relation between downstream and upstream speeds. An answer of 8 hours would be too short, implying a still-water speed greater than either downstream or upstream speed, which is impossible since the stream only adds or subtracts speed. Twelve hours corresponds to the downstream time only, not the balanced still-water time for the same distance. Common Pitfalls: Trying to simply average 12 and 24 to get 18 hours, which is incorrect. Assuming that still-water speed is the arithmetic mean of downstream and upstream speeds without relating them to actual distances. Not using the fact that the distance is the same in both directions. Algebraic slip-ups when adding or subtracting (b + s) and (b − s). Final Answer:The boat would take 16 hours to cover the distance in still water.

Question 5

The speed of the current of a stream is 1 km/h. A motor boat goes 35 km upstream and then back 35 km downstream to its starting point in a total of 12 hours. What is the speed of the motor boat in still water?

Accepted Answer

Introduction / Context:This boats and streams problem deals with a round trip: the boat travels upstream a certain distance and then returns downstream to the starting point. You are given the speed of the current and the total time for the entire trip, and asked to find the speed of the boat in still water. Such questions strengthen understanding of relative speed concepts and equation formation. Given Data / Assumptions: Distance upstream = 35 km; distance downstream back = 35 km. Speed of current c = 1 km/h. Let speed of boat in still water be b km/h. Upstream speed = b − 1 km/h. Downstream speed = b + 1 km/h. Total time for upstream and downstream together = 12 hours. Concept / Approach:We use time = distance / speed for each leg of the journey and add the times. This gives an equation involving b, which we then solve. The main idea is that the same distance is traveled twice, but at different effective speeds due to the current, giving different times. Step-by-Step Solution:Step 1: Time taken to go upstream 35 km = 35 / (b − 1) hours.Step 2: Time taken to return downstream 35 km = 35 / (b + 1) hours.Step 3: Total time is given as 12 hours, so we write: 35 / (b − 1) + 35 / (b + 1) = 12.Step 4: Factor 35: 35[1/(b − 1) + 1/(b + 1)] = 12.Step 5: Combine the fractions inside brackets: 1/(b − 1) + 1/(b + 1) = (b + 1 + b − 1) / (b^2 − 1) = 2b / (b^2 − 1).Step 6: So the equation becomes 35 * (2b / (b^2 − 1)) = 12.Step 7: Simplify: 70b / (b^2 − 1) = 12.Step 8: Cross multiply: 70b = 12(b^2 − 1) = 12b^2 − 12.Step 9: Rearranging: 12b^2 − 70b − 12 = 0.Step 10: Divide the whole equation by 2 to simplify: 6b^2 − 35b − 6 = 0.Step 11: Solve the quadratic 6b^2 − 35b − 6 = 0 (for example by factoring or quadratic formula). The positive root is b = 6.Step 12: Therefore, the speed of the motor boat in still water is 6 km/h. Verification / Alternative check:With b = 6 km/h, upstream speed = 6 − 1 = 5 km/h and downstream speed = 6 + 1 = 7 km/h. Upstream time for 35 km = 35 / 5 = 7 hours. Downstream time for 35 km = 35 / 7 = 5 hours. Total time = 7 + 5 = 12 hours, which matches the given total time, confirming that b = 6 km/h is correct. Why Other Options Are Wrong:Speeds like 8 or 7.5 km/h do not give a total time of 12 hours when substituted into the equation. A speed of 5.5 km/h yields non-integer and incorrect total time compared to 12 hours. A speed of 9 km/h would make the boat too fast and total time too short. Only 6 km/h leads to a perfect match with the given total round-trip time. Common Pitfalls: Using 35 km as the total distance instead of 70 km (35 upstream + 35 downstream). Writing upstream speed as b + 1 instead of b − 1. Algebraic errors when combining fractions or when solving the quadratic equation. Trying to guess the answer without verifying that the total time is exactly 12 hours. Final Answer:The speed of the motor boat in still water is 6 km/h.

Question 6

A boy can swim at 4.5 km/h in still water. It takes him twice as long to swim a certain distance upstream as it takes to swim the same distance downstream. What is the speed of the stream?

Accepted Answer

Introduction / Context:This swimming-in-a-stream question is another relative speed problem. The swimmer's speed in still water is known and you are told that the time taken to swim upstream is twice the time taken downstream for the same distance. From this information, you must find the speed of the stream. The structure closely resembles standard boats and streams problems, just with a swimmer instead of a boat. Given Data / Assumptions: Speed of boy in still water b = 4.5 km/h. Let speed of stream be c km/h. Downstream speed = b + c km/h. Upstream speed = b − c km/h. Time upstream is twice the time downstream for a fixed distance. Concept / Approach:Let the distance in each direction be D. Then: Downstream time td = D / (b + c). Upstream time tu = D / (b − c). The condition tu = 2 td leads to an equation in b and c. Since b is known, this equation allows us to solve for c. Step-by-Step Solution:Step 1: Write the time relation: D / (b − c) = 2 * D / (b + c).Step 2: Cancel D from both sides: 1 / (b − c) = 2 / (b + c).Step 3: Cross multiply: b + c = 2(b − c).Step 4: Expand right side: b + c = 2b − 2c.Step 5: Rearranging: b + c − 2b + 2c = 0 ⇒ −b + 3c = 0.Step 6: Therefore, 3c = b ⇒ c = b / 3.Step 7: Substitute b = 4.5 km/h: c = 4.5 / 3 = 1.5 km/h.Step 8: Thus, the speed of the stream is 1.5 km/h. Verification / Alternative check:With c = 1.5 km/h, downstream speed = 4.5 + 1.5 = 6 km/h and upstream speed = 4.5 − 1.5 = 3 km/h. For any distance D, downstream time is D/6 and upstream time is D/3. The ratio D/3 : D/6 simplifies to 2 : 1, which exactly matches the condition that upstream time is twice downstream time. Why Other Options Are Wrong:Speeds like 1.8, 2, 2.2 or 2.5 km/h would produce downstream and upstream speeds that do not satisfy the exact 2:1 ratio of times. Only when c is 1.5 km/h do the effective speeds (6 and 3) yield an upstream time that is double the downstream time for any fixed distance. Common Pitfalls: Reversing the roles of upstream and downstream times, leading to the wrong equation. Taking c = b/2 instead of b/3, perhaps by misreading the condition. Neglecting to cancel the distance D before manipulating the equation. Confusing the swimmer's speed in still water with downstream speed. Final Answer:The speed of the stream is 1.5 km/h.

Question 7

A man rows downstream from his starting point to a destination and then rows back upstream to the starting point in a total of 5 hours. If the speed of the boat in still water is 10 km/h and the speed of the stream is 4 km/h, what is the distance be…

Accepted Answer

Introduction / Context:This question describes a round trip on a river: a man travels from his starting point to a destination downstream and then returns upstream. You are given the total time for the complete up-and-down journey, as well as the speed of the boat in still water and the speed of the stream. You must determine the one-way distance to the destination. This is a standard application of relative speed and time equations in boats and streams problems. Given Data / Assumptions: Speed of boat in still water b = 10 km/h. Speed of stream s = 4 km/h. Downstream speed = b + s = 14 km/h. Upstream speed = b − s = 6 km/h. Total time for downstream plus upstream journey = 5 hours. Let one-way distance to the destination be D km. Concept / Approach:We express the total time as the sum of the downstream and upstream travel times for the same distance D: Downstream time = D / 14 hours. Upstream time = D / 6 hours. The sum of these two times is given as 5 hours. So we solve the equation D/14 + D/6 = 5 for D. Step-by-Step Solution:Step 1: Write the total time equation: D/14 + D/6 = 5.Step 2: Factor D: D(1/14 + 1/6) = 5.Step 3: Compute 1/14 + 1/6 = (3/42 + 7/42) = 10/42 = 5/21.Step 4: So D * (5/21) = 5.Step 5: Divide both sides by (5/21): D = 5 * (21/5) = 21.Step 6: Therefore, the distance between the starting point and the destination is 21 km. Verification / Alternative check:Substitute D = 21 back into the times. Downstream time = 21 / 14 = 1.5 hours. Upstream time = 21 / 6 = 3.5 hours. Total time = 1.5 + 3.5 = 5 hours, exactly as given in the problem. This confirms that D = 21 km is correct. Why Other Options Are Wrong:If D were 16, 18, 25 or 12 km, the total time D/14 + D/6 would not equal 5 hours. For instance, with D = 16, the total time is less than 5 hours, and with D = 25, it exceeds 5 hours. Only D = 21 satisfies the equation derived from the given speeds and total time. Common Pitfalls: Using 10 km/h and 4 km/h directly as the upstream and downstream speeds, instead of adding or subtracting the stream speed. Forgetting that both the downstream and upstream distances are D, not different values. Arithmetic mistakes when adding the fractions 1/14 and 1/6. Trying to approximate instead of forming and solving the exact equation. Final Answer:The destination is 21 km away from the starting point.

Question 8

A man rows his boat 60 km downstream and 30 km upstream, taking 3 hours for each trip. What is the speed of the stream?

Accepted Answer

Introduction / Context:This boats and streams problem gives different distances downstream and upstream but the same time for each journey. That information allows you to calculate downstream and upstream speeds directly, and then deduce the speed of the stream and the speed of the boat in still water. The question specifically asks for the speed of the stream, making it a simple but instructive exercise in relative speed. Given Data / Assumptions: Downstream distance = 60 km; time taken = 3 hours. Upstream distance = 30 km; time taken = 3 hours. Let speed of stream be s km/h. Let speed of boat in still water be b km/h. Downstream speed = b + s. Upstream speed = b − s. Concept / Approach:We first compute the effective speeds from the given distance and time for each direction: Downstream speed = distance / time = 60 / 3 = 20 km/h. Upstream speed = distance / time = 30 / 3 = 10 km/h. Then we use the standard relations b + s = 20 and b − s = 10 to solve for b and s. Finally, we report s, the speed of the stream. Step-by-Step Solution:Step 1: Downstream speed = 60 / 3 = 20 km/h.Step 2: Upstream speed = 30 / 3 = 10 km/h.Step 3: Let b be speed in still water and s be speed of stream. Then b + s = 20 and b − s = 10.Step 4: Add the two equations: (b + s) + (b − s) = 20 + 10 ⇒ 2b = 30.Step 5: So b = 30 / 2 = 15 km/h.Step 6: Substitute back into b + s = 20: 15 + s = 20 ⇒ s = 5 km/h.Step 7: Thus, the speed of the stream is 5 km/h. Verification / Alternative check:With b = 15 and s = 5, downstream speed = 15 + 5 = 20 km/h, so time for 60 km downstream is 60 / 20 = 3 hours. Upstream speed = 15 − 5 = 10 km/h, so time for 30 km upstream is 30 / 10 = 3 hours. Both match the conditions given in the question, confirming that s = 5 km/h is correct. Why Other Options Are Wrong:10 km/h or 15 km/h as stream speeds would imply impossible or negative still-water speeds when solving b + s = 20 and b − s = 10. A stream speed of 45 km/h is clearly impossible because it would exceed the downstream speed itself. 7.5 km/h does not satisfy both equations simultaneously. Only 5 km/h yields consistent downstream and upstream speeds with the given travel times. Common Pitfalls: Assuming upstream distance is also 60 km instead of 30 km. Mixing up which speed corresponds to downstream or upstream motion. Arithmetic mistakes when adding or subtracting the equations b + s = 20 and b − s = 10. Guessing the answer without verifying that both given times are satisfied. Final Answer:The speed of the stream is 5 km/h.

Question 9

A boatman can row 3 km upstream against the current in 20 minutes and then return the same 3 km downstream in 18 minutes. What is the speed of the river current (in km/h)?

Accepted Answer

Introduction / Context: In this boats and streams problem, we are given the time taken by a boatman to cover the same distance upstream and downstream. The question asks for the speed of the current. Such questions test understanding of relative speed concepts, where the effective speed of a boat changes depending on whether it is moving with or against the current. By working with distance, time, and speed relations, we can isolate and calculate the speed of the stream in km/h. Given Data / Assumptions: The distance between the two points is 3 km each way. Time taken upstream (against the current) is 20 minutes. Time taken downstream (with the current) is 18 minutes. We assume uniform speed of the boat in still water and a uniform stream speed. We must express time in hours to find speeds in km/h. Concept / Approach: The core idea is that: Upstream speed = boat speed in still water - stream speed. Downstream speed = boat speed in still water + stream speed. Once we compute both effective speeds from distance and time, we can use simple algebra. The average of upstream and downstream speeds gives the boat speed in still water, and half the difference of those speeds gives the stream speed. Step-by-Step Solution: Distance each way = 3 km. Upstream time = 20 minutes = 20 / 60 hours = 1/3 hours. Downstream time = 18 minutes = 18 / 60 hours = 3/10 hours. Upstream speed = distance / time = 3 / (1/3) = 9 km/h. Downstream speed = distance / time = 3 / (3/10) = 10 km/h. Let b = speed of boat in still water, c = speed of current. Then b - c = 9 and b + c = 10. Add the equations: 2b = 19 so b = 19 / 2 = 9.5 km/h. Subtract the equations: 2c = 1 so c = 1/2 = 0.5 km/h. Verification / Alternative check: If the stream speed is 0.5 km/h, then upstream speed is 9.5 - 0.5 = 9 km/h and downstream speed is 9.5 + 0.5 = 10 km/h. Time upstream is 3 / 9 = 1/3 hours, which is 20 minutes. Time downstream is 3 / 10 hours, which is 0.3 hours or 18 minutes. These match the given data, so the calculated stream speed is consistent and correct. Why Other Options Are Wrong: Option 1/3 km/h underestimates the current because it would not create the observed difference between 20 minutes and 18 minutes travel times. Option 2/3 km/h is too large and would make upstream travel significantly slower than given. Option 1/4 km/h is also too small and would not adequately explain the 2 minute time difference. Only 1/2 km/h reproduces both the upstream and downstream times correctly. Common Pitfalls: Many learners forget to convert minutes into hours before calculating speeds. Others incorrectly average the times instead of the speeds, which leads to wrong values for the boat and stream speeds. Another frequent mistake is to treat the boat speed in still water as the average of distance and time directly, without separating upstream and downstream situations. Careful unit handling and correct use of the upstream and downstream formulas are essential. Final Answer: The speed of the current is 1/2 km/h.

Question 10

A man can row at 6 km/h in still water. The river flows at 1.2 km/h. It takes him a total of 1 hour to row from his starting point to a place and then return. What is the total distance he travels (in km)?

Accepted Answer

Introduction / Context: This question involves a boat travelling in a river where there is a current. The speed of the boat in still water and the speed of the stream are known, and the total time for a round trip is given. We are asked to compute the total distance covered. The idea is to work out the downstream and upstream speeds, express the one way distance as a variable, and then use the total time equation to find that distance and hence the round trip distance. Given Data / Assumptions: Speed of boat in still water = 6 km/h. Speed of river current = 1.2 km/h. Total time for going to the destination and coming back = 1 hour. The distance between the two points is the same in both directions. We assume uniform speeds throughout the motion. Concept / Approach: When a boat moves with the current, its effective speed is boat speed plus stream speed. When it moves against the current, its effective speed is boat speed minus stream speed. If the one way distance is d km, the total time is d divided by downstream speed plus d divided by upstream speed. This total is given as 1 hour, which allows us to solve for d and hence find the total distance 2d. Step-by-Step Solution: Let d be the distance (in km) from the starting point to the place. Downstream speed = 6 + 1.2 = 7.2 km/h. Upstream speed = 6 - 1.2 = 4.8 km/h. Time downstream = d / 7.2 hours. Time upstream = d / 4.8 hours. Total time = d / 7.2 + d / 4.8 = 1 hour. Compute d / 7.2 + d / 4.8 = 1. Take common denominator: 7.2 * 4.8. This simplifies to d * (4.8 + 7.2) / (7.2 * 4.8) = 1. So d * 12 / (34.56) = 1 giving d = 34.56 / 12 = 2.88 km. Total distance travelled = 2d = 2 * 2.88 = 5.76 km. Verification / Alternative check: Using d = 2.88 km, time downstream is 2.88 / 7.2 = 0.4 hours (24 minutes). Time upstream is 2.88 / 4.8 = 0.6 hours (36 minutes). The sum 0.4 + 0.6 = 1 hour matches the given total time. Therefore the distance calculation is correct and the total distance travelled is 5.76 km. Why Other Options Are Wrong: A total distance of 4.58 km or 5.24 km would produce total times different from 1 hour when substituted into the same calculation. A distance of 6.35 km would also result in a longer total time because the speeds are fixed and the distance is larger. Only 5.76 km makes the downstream and upstream times sum to the given 1 hour. Common Pitfalls: Students sometimes confuse total distance with one way distance, mistakenly using d instead of 2d. Another common mistake is to directly average the speeds instead of correctly using time = distance / speed for each leg of the journey. Forgetting to add and subtract the current speed correctly also leads to errors. Careful use of the formulas is important to avoid those traps. Final Answer: The man travels a total distance of 5.76 km during the round trip.

Question 11

A boat takes 8 hours 48 minutes to travel a certain distance upstream and 4 hours to cover the same distance downstream. What is the ratio of the speed of the boat in still water to the speed of the water current?

Accepted Answer

Introduction / Context: This question deals with the relationship between upstream time, downstream time, and the speeds of a boat and the current. We are not asked for the numerical values of the speeds, but rather for the ratio of the speed of the boat in still water to the speed of the stream. This is a common pattern in aptitude tests and is easy to handle using the standard boats and streams formulas and time relations. Given Data / Assumptions: Upstream time to cover a certain distance = 8 hours 48 minutes. Downstream time to cover the same distance = 4 hours. Distance is the same in both directions. Boat and stream speeds are constant for the entire travel. We need the ratio boat speed : current speed. Concept / Approach: Let the distance be D, boat speed in still water be b, and current speed be c. Then: Upstream speed = b - c = D / T_up. Downstream speed = b + c = D / T_down. Although we do not know D, by expressing b and c in terms of D and times, we can eliminate D and find a ratio b : c using a time based shortcut. The ratio b : c can be written as (T_up + T_down) : (T_up - T_down). Step-by-Step Solution: Convert 8 hours 48 minutes into hours: 48 minutes = 48 / 60 = 0.8 hours. So T_up = 8.8 hours and T_down = 4 hours. Let D be the distance. Then upstream speed = D / 8.8 and downstream speed = D / 4. Boat speed b = (upstream speed + downstream speed) / 2. Stream speed c = (downstream speed - upstream speed) / 2. Thus b : c = (T_up + T_down) : (T_up - T_down). Compute T_up + T_down = 8.8 + 4 = 12.8. Compute T_up - T_down = 8.8 - 4 = 4.8. So b : c = 12.8 : 4.8. Divide both terms by 0.8 to simplify: 12.8 / 0.8 = 16 and 4.8 / 0.8 = 6, so ratio = 16 : 6. Further simplify by dividing by 2 to get 8 : 3. Verification / Alternative check: If we assume D = 1 unit for convenience, upstream speed is 1 / 8.8 and downstream speed is 1 / 4. Then b = (1 / 8.8 + 1 / 4) / 2 and c = (1 / 4 - 1 / 8.8) / 2. Computing these values and then forming b : c yields the same ratio 8 : 3 after simplification. Therefore the shortcut using times is reliable and confirms the result. Why Other Options Are Wrong: Ratios such as 7:4, 11:4, and 4:7 do not correspond to the time combination pattern. If we tried to reconstruct speeds using these ratios, the implied upstream and downstream times would not match 8 hours 48 minutes and 4 hours. Only 8:3 is consistent with the given timings. Common Pitfalls: A typical mistake is to attempt to find absolute speeds with an assumed distance but then lose track of simplifying to a ratio. Another error is to forget converting 48 minutes into hours, which changes the value of T_up. Some learners also guess that the ratio must be T_up : T_down, which is not correct. Remember that the correct time based ratio for b : c is (T_up + T_down) : (T_up - T_down). Final Answer: The required ratio of the speed of the boat in still water to the speed of the current is 8:3.

Question 12

A boat has a speed of 9 km/h in still water. It goes 12 km downstream and then returns 12 km upstream, taking a total of 3 hours for the round trip. What is the speed of the stream (in km/h)?

Accepted Answer

Introduction / Context: This problem is about a boat that travels downstream and then upstream over the same distance. We know the speed of the boat in still water and the total time for the round trip. The objective is to find the speed of the river current. The concept relies on understanding that downstream and upstream speeds are different due to the current and that total time is the sum of times for each leg. Given Data / Assumptions: Boat speed in still water = 9 km/h. Distance downstream = 12 km. Distance upstream = 12 km. Total time for the trip = 3 hours. Speed of stream = c km/h (unknown). Speeds are constant and the river current is uniform. Concept / Approach: If b is the speed of the boat in still water and c is the speed of the current, then: Downstream speed = b + c. Upstream speed = b - c. Time taken downstream is distance divided by downstream speed, and time upstream is distance divided by upstream speed. Their sum is the given total time. With b known and total time known, we can set up an equation in c and solve it. Step-by-Step Solution: Let b = 9 km/h and c be the speed of the stream. Downstream speed = 9 + c km/h. Upstream speed = 9 - c km/h. Time downstream = 12 / (9 + c) hours. Time upstream = 12 / (9 - c) hours. Total time = 12 / (9 + c) + 12 / (9 - c) = 3 hours. Multiply both sides by (9 + c)(9 - c): 12(9 - c) + 12(9 + c) = 3(81 - c^2). Left side simplifies to 108 - 12c + 108 + 12c = 216. So 216 = 3(81 - c^2). Divide both sides by 3: 72 = 81 - c^2. So c^2 = 81 - 72 = 9 and hence c = 3 km/h. Verification / Alternative check: If c = 3 km/h, downstream speed is 9 + 3 = 12 km/h and upstream speed is 9 - 3 = 6 km/h. Downstream time is 12 / 12 = 1 hour and upstream time is 12 / 6 = 2 hours. The total is 1 + 2 = 3 hours, which matches the given total time. Therefore the calculated current speed 3 km/h is correct. Why Other Options Are Wrong: If the stream were only 1 km/h or 4 km/h, the total time would not match 3 hours when recalculated. A current speed of 5 km/h would produce an upstream speed of only 4 km/h, which would make the upstream leg take too long. Thus 3 km/h is the only value that produces a consistent round trip time. Common Pitfalls: One common error is to simply average the speeds or times without using the correct equation. Another mistake is to treat 3 hours as the time for one way instead of the round trip. Some students also forget that upstream speed is boat speed minus stream speed. Setting up the time equation correctly and solving the resulting quadratic in a careful way avoids these mistakes. Final Answer: The speed of the stream is 3 km/h.

Question 13

A man can row at 8 km/h in still water. The river current flows at 2 km/h. For the same distance, it takes him 4 hours longer to row upstream than to row downstream. What is that distance (in km)?

Accepted Answer

Introduction / Context: This problem is about comparing the time taken to travel the same distance upstream and downstream when both the boat speed and stream speed are known. We are told that the upstream journey takes 4 hours more than the downstream journey, and we need to find the common distance. Such questions require forming equations using the basic relation time = distance / speed for both directions. Given Data / Assumptions: Boat speed in still water = 8 km/h. River current speed = 2 km/h. Distance covered upstream and downstream is the same (call it d km). Upstream journey takes 4 hours more than downstream journey. Speeds and current are uniform. Concept / Approach: The effective speeds are: Downstream speed = boat speed + current = 8 + 2 = 10 km/h. Upstream speed = boat speed - current = 8 - 2 = 6 km/h. If the distance is d km, time downstream is d / 10 hours and time upstream is d / 6 hours. The difference between these times is 4 hours. This leads to a simple linear equation in d that we can solve to obtain the distance. Step-by-Step Solution: Let the distance be d km. Downstream speed = 10 km/h, upstream speed = 6 km/h. Downstream time = d / 10 hours. Upstream time = d / 6 hours. Given that upstream time is 4 hours more: d / 6 - d / 10 = 4. Find a common denominator: 30. Then (5d - 3d) / 30 = 4, so 2d / 30 = 4. Simplify to d / 15 = 4, thus d = 4 * 15 = 60 km. Verification / Alternative check: If the distance is 60 km, downstream time is 60 / 10 = 6 hours. Upstream time is 60 / 6 = 10 hours. The difference 10 - 6 = 4 hours matches the given condition. So the calculated distance 60 km is consistent with all the data and is therefore correct. Why Other Options Are Wrong: For 32 km, the times would be 3.2 hours downstream and about 5.33 hours upstream, a difference of only 2.13 hours. For 45 km, the times would be 4.5 hours and 7.5 hours, a difference of 3 hours. For 54 km, the difference is 3.6 hours. None of these match the required 4 hour difference. Only 60 km gives exactly a 4 hour difference between upstream and downstream travel times. Common Pitfalls: A frequent error is to subtract the speeds first and incorrectly think that time difference is distance divided by speed difference. That approach does not correctly reflect the relation between time, distance, and speed in each direction. Another pitfall is misreading the statement and treating 4 hours as the total time rather than the difference in times. Setting up the equation d / 6 - d / 10 = 4 is the key step to solving such problems correctly. Final Answer: The distance for which the upstream time is 4 hours more than the downstream time is 60 km.

Question 14

A motorboat can travel 10 miles downstream on a river in 20 minutes and takes 30 minutes to travel the same 10 miles upstream. What is the speed of the river current (in miles per hour)?

Accepted Answer

Introduction / Context: Here we have a motorboat moving in a river. We know the time it takes to travel a fixed distance downstream and the time to travel the same distance upstream. From these times we can derive the downstream and upstream speeds, and then find the speed of the current by using the basic boats and streams relations. The problem uses miles and minutes, so careful unit conversion is important. Given Data / Assumptions: Distance downstream = 10 miles. Distance upstream = 10 miles. Time downstream = 20 minutes. Time upstream = 30 minutes. Speeds of boat and current are constant. We need the speed of the current in miles per hour (mph). Concept / Approach: Let b be the boat speed in still water and c be the current speed. Then: Downstream speed = b + c. Upstream speed = b - c. Using distance = 10 miles and the times in hours, we can find numerical values for b + c and b - c. Solving these two linear equations gives both b and c. The answer required is c, the speed of the current. Step-by-Step Solution: Convert times to hours: 20 minutes = 20 / 60 = 1/3 hour, 30 minutes = 30 / 60 = 1/2 hour. Downstream speed = distance / time = 10 / (1/3) = 30 mph. Upstream speed = distance / time = 10 / (1/2) = 20 mph. Let b be boat speed in still water and c be current speed. Then b + c = 30 and b - c = 20. Add the equations: 2b = 50 so b = 25 mph. Subtract the equations: 2c = 10 so c = 5 mph. Verification / Alternative check: With b = 25 mph and c = 5 mph, downstream speed is 25 + 5 = 30 mph, giving time 10 / 30 hours, which is 1/3 hour or 20 minutes. Upstream speed is 25 - 5 = 20 mph, giving time 10 / 20 = 1/2 hour or 30 minutes. These match the given times exactly, confirming that 5 mph is the correct current speed. Why Other Options Are Wrong: If the current were 6 mph or 7 mph, the implied upstream and downstream speeds would not produce the given travel times for 10 miles. A current of 8 mph would be even more inconsistent. Only 5 mph leads to the correct pair of effective speeds, 30 mph and 20 mph, that match both the downstream and upstream journeys. Common Pitfalls: Learners sometimes forget to convert minutes into hours and end up with speeds in mixed units. Another common mistake is to confuse downstream and upstream formulas and write b - c for downstream speed. A third error is to average the two speeds incorrectly rather than solving the simple system of equations. Keeping track of units and using the linear equations b + c and b - c is the safest method. Final Answer: The speed of the river current is 5 mph.

Question 15

In one hour, a boat covers 12 km along the stream and 6 km against the stream. What is the speed of the boat in still water (in km/h)?

Accepted Answer

Introduction / Context: This is a straightforward boats and streams problem where the effective downstream and upstream speeds are directly given as distances covered in one hour. From these effective speeds, we can easily compute the speed of the boat in still water by using the relation between upstream speed, downstream speed, and the boat and stream speeds. The question checks understanding of average of two opposite direction speeds in still water. Given Data / Assumptions: Distance covered downstream in one hour = 12 km, so downstream speed = 12 km/h. Distance covered upstream in one hour = 6 km, so upstream speed = 6 km/h. Let b be the speed of the boat in still water. Let c be the speed of the stream. Speeds are assumed constant. Concept / Approach: For a boat in a stream: Downstream speed = b + c. Upstream speed = b - c. We are given both downstream and upstream speeds numerically, so we can directly set up two linear equations in b and c and solve. The boat speed in still water is simply the average of the downstream and upstream speeds. Step-by-Step Solution: Downstream speed = 12 km/h, upstream speed = 6 km/h. Let b be the speed of the boat in still water and c be the speed of the stream. Then b + c = 12 and b - c = 6. Add the two equations: (b + c) + (b - c) = 12 + 6. This gives 2b = 18. So b = 18 / 2 = 9 km/h. Verification / Alternative check: If the boat speed in still water is 9 km/h, then the sum of upstream and downstream speeds must be 2b = 18 km/h. The given speeds 12 km/h and 6 km/h indeed add up to 18 km/h. The stream speed is then (downstream - upstream) / 2 = (12 - 6) / 2 = 3 km/h, which is a reasonable value. The given distances covered in one hour match these speeds, so the solution is consistent. Why Other Options Are Wrong: Options 8 km/h, 7 km/h, and 7.5 km/h would yield different sums of upstream and downstream speeds. For example, if b were 8 km/h, the sum of downstream and upstream speeds would be 16 km/h, which does not match the observed 12 + 6 = 18 km/h. Therefore only 9 km/h correctly fits the given data. Common Pitfalls: Students sometimes confuse the formula and think that the boat speed equals the difference between downstream and upstream speeds divided by 2, which actually gives the stream speed. It is important to remember that boat speed is (downstream + upstream) / 2 and stream speed is (downstream - upstream) / 2. Keeping the formulas straight avoids this common confusion. Final Answer: The speed of the boat in still water is 9 km/h.

Question 16

The sum of a boat’s upstream and downstream speeds is 82 km/h. The boat travels 105 km upstream in 3 hours. How much time (in hours) will it take to cover 126 km downstream?

Accepted Answer

Introduction / Context: This question combines upstream and downstream speeds with information about actual travel time. We are given the sum of upstream and downstream speeds and the time taken to travel a known upstream distance. From that we find the individual upstream speed and obtain the downstream speed. Finally we use the downstream speed to compute how long it will take to travel another known distance with the current. This kind of question tests algebraic manipulation and application of the basic speed relations. Given Data / Assumptions: Upstream distance covered = 105 km. Time taken upstream = 3 hours. Sum of upstream and downstream speeds = 82 km/h. Downstream distance to be covered = 126 km. Speeds are constant and current is uniform. Concept / Approach: We first compute upstream speed from distance and time. Then we use the given sum of upstream and downstream speeds to find downstream speed. Finally, time downstream is simply distance divided by downstream speed. Rounding to one decimal place is enough to match the options given. Step-by-Step Solution: Upstream speed = distance / time = 105 / 3 = 35 km/h. Let u be upstream speed and d be downstream speed. Given u + d = 82 km/h. We have u = 35, so 35 + d = 82. Thus d = 82 - 35 = 47 km/h. Downstream speed = 47 km/h. Downstream distance to be covered = 126 km. Time downstream = distance / speed = 126 / 47 hours. Compute 126 / 47 which is approximately 2.68 hours. Rounding to one decimal place gives about 2.7 hours. Verification / Alternative check: We can estimate 47 * 2.7 = 47 * (2 + 0.7) = 94 + 32.9 = 126.9 km, which is very close to 126 km. Using the exact fraction 126 / 47 is about 2.68 hours, so option 2.7 hours is the best match among the choices. The upstream speed of 35 km/h is consistent with 105 km in 3 hours and the sum of speeds being 82 km/h, confirming the calculations. Why Other Options Are Wrong: If the time were 2.5 hours, the distance covered at 47 km/h would be only 117.5 km, which is too short. If the time were 2.6 hours, the distance would be 122.2 km, still less than 126 km. If the time were 2.8 hours, the distance would be 131.6 km, which is too large. Only around 2.7 hours gives a distance close to 126 km, matching the required value. Common Pitfalls: A common mistake is to misinterpret the sum of speeds 82 km/h as twice the still water speed without checking the upstream data. Another error is rounding the intermediate speeds too early, which can produce a noticeably different final time. It is good practice to keep fractions exact until the last step and then round only when comparing with the given options. Final Answer: The boat will take approximately 2.7 hours to travel 126 km downstream.

Question 17

A boat travels 72 km downstream in 8 hours and 84 km upstream in 12 hours. Find the speed of the boat in still water and the speed of the stream (in km/h).

Accepted Answer

Introduction / Context: This question gives different distances and times for downstream and upstream travel. From these we can calculate the effective upstream and downstream speeds, then use them to determine the boat speed in still water and the stream speed. It is a classic application of boats and streams formulas and reinforces the idea that effective speed depends on whether the boat moves with or against the current. Given Data / Assumptions: Downstream distance = 72 km, time = 8 hours. Upstream distance = 84 km, time = 12 hours. Let b be the speed of the boat in still water. Let c be the speed of the current. Speeds are constant in both directions. Concept / Approach: We first find effective speeds: Downstream speed = distance / time = 72 / 8. Upstream speed = distance / time = 84 / 12. Then we use the relations: b + c = downstream speed. b - c = upstream speed. Solving these equations gives b and c. Finally, we match our answer to the option written as boat speed and stream speed. Step-by-Step Solution: Downstream speed = 72 / 8 = 9 km/h. Upstream speed = 84 / 12 = 7 km/h. Let b be boat speed, c be stream speed. Then b + c = 9 and b - c = 7. Add the equations: 2b = 16 so b = 8 km/h. Subtract the equations: 2c = 2 so c = 1 km/h. Therefore, boat speed = 8 km/h and stream speed = 1 km/h. Verification / Alternative check: Check downstream: effective speed 8 + 1 = 9 km/h; time for 72 km is 72 / 9 = 8 hours, matching the given time. Check upstream: effective speed 8 - 1 = 7 km/h; time for 84 km is 84 / 7 = 12 hours, which matches as well. This confirms that the values b = 8 km/h and c = 1 km/h satisfy all conditions. Why Other Options Are Wrong: Option 9 and 3 km/h would imply downstream and upstream speeds that do not produce the given times for 72 km and 84 km. Similarly, 6 and 7 km/h or 7 and 2 km/h do not satisfy both distance time pairs simultaneously. Only the pair 8 and 1 km/h correctly reproduces the 8 hours downstream and 12 hours upstream travel times. Common Pitfalls: Some students mistakenly average the distances or times incorrectly or mix up which pair corresponds to boat speed and which to stream speed. Another frequent issue is forgetting that different distances require separate speed calculations before forming equations. Always compute downstream and upstream speeds first and only then set up the equations b + c and b - c to avoid confusion. Final Answer: The speed of the boat in still water is 8 km/h and the speed of the stream is 1 km/h.

Question 18

A boatman can row 96 km downstream in 8 hours. If the speed of the current is 4 km/h, then in how much time (in hours) will he cover 8 km upstream?

Accepted Answer

Introduction / Context: In this problem, we are given the time and distance for a downstream journey and the speed of the current. From this, we must first determine the speed of the boat in still water, and then use it to find the time needed to travel a shorter distance upstream. This exercise reinforces the relationships between downstream speed, upstream speed, boat speed, and stream speed. Given Data / Assumptions: Downstream distance = 96 km. Downstream time = 8 hours. Speed of the current = 4 km/h. We need time to travel 8 km upstream. Speeds are uniform, and the river flow is steady. Concept / Approach: First we compute the downstream speed from distance and time. Then we use the relation downstream speed = boat speed + current speed to find the boat speed in still water. Once we know the boat speed, upstream speed is boat speed minus current speed. Finally, time upstream is distance divided by upstream speed. Step-by-Step Solution: Downstream speed = distance / time = 96 / 8 = 12 km/h. Let b be boat speed in still water and c be current speed. Given c = 4 km/h and b + c = 12 km/h. So b = 12 - 4 = 8 km/h. Upstream speed = b - c = 8 - 4 = 4 km/h. Upstream distance to be covered = 8 km. Time upstream = distance / speed = 8 / 4 = 2 hours. Verification / Alternative check: We can verify these speeds with the original downstream journey. With b = 8 km/h and c = 4 km/h, downstream speed is 12 km/h. Time to cover 96 km is 96 / 12 = 8 hours, which matches the given time. For the upstream leg, going at 4 km/h over 8 km clearly takes 2 hours. The calculations are consistent with all the given information. Why Other Options Are Wrong: A time of 1 hour would correspond to an upstream speed of 8 km/h, which contradicts the calculated upstream speed of 4 km/h. Times like 1.5 hours or 2.5 hours would also imply inconsistent speeds or contradict the relationship between boat speed, current speed, and downstream speed. Only 2 hours matches the correct upstream speed and the target distance. Common Pitfalls: Sometimes learners directly average the known downstream speed and current speed without properly solving for the boat speed. Others mistakenly subtract in the wrong direction when forming upstream and downstream speeds. To avoid such errors, always use the formulas downstream = b + c and upstream = b - c and then carefully substitute values and compute the required time. Final Answer: The boatman will take 2 hours to cover 8 km upstream.

Question 19

Rajesh rows a boat in still water at 4.5 km/h to go to a place and then return. If the river flows at 1.5 km/h, what is his average speed (in km/h) for the entire round trip?

Accepted Answer

Introduction / Context: This problem asks for the average speed of a round trip in a river, where Rajesh rows a boat to a certain point and returns. His speed in still water and the speed of the stream are given. Because downstream and upstream speeds differ, the effective average speed is not a simple average of the two speeds but rather total distance divided by total time. This illustrates the classic concept of harmonic type averaging in speed questions. Given Data / Assumptions: Boat speed in still water = 4.5 km/h. Stream speed = 1.5 km/h. Distance from start to destination is d km (same on return). He rows the same route downstream and then upstream. We assume steady speeds throughout. Concept / Approach: We first compute downstream and upstream speeds. Then we find the time taken for each leg in terms of d. Total distance is 2d and total time is the sum of downstream and upstream times. Average speed for the whole journey is total distance divided by total time. Since d cancels, we get a numeric answer independent of the actual one way distance. Step-by-Step Solution: Downstream speed = 4.5 + 1.5 = 6 km/h. Upstream speed = 4.5 - 1.5 = 3 km/h. Let d be the one way distance in km. Time downstream = d / 6 hours. Time upstream = d / 3 hours. Total distance = 2d km. Total time = d / 6 + d / 3 = d / 6 + 2d / 6 = 3d / 6 = d / 2 hours. Average speed = total distance / total time = 2d / (d / 2) = 2d * 2 / d = 4 km/h. Verification / Alternative check: We can test with a specific value, for example d = 6 km. Downstream, Rajesh would take 6 / 6 = 1 hour. Upstream, he would take 6 / 3 = 2 hours. Total distance is 12 km and total time is 3 hours, giving an average speed of 12 / 3 = 4 km/h. This matches the value obtained algebraically and confirms the correctness of the solution. Why Other Options Are Wrong: An average speed of 6 km/h would suggest that the boat travelled at downstream speed the whole way, ignoring the slower upstream leg. An average of 2 km/h or 8 km/h has no basis in the given speeds and results in inconsistent time calculations. Only 4 km/h balances the downstream and upstream segments correctly over the entire journey. Common Pitfalls: Students often incorrectly take the simple arithmetic average of 6 km/h and 3 km/h as (6 + 3) / 2 = 4.5 km/h, which is not the correct average speed for a round trip over equal distances. The correct method always uses total distance divided by total time. Remember that when distances are equal, the average speed is closer to the smaller speed because more time is spent at that speed. Final Answer: Rajesh’s average speed for the entire trip is 4 km/h.

Question 20

A boat covers 36 km downstream in 6 hours, which is 3 hours less than the time taken to cover the same distance upstream. What is the speed of the stream (in km/h)?

Accepted Answer

Introduction / Context: This question compares downstream and upstream travel times for the same distance. The downstream time is given, and the upstream time is said to be 3 hours more. Using this, we can find the effective downstream and upstream speeds, then work back to the boat speed in still water and the stream speed. This is a typical relative speed problem in river boat scenarios. Given Data / Assumptions: Downstream distance = 36 km. Time downstream = 6 hours. Upstream time is 3 hours more than downstream time. Thus upstream time = 6 + 3 = 9 hours. We assume uniform speeds and steady current. Concept / Approach: We first find the downstream and upstream speeds using distance divided by time. Then we use the standard relations: Downstream speed = b + c. Upstream speed = b - c. Where b is boat speed in still water and c is stream speed. Solving these two equations will give c, the required stream speed. Step-by-Step Solution: Downstream speed = 36 / 6 = 6 km/h. Upstream time = 9 hours, so upstream speed = 36 / 9 = 4 km/h. Let b be boat speed and c be stream speed. Then b + c = 6 and b - c = 4. Add the equations: 2b = 10 so b = 5 km/h. Subtract the equations: 2c = 2 so c = 1 km/h. Therefore, the speed of the stream is 1 km/h. Verification / Alternative check: If the boat speed is 5 km/h and the stream speed is 1 km/h, downstream speed is 5 + 1 = 6 km/h. Time to cover 36 km is 36 / 6 = 6 hours, which matches the given data. Upstream speed is 5 - 1 = 4 km/h, and time to cover 36 km is 36 / 4 = 9 hours, which is indeed 3 hours more than 6 hours. This confirms consistency. Why Other Options Are Wrong: If the stream speed were 1.5 km/h, the speeds and times would no longer be 6 hours and 9 hours. For 0.75 km/h or 0.5 km/h, recalculated upstream times would not differ by exactly 3 hours from the downstream time. Only a stream speed of 1 km/h produces the required difference in travel times. Common Pitfalls: A frequent misunderstanding is to subtract times directly from speeds without using the distance relation. Another error is to misread the phrase "3 hours less" or "3 hours more" and reverse the relationship between upstream and downstream times. Carefully translating the verbal condition into an equation and then applying the boats and streams formulas avoids these issues. Final Answer: The speed of the stream is 1 km/h.

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