Question 1

A steamer moves with a speed of 4.5 km/h in still water, travelling to a point upstream and back to the starting point in a river flowing at 1.5 km/h. What is the average speed (in km/h) of the steamer for the entire journey?

Accepted Answer

Introduction / Context: In this question, a steamer travels from a starting point to a point upstream and then back downstream to the starting point. We are given the speed of the steamer in still water and the speed of the river current. The task is to find the average speed for the entire round trip. Since the distance upstream and downstream is the same, average speed is determined by the harmonic mean of the upstream and downstream speeds. Given Data / Assumptions: Steamer speed in still water b = 4.5 km/h. Speed of the current c = 1.5 km/h. Upstream speed = b - c = 4.5 - 1.5 = 3 km/h. Downstream speed = b + c = 4.5 + 1.5 = 6 km/h. Let the one way distance be d km. We must find the average speed for the round trip. Concept / Approach: For a journey where equal distances are covered at speeds u and v, the average speed is: Average speed = 2 * u * v / (u + v). Here u = 3 km/h and v = 6 km/h. This formula comes from total distance divided by total time, taking into account different speeds for equal distances. Alternatively, we can compute the time upstream and downstream separately and then find the total distance divided by total time. Step-by-Step Solution: Step 1: Let the one way distance be d km. Total distance = 2d km. Step 2: Compute the time upstream and downstream. Upstream time = d / 3 hours. Downstream time = d / 6 hours. Total time = d / 3 + d / 6 = (2d + d) / 6 = 3d / 6 = d / 2 hours. Step 3: Compute average speed. Average speed = total distance / total time = 2d / (d / 2) = 2d * 2 / d = 4 km/h. Verification / Alternative check: Using the harmonic mean formula: Average speed = 2 * 3 * 6 / (3 + 6) = 36 / 9 = 4 km/h. Both methods confirm the same result. Why Other Options Are Wrong: Speeds like 6 km/h, 10 km/h or 12 km/h ignore the fact that the steamer spends a longer time moving upstream at the slower speed of 3 km/h than downstream at 6 km/h. An average speed of 5 km/h would not match the total distance divided by the total time for the combination of the two legs. Common Pitfalls: A typical mistake is to average the speeds 3 km/h and 6 km/h directly to get 4.5 km/h, which is incorrect for equal distances. Another pitfall is to confuse the still water speed with the average speed, even though the current slows the boat upstream and speeds it up downstream. Final Answer: The average speed of the steamer for the total journey is 4 km/h.

Question 2

The ratio of the speed of a boat downstream to the speed of the stream is 9 : 1. If the speed of the current is 3 km/h, how far (in km) will the boat travel upstream in 5 hours?

Accepted Answer

Introduction / Context: This problem gives a ratio between the downstream speed of a boat and the speed of the stream. We also know the numerical value of the stream speed and the time the boat travels upstream. The question asks for the upstream distance covered in that time. To solve it, we translate the ratio into actual speeds, determine the boat speed in still water, find the upstream speed, and then use distance = speed * time. Given Data / Assumptions: Ratio of downstream speed to stream speed = 9 : 1. Stream speed c = 3 km/h. Let downstream speed be d_s km/h and boat speed in still water be b km/h. d_s : c = 9 : 1. Upstream speed = b - c. Time travelled upstream = 5 hours. We must find upstream distance in km. Concept / Approach: From the ratio d_s : c = 9 : 1 and the known stream speed, we find the downstream speed. Downstream speed is b + c, so we can compute b. Then, using b and c, we obtain the upstream speed b - c. Finally, the upstream distance covered in 5 hours is simply upstream speed multiplied by time. The steps are algebraic but straightforward once the ratio is interpreted correctly. Step-by-Step Solution: Step 1: Use the ratio to find downstream speed. d_s : c = 9 : 1 and c = 3 km/h. So d_s = 9 * c = 9 * 3 = 27 km/h. Step 2: Express downstream speed in terms of boat and stream speeds. Downstream speed = b + c = 27. So b + 3 = 27, giving b = 24 km/h. Step 3: Compute upstream speed. Upstream speed = b - c = 24 - 3 = 21 km/h. Step 4: Compute distance covered upstream in 5 hours. Distance = speed * time = 21 * 5 = 105 km. Verification / Alternative check: Check the ratio condition. Downstream speed with b = 24 and c = 3 is 24 + 3 = 27 km/h. Ratio of downstream speed to stream speed = 27 : 3 = 9 : 1, which matches the given ratio. Thus the calculated speeds and the resulting distance are consistent. Why Other Options Are Wrong: Distances like 100, 98, 109 or 90 km would correspond to upstream speeds that are different from 21 km/h when divided by 5 hours. Those speeds in turn would not match the given downstream to stream speed ratio of 9 : 1. Common Pitfalls: A common error is to misinterpret the ratio as boat speed in still water to stream speed, rather than downstream speed to stream speed. Another pitfall is to forget to subtract the stream speed to get the upstream speed, leading to using the downstream speed directly in the distance calculation. Final Answer: The boat will travel 105 km upstream in 5 hours.

Question 3

A boat goes a certain distance at 40 km/h and comes back the same distance at 24 km/h. What is the average speed (in km/h) for the total journey?

Accepted Answer

Introduction / Context: Here we are again dealing with average speed for a round trip where the distances are equal but the speeds are different. The boat travels one way at 40 km/h and returns at 24 km/h. To find the overall average speed, we must use total distance divided by total time, which for equal distances leads to the harmonic mean of the two speeds, not the simple arithmetic average. Given Data / Assumptions: Speed going from starting point to destination = 40 km/h. Speed coming back = 24 km/h. Let the one way distance be d km. Total distance for the round trip = 2d km. We must find the average speed over the total journey. Concept / Approach: If two equal distances are covered at speeds v1 and v2, then: Average speed = 2 * v1 * v2 / (v1 + v2). This formula is derived from total distance divided by total time. Here v1 = 40 and v2 = 24. Alternatively, we can compute time for each leg separately, add them, and then use total distance / total time to get the same result. Step-by-Step Solution: Step 1: Let the one way distance be d km. Total distance = 2d km. Step 2: Compute the time for each leg. Time forward = d / 40 hours. Time return = d / 24 hours. Total time = d / 40 + d / 24. Step 3: Combine the times using a common denominator. d / 40 + d / 24 = (3d + 5d) / 120 = 8d / 120 = d / 15. Step 4: Compute average speed = total distance / total time. Average speed = (2d) / (d / 15) = 2d * 15 / d = 30 km/h. Verification / Alternative check: Using the harmonic mean formula directly: Average speed = 2 * 40 * 24 / (40 + 24) = 1920 / 64 = 30 km/h. Both methods confirm the same value. Why Other Options Are Wrong: The arithmetic mean (40 + 24) / 2 = 32 km/h is not correct because the boat spends longer time at the slower speed. Other values such as 28, 34, 36 km/h do not match the correct ratio of total distance to total time for the given speeds. Common Pitfalls: The most common mistake is using the simple average of 40 and 24 instead of total distance divided by total time. Another error is mishandling the fractions when adding times, leading to an incorrect denominator and hence wrong average speed. Final Answer: The average speed for the total journey is 30 km/h.

Question 4

A boat moves downstream at 8 km/h and upstream at 4 km/h. What is the speed (in km/h) of the boat in still water?

Accepted Answer

Introduction / Context: This question provides the effective speeds of a boat while travelling downstream and upstream and asks for the speed of the boat in still water. In boats and streams problems, the downstream speed is increased by the current, while the upstream speed is reduced by it. Knowing both effective speeds lets us compute both the boat speed in still water and the current speed. Given Data / Assumptions: Downstream speed = 8 km/h. Upstream speed = 4 km/h. Let boat speed in still water be b km/h. Let current speed be c km/h. Downstream speed = b + c and upstream speed = b - c. We must find b. Concept / Approach: From the two equations b + c = 8 and b - c = 4, we can solve for b and c. The standard formulas are: b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 So we simply substitute the given numerical values and compute b. The distances are not needed because the speeds already reflect the effect of the current. Step-by-Step Solution: Step 1: Write the system of equations. b + c = 8. b - c = 4. Step 2: Add the equations to eliminate c. (b + c) + (b - c) = 8 + 4. 2b = 12. b = 12 / 2 = 6 km/h. Step 3: If needed, compute current speed. From b + c = 8, c = 8 - 6 = 2 km/h. Verification / Alternative check: With b = 6 and c = 2, downstream speed is 6 + 2 = 8 km/h, matching the given value. Upstream speed is 6 - 2 = 4 km/h, again matching the given value. This confirms that b = 6 km/h is correct. Why Other Options Are Wrong: If b were 4.5, 5, 6.4 or 7 km/h, it would be impossible to choose a single current speed c such that b + c = 8 and b - c = 4 simultaneously. Checking each option quickly shows contradictions in at least one of the equations. Common Pitfalls: One common error is to average 8 and 4 incorrectly or to forget that the boat speed is the midpoint between the downstream and upstream speeds. Another pitfall is to mistake the current speed for the boat speed in still water and report c instead of b. Final Answer: The speed of the boat in still water is 6 km/hr.

Question 5

The speed of a boat along the current (downstream) is 14 km/h, and the speed of the same boat against the current (upstream) is 7 km/h. Based on these two given speeds, what is the speed of the boat in still water (in km/h)?

Accepted Answer

Introduction / Context: This question tests the standard boats-and-streams concept: the speed you observe downstream and upstream is affected by the stream (current). Downstream speed is increased by the current, while upstream speed is decreased by the current. Using these two observed speeds, we can recover the boat's actual speed in still water, which is the base speed of the boat when there is no current at all. Given Data / Assumptions: Downstream (along the current) speed = 14 km/h Upstream (against the current) speed = 7 km/h Let boat speed in still water = b km/h Let speed of current = c km/h We need to find b Concept / Approach: In boats and streams problems: Downstream speed = b + c Upstream speed = b - c If we add the downstream and upstream speeds, the current cancels out: (b + c) + (b - c) = 2b So the boat's still-water speed is simply the average of the downstream and upstream speeds: b = (downstream + upstream) / 2 This is the quickest and most reliable method here. Step-by-Step Solution: Step 1: Write the two speed equations. b + c = 14 b - c = 7 Step 2: Add both equations to eliminate c. (b + c) + (b - c) = 14 + 7 2b = 21 Step 3: Solve for b. b = 21 / 2 = 10.5 Verification / Alternative check: If b = 10.5, then the current speed is c = 14 - 10.5 = 3.5. Check upstream: b - c = 10.5 - 3.5 = 7, which matches the given upstream speed. So the computed still-water speed is consistent with both directions. Why Other Options Are Wrong: 9.5 would imply current c = 14 - 9.5 = 4.5, giving upstream 9.5 - 4.5 = 5, not 7. 8.5 would imply current c = 5.5, giving upstream 3, not 7. 11 would imply current c = 3, giving upstream 8, not 7. 7.5 would imply current c = 6.5, giving upstream 1, not 7. Common Pitfalls: Many learners incorrectly subtract the two speeds and divide by 2, which actually gives the current speed, not the boat speed. Another common mistake is to assume the still-water speed equals the downstream speed, ignoring the effect of the current. Final Answer: The speed of the boat in still water is 10.5 km/h.

Question 6

A boat travels upstream at a speed of 24 km/h and returns downstream over the same distance at a speed of 40 km/h. What is the average speed (in km/h) for the entire round trip (going and coming back over equal distances)?

Accepted Answer

Introduction / Context: This problem tests the concept of average speed over a round trip when the distances in the two legs are equal but the speeds are different. A very common trap is to take the simple arithmetic mean of the two speeds, but that is incorrect for equal-distance travel. The correct method is to use total distance divided by total time, which naturally leads to the harmonic mean for two equal distances. Given Data / Assumptions: Upstream speed = 24 km/h Downstream speed = 40 km/h The boat returns over the same distance (equal distances each way) Let one-way distance = d km We must find the average speed for the full journey Concept / Approach: Average speed is defined as: Average speed = (total distance) / (total time) Here the total distance is: d + d = 2d Time taken upstream is: t1 = d / 24 Time taken downstream is: t2 = d / 40 So total time is: t1 + t2 = d/24 + d/40 Then: Average speed = 2d / (d/24 + d/40) The d cancels, so we get a clean numeric answer without needing the actual distance. Step-by-Step Solution: Step 1: Let the one-way distance be d km. Step 2: Write the times for each leg. Upstream time = d / 24 Downstream time = d / 40 Step 3: Write total distance and total time. Total distance = 2d Total time = d/24 + d/40 Step 4: Compute average speed. Average speed = 2d / (d/24 + d/40) = 2d / (d * (1/24 + 1/40)) = 2 / (1/24 + 1/40) Step 5: Add the fractions. 1/24 + 1/40 = (40 + 24) / (24 * 40) = 64 / 960 = 1 / 15 Step 6: Finish the calculation. Average speed = 2 / (1/15) = 2 * 15 = 30 Verification / Alternative check: Pick an easy distance, say d = 120 km. Upstream time = 120/24 = 5 hours. Downstream time = 120/40 = 3 hours. Total distance = 240 km, total time = 8 hours. Average speed = 240/8 = 30 km/h, confirming the result. Why Other Options Are Wrong: 32 is the simple average (24 + 40) / 2, which is incorrect because time spent at 24 km/h is longer than time spent at 40 km/h. 31 and 33 are near the wrong arithmetic mean idea and do not match total distance divided by total time. 28 would imply even more time than the upstream leg justifies, and does not satisfy the equal-distance condition when verified with any sample distance. Common Pitfalls: The biggest mistake is using arithmetic mean for equal distances. For equal distances, you must use total distance / total time, which becomes the harmonic mean. Another mistake is assuming the distance matters; in equal-distance two-speed problems, distance cancels out if set up correctly. Final Answer: The average speed for the full round trip is 30 km/h.

Question 7

Ashok can row a boat upstream at 8 km/h and downstream at 12 km/h. What is the speed of the stream (in km/h)?

Accepted Answer

Introduction / Context: This is a standard boats-and-streams question. You are given the speed of rowing upstream and downstream and asked to find the speed of the stream. The key idea is that the stream either helps or opposes the boat, changing the effective speed. Given Data / Assumptions: • Upstream speed (against current) = 8 km/h.• Downstream speed (with current) = 12 km/h.• Let the speed of the boat in still water be u km/h.• Let the speed of the stream be v km/h. Concept / Approach: When the boat moves upstream, the current opposes it, so the effective speed is (u − v). When it moves downstream, the current aids it, so the effective speed is (u + v). These two linear equations can be solved simultaneously to find u and v. The question only asks for v, the speed of the stream. Step-by-Step Solution: Step 1: Upstream: u − v = 8.Step 2: Downstream: u + v = 12.Step 3: Add the two equations: (u − v) + (u + v) = 8 + 12.Step 4: This gives 2u = 20 ⇒ u = 10 km/h.Step 5: Substitute u = 10 into u + v = 12 ⇒ 10 + v = 12.Step 6: Therefore, v = 2 km/h. Verification / Alternative check: If the boat's own speed is 10 km/h and the stream is 2 km/h, then upstream speed = 10 − 2 = 8 km/h and downstream speed = 10 + 2 = 12 km/h, matching the given values exactly. Why Other Options Are Wrong: 6 km/hr: Would give impossible values for the boat's own speed when combined with the given upstream speed.3 km/hr or 4.5 km/hr: These do not satisfy both equations simultaneously when checked. Common Pitfalls: Students sometimes average 8 and 12 directly and call that the stream speed, forgetting that the average gives the still-water speed, not the stream speed. The stream speed is half the difference of downstream and upstream speeds. Final Answer: The speed of the stream is 2 km/h.

Question 8

A boat covers 63 km upstream in 9 hours and the same distance downstream in 7 hours. What is the speed (in km/h) of the boat in still water?

Accepted Answer

Introduction / Context: This problem tests your understanding of upstream and downstream speeds and how to extract the boat's own speed from them. Instead of being given the speeds directly, you are given distance and time, so you must compute the effective speeds first. Given Data / Assumptions: • Distance each way = 63 km.• Time upstream = 9 hours.• Time downstream = 7 hours.• Let boat speed in still water be u km/h.• Let stream speed be v km/h. Concept / Approach: Speed = distance ÷ time. So we first compute the upstream and downstream speeds from the given distances and times. For boats-and-streams questions: upstream speed = u − v, downstream speed = u + v. Once we know those two values, we can find u as the average of them. Step-by-Step Solution: Step 1: Upstream speed = 63 ÷ 9 = 7 km/h.Step 2: Downstream speed = 63 ÷ 7 = 9 km/h.Step 3: Set up equations: u − v = 7 and u + v = 9.Step 4: Add the equations: (u − v) + (u + v) = 7 + 9.Step 5: This gives 2u = 16 ⇒ u = 16 ÷ 2 = 8 km/h. Verification / Alternative check: If u = 8 and upstream speed is 7, then v = u − 7 = 1 km/h. Check downstream: u + v = 8 + 1 = 9 km/h, which matches the found downstream speed. So our value of u is consistent. Why Other Options Are Wrong: 7: This is the upstream speed, not the still-water speed.8.5 or 8.7: These values do not produce upstream and downstream speeds of exactly 7 and 9 when combined with a single current speed. Common Pitfalls: Many students try to average the times instead of speeds or forget to compute speeds before applying formulas. Always convert distance–time information into speeds first, then use the standard upstream/downstream relations. Final Answer: The speed of the boat in still water is 8 km/h.

Question 9

The speed of a boat downstream is 14 km/h and upstream is 8 km/h. What is the speed (in km/h) of the current?

Accepted Answer

Introduction / Context: This question again uses the basic boats-and-streams model. You are given the effective speeds with and against the current and are asked to find the speed of the current alone. Given Data / Assumptions: • Downstream speed = 14 km/h.• Upstream speed = 8 km/h.• Let boat speed in still water = u km/h.• Let stream speed = v km/h. Concept / Approach: The standard relations are: downstream speed = u + v and upstream speed = u − v. Adding them gives 2u; subtracting them gives 2v. The stream speed is therefore half the difference between downstream and upstream speeds. Step-by-Step Solution: Step 1: Write equations: u + v = 14 and u − v = 8.Step 2: Subtract the second from the first: (u + v) − (u − v) = 14 − 8.Step 3: This simplifies to 2v = 6.Step 4: So v = 6 ÷ 2 = 3 km/h. Verification / Alternative check: Using v = 3, we can quickly find u. Substitute into u − v = 8 ⇒ u − 3 = 8 ⇒ u = 11 km/h. Check downstream: u + v = 11 + 3 = 14 km/h, which matches the given value, confirming v = 3 km/h is correct. Why Other Options Are Wrong: 11 kms/hr: This is the boat's own speed in still water, not the stream speed.6 kms/hr: Equal to the full difference (14 − 8), but the stream speed is half this difference.5.5 kms/hr: A random average-like value that does not fit the equations. Common Pitfalls: The usual mistake is to take the difference of speeds (14 − 8 = 6) as the stream speed directly. Remember that the difference equals 2v, so you must divide by 2 to get the current's velocity. Final Answer: The speed of the current is 3 km/h.

Question 10

A boat has a speed of 20 km/h in still water. The speed of the current is 5 km/h. How much time (in hours) will the boat take to travel 100 km downstream (with the current)?

Accepted Answer

Introduction / Context: This question asks you to compute travel time when both the boat's own speed and the stream's speed are given. You must first determine the effective downstream speed and then apply the basic distance–speed–time relation. Given Data / Assumptions: • Boat speed in still water = 20 km/h.• Stream speed = 5 km/h.• Distance to be travelled downstream = 100 km. Concept / Approach: Downstream, the current helps the boat. So the downstream speed is the sum of the boat's speed in still water and the speed of the current. Once we know the effective speed, time taken is simply distance divided by speed. Step-by-Step Solution: Step 1: Downstream speed = 20 + 5 = 25 km/h.Step 2: Distance to be covered = 100 km.Step 3: Time = distance ÷ speed = 100 ÷ 25 hours.Step 4: 100 ÷ 25 = 4 hours. Verification / Alternative check: In 1 hour the boat travels 25 km downstream. In 4 hours it will cover 4 × 25 = 100 km, which matches the required distance, confirming that 4 hours is correct. Why Other Options Are Wrong: 2 hrs or 3 hrs: These would correspond to speeds of 50 km/h or about 33.33 km/h, which are higher than the available downstream speed of 25 km/h.7 hrs: This would imply a downstream speed of less than 15 km/h, too slow for a 20 km/h boat helped by a 5 km/h current. Common Pitfalls: Some learners incorrectly subtract the stream speed even for downstream motion or forget to add the speeds. Always remember: downstream = boat + stream; upstream = boat − stream. Then apply the distance–speed–time formula accurately. Final Answer: The boat takes 4 hours to travel 100 km downstream.

Question 11

A man can row a boat upstream at 12 km/h and downstream at 18 km/h. What is his rowing speed (in km/h) in still water?

Accepted Answer

Introduction / Context: This is another classical upstream–downstream question, where you are asked to find the boatman's own rowing speed in still water. You are given effective speeds in both directions. Given Data / Assumptions: • Upstream speed = 12 km/h.• Downstream speed = 18 km/h.• Let speed in still water be u km/h.• Let stream speed be v km/h. Concept / Approach: The upstream and downstream speeds are related to u and v as: upstream = u − v, downstream = u + v. Adding these two equations eliminates v and directly gives 2u. So the speed in still water is simply the average of upstream and downstream speeds. Step-by-Step Solution: Step 1: Write equations: u − v = 12 and u + v = 18.Step 2: Add the equations: (u − v) + (u + v) = 12 + 18.Step 3: This simplifies to 2u = 30.Step 4: Therefore, u = 30 ÷ 2 = 15 km/h. Verification / Alternative check: If u = 15, then v can be found from u + v = 18 ⇒ v = 3 km/h. Check upstream: u − v = 15 − 3 = 12 km/h, which matches the given upstream speed. Both directions are consistent. Why Other Options Are Wrong: 5 km/hr or 3 km/hr: These values are closer to the speed of the stream than to the boat's own speed.10 km/hr: This is less than both upstream and downstream speeds, which is impossible for the boat's still-water speed. Common Pitfalls: Sometimes students mistakenly think the still-water speed is the difference or half the difference of the given speeds. Remember: still-water speed = (upstream + downstream) ÷ 2, and stream speed = (downstream − upstream) ÷ 2. Final Answer: The man's rowing speed in still water is 15 km/h.

Question 12

A boat goes a certain distance at 30 km/h and returns the same distance at 20 km/h. What is the average speed (in km/h) for the entire journey?

Accepted Answer

Introduction / Context: Here we deal with average speed over a round trip. The boat travels the same distance twice but at different speeds: faster one way and slower on the way back. The average speed is not the simple average of 30 and 20. Given Data / Assumptions: • Speed going out = 30 km/h.• Speed coming back = 20 km/h.• Distances in both directions are equal. Concept / Approach: Average speed is defined as total distance divided by total time. When the distance each way is the same, we can either compute it directly or use the harmonic mean formula for two speeds: average speed = (2uv) / (u + v), where u and v are the two speeds. Step-by-Step Solution (distance–time method): Step 1: Let the one-way distance be D km.Step 2: Time for onward journey = D ÷ 30 hours.Step 3: Time for return journey = D ÷ 20 hours.Step 4: Total distance = 2D.Step 5: Total time = D/30 + D/20 = (2D + 3D) ÷ 60 = 5D ÷ 60 = D/12 hours.Step 6: Average speed = total distance ÷ total time = 2D ÷ (D/12) = 2D × (12/D) = 24 km/h. Verification / Harmonic mean check: Using the formula: average = (2uv) / (u + v) = 2 × 30 × 20 ÷ (30 + 20) = 1200 ÷ 50 = 24 km/h, which matches the result obtained earlier. Why Other Options Are Wrong: 25 or 24.5 or 25.5: These are close to the arithmetic mean (25), but average speed for equal distances must be the harmonic mean, which is always less than or equal to the arithmetic mean when speeds differ. Common Pitfalls: A common error is to simply average the two speeds: (30 + 20) ÷ 2 = 25 km/h. This ignores the fact that more time is spent at the slower speed, pulling the overall average down. Final Answer: The average speed for the entire journey is 24 km/h.

Question 13

A boat goes upstream at 18 km/h and comes back downstream at 30 km/h over the same distance. What is the average speed (in km/h) for the entire journey?

Accepted Answer

Introduction / Context: This is another average speed question over a round trip, now with upstream and downstream speeds given. The distance each way is the same, but the speeds differ, so we must use the definition of average speed or the harmonic mean formula. Given Data / Assumptions: • Upstream speed = 18 km/h.• Downstream speed = 30 km/h.• Distance upstream = distance downstream. Concept / Approach: Average speed = total distance ÷ total time. For equal distances at speeds u and v, the average speed is the harmonic mean: (2uv) ÷ (u + v). This accounts for the extra time spent travelling at the slower speed. Step-by-Step Solution (harmonic mean): Step 1: Let u = 18 km/h (upstream) and v = 30 km/h (downstream).Step 2: Average speed = (2uv) ÷ (u + v).Step 3: Substitute values: (2 × 18 × 30) ÷ (18 + 30) = (1080) ÷ 48.Step 4: 1080 ÷ 48 = 22.5 km/h. Alternative distance–time method: Step 1: Let one-way distance be D km.Step 2: Time upstream = D ÷ 18 hours.Step 3: Time downstream = D ÷ 30 hours.Step 4: Total distance = 2D.Step 5: Total time = D/18 + D/30 = (5D + 3D) ÷ 90 = 8D ÷ 90 = 4D/45 hours.Step 6: Average speed = 2D ÷ (4D/45) = 2D × 45 ÷ 4D = 90 ÷ 4 = 22.5 km/h. Why Other Options Are Wrong: 24, 20.5, 25: These do not match the harmonic mean and arise from incorrect averaging or arithmetic mistakes. Common Pitfalls: The most frequent mistake is taking the simple average (18 + 30) ÷ 2 = 24 km/h, which ignores the unequal time spent at each speed. Always use total distance divided by total time, especially when dealing with different speeds over equal distances. Final Answer: The average speed for the entire journey is 22.5 km/h.

Boats and Streams Questions