Question 1

A boy can swim in still water at 4.5 km/h, but it takes him twice as long to swim upstream as it takes to swim downstream over the same distance. What is the speed of the stream (in km/h)?

Accepted Answer

Introduction / Context: Swimming problems in a river are analogous to boat and stream questions. The swimmer's speed in still water is known, and the current either helps or hinders him depending on direction. Here, we are told that the time taken upstream is twice the time taken downstream for the same distance, and we must find the speed of the stream. Given Data / Assumptions: Swimmer's speed in still water, b = 4.5 km/h. Let the speed of the stream be c km/h. Upstream speed = (b - c) km/h. Downstream speed = (b + c) km/h. For the same distance d, time upstream is twice time downstream. Distance d cancels out in the ratio calculations. Concept / Approach: Using time = distance / speed, if the time upstream is twice the time downstream for the same distance d, we have: d / (b - c) = 2 * [d / (b + c)]. The distance cancels, leaving an equation involving b and c. Substituting b = 4.5 and solving this equation gives c, the speed of the stream. Step-by-Step Solution: Step 1: Set up the time relation. d / (b - c) = 2d / (b + c). Cancel d: 1 / (b - c) = 2 / (b + c). Step 2: Cross-multiply. b + c = 2(b - c). b + c = 2b - 2c. Step 3: Solve for c. Rearrange: b + c = 2b - 2c ⇒ 3c = b ⇒ c = b / 3. Given b = 4.5 km/h, c = 4.5 / 3 = 1.5 km/h. Verification / Alternative check: With c = 1.5 km/h, downstream speed = 4.5 + 1.5 = 6 km/h, upstream speed = 4.5 - 1.5 = 3 km/h. For distance d, time downstream = d / 6, time upstream = d / 3. Since d / 3 = 2 * (d / 6), the upstream time is exactly twice the downstream time, which matches the condition perfectly. Why Other Options Are Wrong: 1.8 km/h or 2 km/h as the stream speed would change the ratio of upstream to downstream times and would not give exactly a 2:1 ratio. 2.2 km/h is too high and would make upstream speed too slow relative to downstream, creating a different time ratio than specified in the problem. Common Pitfalls: A common mistake is to try to average times or speeds, or to use b ± c incorrectly (sometimes students swap them). It is essential to remember that upstream speed is b - c and downstream speed is b + c. Setting up the correct ratio of times using time = distance / speed and cancelling the common distance term leads to a simple and accurate solution. Final Answer: The speed of the stream is 1.5 km/h.

Question 2

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 (in km/h)?

Accepted Answer

Introduction / Context: This question tests your understanding of downstream and upstream speeds and how to extract the speed of the current from given distances and times. Here, the man takes the same time for two trips but covers different distances downstream and upstream, giving you enough information to find the stream speed. Given Data / Assumptions: Downstream distance = 60 km, time taken = 3 hours. Upstream distance = 30 km, time taken = 3 hours. Let the speed of the boat in still water be b km/h. Let the speed of the stream be c km/h. Downstream speed = (b + c) km/h. Upstream speed = (b - c) km/h. Concept / Approach: From distance and time, we can compute effective downstream and upstream speeds. Once we have these two effective speeds, we can find b and c using the relationships: b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 Our main interest is c, the speed of the stream. Step-by-Step Solution: Step 1: Find downstream speed. Downstream speed v_d = distance / time = 60 / 3 = 20 km/h. Step 2: Find upstream speed. Upstream speed v_u = 30 / 3 = 10 km/h. Step 3: Compute the speed of the stream. c = (v_d - v_u) / 2 = (20 - 10) / 2 = 10 / 2 = 5 km/h. Verification / Alternative check: Now compute the still-water speed: b = (v_d + v_u) / 2 = (20 + 10) / 2 = 30 / 2 = 15 km/h. Using b = 15 and c = 5, downstream speed = 20 km/h and upstream speed = 10 km/h, which reproduce the original times of 60/20 = 3 h downstream and 30/10 = 3 h upstream. This consistency confirms our answer. Why Other Options Are Wrong: 10 km/h would imply a still-water speed of 15 km/h and give upstream speed = 5 km/h, leading to an upstream time of 30 / 5 = 6 hours, not 3 hours. 15 km/h and 45 km/h as current speeds are unrealistic here; they lead to negative or inconsistent upstream speeds and do not match the given trip times. Common Pitfalls: Some students mistakenly average distances or times instead of calculating speeds separately. Others mix up the formulas for b and c or forget to divide the difference of speeds by 2. Always compute effective downstream and upstream speeds first, and then use the standard formulas to find the still-water and current speeds. Final Answer: The speed of the stream is 5 km/h.

Question 3

A boatman can row 3 km against the stream in 20 minutes and return the same distance with the stream in 18 minutes. What is the rate of the current (in km/h)?

Accepted Answer

Introduction / Context: This problem involves a boatman travelling the same distance upstream and downstream in different times. From these times and the known distance, we can determine the effective upstream and downstream speeds and then compute the rate of the current in the river. Given Data / Assumptions: Distance upstream = 3 km; distance downstream = 3 km (same path back). Time upstream = 20 minutes = 20/60 = 1/3 hour. Time downstream = 18 minutes = 18/60 = 3/10 hour. Let boat speed in still water be b km/h. Let speed of current be c km/h. Upstream speed = (b - c) km/h; downstream speed = (b + c) km/h. Concept / Approach: We compute the effective speeds from distance and time. For upstream, speed = distance / time, and similarly for downstream. Then we use the standard relationships: b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 Our target is c, the rate of the current. Step-by-Step Solution: Step 1: Compute upstream speed. Upstream speed v_u = distance / time = 3 / (1/3) = 9 km/h. Step 2: Compute downstream speed. Downstream time = 18/60 = 3/10 hour. Downstream speed v_d = 3 / (3/10) = 3 * (10/3) = 10 km/h. Step 3: Find the rate of current. c = (v_d - v_u) / 2 = (10 - 9) / 2 = 1/2 km/h. Verification / Alternative check: The still-water speed b = (v_d + v_u) / 2 = (10 + 9) / 2 = 19/2 = 9.5 km/h. Check upstream: b - c = 9.5 - 0.5 = 9 km/h, giving time 3/9 = 1/3 h = 20 min. Check downstream: b + c = 9.5 + 0.5 = 10 km/h, giving time 3/10 h = 0.3 h = 18 min. Both match the given data, confirming the correctness of c = 1/2 km/h. Why Other Options Are Wrong: 1/3 km/h or 2/3 km/h would not yield the observed upstream and downstream times when combined with any consistent still-water speed. 1/4 km/h is too small, leading to almost equal upstream and downstream speeds, which is inconsistent with the clearly different times of 20 and 18 minutes. Common Pitfalls: Learners often forget to convert minutes into hours, causing incorrect speed calculations. Another common slip is to subtract upstream speed from downstream speed but forget to divide by 2 when computing current speed. Maintaining clear units and applying the formulas for b and c carefully will avoid these issues. Final Answer: The rate of the current is 1/2 km/h.

Question 4

A man can row at 6 km/h in still water. When the river is running at 1.2 km/h, it takes him a total of 1 hour to row from one point to another and then back to the starting point. What is the total distance travelled by the man (in km)?

Accepted Answer

Introduction / Context: This question asks you to find the total distance travelled in a round trip on a river, given the boat's speed in still water, the speed of the current, and the total time taken for going and returning. It is a typical boats and streams problem that uses relative speeds and time calculations. Given Data / Assumptions: Boat speed in still water, b = 6 km/h. River current speed, c = 1.2 km/h. Downstream speed = b + c = 6 + 1.2 = 7.2 km/h. Upstream speed = b - c = 6 - 1.2 = 4.8 km/h. Let the one-way distance between the two points be d km. Total time for going downstream and returning upstream = 1 hour. We are asked for the total distance travelled, which is 2d. Concept / Approach: Time is distance divided by speed. The total time is the sum of the time downstream and the time upstream. We create an equation with d as the unknown: d/7.2 + d/4.8 = 1. Solving for d yields the one-way distance, and doubling it gives the total distance travelled. Step-by-Step Solution: Step 1: Write the time equation. Downstream time = d / 7.2 hours. Upstream time = d / 4.8 hours. Total time: d/7.2 + d/4.8 = 1. Step 2: Simplify by using fractions for speeds. 7.2 = 36/5, so d/7.2 = d / (36/5) = 5d/36. 4.8 = 24/5, so d/4.8 = d / (24/5) = 5d/24. Thus 5d/36 + 5d/24 = 1. Step 3: Combine the fractions. Factor out 5d: 5d(1/36 + 1/24) = 1. 1/36 + 1/24 = (2 + 3) / 72 = 5 / 72. So 5d * (5/72) = 1 ⇒ 25d / 72 = 1. d = 72 / 25 = 2.88 km (one way). Step 4: Compute total distance. Total distance = 2d = 2 * 2.88 = 5.76 km. Verification / Alternative check: Check time with d = 2.88 km. Downstream time = 2.88 / 7.2 = 0.4 h (24 minutes). Upstream time = 2.88 / 4.8 = 0.6 h (36 minutes). Total = 0.4 + 0.6 = 1 hour, which matches the given total time exactly. Thus 5.76 km is correct. Why Other Options Are Wrong: 4.58 km, 5.24 km or 6.35 km would give total times that are not equal to 1 hour when you recompute downstream and upstream times with speeds 7.2 km/h and 4.8 km/h. Common Pitfalls: Some students incorrectly assume that the average speed for the round trip is simply the average of 7.2 and 4.8, which is not valid when time in each direction is unknown. Others make arithmetic mistakes converting decimal speeds to fractions. Always solve for distance using the time equation and double-check by substituting back into the problem. Final Answer: The total distance travelled by the man is 5.76 km.

Question 5

A boat running upstream takes 8 hours 48 minutes to cover a certain distance, while it takes 4 hours to cover the same distance running downstream. What is the ratio of the speed of the boat in still water to the speed of the water current, respecti…

Accepted Answer

Introduction / Context: This problem deals with the ratio of the boat's speed in still water to the speed of the current, based on the times needed to travel the same distance upstream and downstream. It is a standard type of boats and streams question involving relative speed and ratios. Given Data / Assumptions: Let the one-way distance be d km. Time upstream = 8 hours 48 minutes = 8 + 48/60 = 8.8 hours = 44/5 hours. Time downstream = 4 hours. Let boat speed in still water be b km/h. Let current speed be c km/h. Upstream speed = b - c; downstream speed = b + c. Distance d is the same in both directions. Concept / Approach: Speed is distance divided by time. We can express upstream and downstream speeds in terms of d and the given times. Then the ratio of upstream to downstream speeds equals the ratio of (b - c) to (b + c). Solving this ratio equation gives b/c, which is the required ratio of the boat's speed in still water to the current speed. Step-by-Step Solution: Step 1: Compute upstream and downstream speeds in terms of d. Upstream speed v_u = d / (44/5) = 5d/44. Downstream speed v_d = d / 4. Step 2: Form the ratio of upstream to downstream speeds. v_u / v_d = (5d/44) / (d/4) = (5d/44) * (4/d) = 20/44 = 5/11. So (b - c) / (b + c) = 5/11. Step 3: Solve for the ratio b : c. (b - c) / (b + c) = 5/11 ⇒ 11(b - c) = 5(b + c). 11b - 11c = 5b + 5c ⇒ 11b - 5b = 11c + 5c ⇒ 6b = 16c. Thus b/c = 16/6 = 8/3. Therefore, the required ratio b : c = 8 : 3. Verification / Alternative check: Let us assume c = 3 units, then b = 8 units (same units, e.g. km/h). Upstream speed = 8 - 3 = 5 units; downstream speed = 8 + 3 = 11 units. Distance d is proportional to speed × time. Using upstream time 44/5 hours and speed 5 units gives d = 5 * 44/5 = 44 units. Using downstream time 4 hours and speed 11 units gives d = 11 * 4 = 44 units again, confirming consistency. Why Other Options Are Wrong: 7:4 and 11:4 do not satisfy the equation (b - c)/(b + c) = 5/11 when you plug in those ratios for b and c. 4:7 implies the boat is slower than the current, which would make upstream motion impossible or extremely slow and does not match the given times. Common Pitfalls: Errors often occur when converting 8 hours 48 minutes into hours; forgetting that 48 minutes is 4/5 of an hour leads to wrong times and speeds. Another mistake is to invert the ratio, taking downstream to upstream instead of upstream to downstream. Carefully setting up the ratio (b - c)/(b + c) and doing the algebra step by step ensures the correct answer. Final Answer: The ratio of the speed of the boat in still water to the speed of the current is 8:3.

Question 6

A boat whose speed in still water is 9 km/h goes 12 km downstream and then returns 12 km upstream to the starting point in a total time of 3 hours. What is the speed of the stream (in km/h)?

Accepted Answer

Introduction / Context: In this boats and streams problem, you know the still-water speed of the boat, the distance covered downstream and upstream, and the total time for the round trip. From this, you must determine the speed of the stream. This involves setting up an equation using time = distance / speed for each leg of the journey. Given Data / Assumptions: Boat speed in still water, b = 9 km/h. Let stream speed be c km/h. Downstream speed = b + c = 9 + c km/h. Upstream speed = b - c = 9 - c km/h (c < 9 so that upstream is possible). Downstream distance = 12 km, upstream distance = 12 km. Total time for downstream plus upstream = 3 hours. Concept / Approach: Time downstream is 12 / (9 + c) hours and time upstream is 12 / (9 - c) hours. The sum of these two times equals 3 hours. This gives a rational equation in c, which we solve to find the stream speed. The algebra simplifies nicely because the distances are equal in both directions. Step-by-Step Solution: Step 1: Write the time equation. 12 / (9 + c) + 12 / (9 - c) = 3. Step 2: Divide both sides by 3 to simplify. 4 / (9 + c) + 4 / (9 - c) = 1. Step 3: Combine the fractions. [4(9 - c) + 4(9 + c)] / (81 - c^2) = 1. Numerator: 4(9 - c + 9 + c) = 4 * 18 = 72. So 72 / (81 - c^2) = 1. Step 4: Solve for c. 81 - c^2 = 72 ⇒ c^2 = 9 ⇒ c = 3 (taking positive value). Thus the stream speed is 3 km/h. Verification / Alternative check: With c = 3 km/h, downstream speed = 9 + 3 = 12 km/h, upstream speed = 9 - 3 = 6 km/h. Downstream time = 12 / 12 = 1 hour. Upstream time = 12 / 6 = 2 hours. Total = 1 + 2 = 3 hours, which matches the given total time, confirming that c = 3 km/h is correct. Why Other Options Are Wrong: 1 km/h, 4 km/h or 5 km/h, when used as stream speeds with b = 9 km/h, do not produce a total time of 3 hours for the 12 km downstream and 12 km upstream journey. For example, with c = 1 km/h, total time would be 12/10 + 12/8 = 1.2 + 1.5 = 2.7 hours, not 3 hours. Common Pitfalls: Errors often occur in simplifying the rational equation: students may forget to factor the denominator as (9 + c)(9 - c) = 81 - c^2 or may make sign errors when combining terms. Another pitfall is choosing the negative root for c, but since speed cannot be negative, only the positive value is acceptable. Writing each algebraic step clearly helps avoid these issues. Final Answer: The speed of the stream is 3 km/h.

Question 7

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

Accepted Answer

Introduction / Context: This is a typical boats and streams time-difference problem. The boat's speed in still water and the current speed are known, and you are told that rowing upstream takes 4 hours more than rowing downstream for the same distance. From this information, you must determine the distance. Given Data / Assumptions: Boat speed in still water, b = 8 km/h. River current speed, c = 2 km/h. Upstream speed = b - c = 8 - 2 = 6 km/h. Downstream speed = b + c = 8 + 2 = 10 km/h. Let the one-way distance be d km. Time upstream is 4 hours more than time downstream. Concept / Approach: Time taken to cover a distance equals distance divided by speed. For distance d, upstream time is d/6 hours and downstream time is d/10 hours. The condition that upstream time is 4 hours more leads to the equation: d/6 = d/10 + 4. Solving this linear equation in d gives the required distance. Step-by-Step Solution: Step 1: Write the time relation. Upstream time - downstream time = 4. So d/6 - d/10 = 4. Step 2: Combine the fractions on the left-hand side. Find LCM of 6 and 10, which is 30. d/6 - d/10 = (5d - 3d) / 30 = 2d / 30 = d / 15. So d / 15 = 4. Step 3: Solve for d. d = 4 * 15 = 60 km. Verification / Alternative check: Check times for d = 60 km. Upstream time = 60 / 6 = 10 hours. Downstream time = 60 / 10 = 6 hours. Difference = 10 - 6 = 4 hours, exactly as stated in the problem. This confirms that 60 km is the correct distance. Why Other Options Are Wrong: For 54 km, upstream time = 54/6 = 9 hours and downstream time = 54/10 = 5.4 hours, difference = 3.6 hours, not 4. For 32 km or 45 km, similar calculations show time differences different from 4 hours, so they cannot satisfy the given condition. Common Pitfalls: A frequent mistake is reversing the time difference (writing d/10 = d/6 + 4 instead of d/6 = d/10 + 4) or mixing up which speed is upstream and which is downstream. Some students also incorrectly average speeds or distances rather than using the time-difference equation. Carefully identifying upstream and downstream speeds and then applying time = distance / speed with the given difference prevents these errors. Final Answer: The required distance is 60 km.

Question 8

A motorboat travels 10 miles downstream on a river in 20 minutes and then takes 30 minutes to travel the same 10 miles upstream. What is the speed of the river current in mph?

Accepted Answer

Introduction / Context: This boats and streams question tests understanding of how a boat speed in still water combines with the river current to give different speeds downstream and upstream. By using distance, time, and speed relationships, we can separate the speed of the current from the speed of the motorboat itself. Given Data / Assumptions: - Distance each way on the river = 10 miles. - Time taken downstream = 20 minutes. - Time taken upstream = 30 minutes. - Let the speed of the boat in still water be b mph. - Let the speed of the current be c mph. - Downstream speed = b + c, upstream speed = b - c. Concept / Approach: Speed is equal to distance divided by time. We first convert the times from minutes to hours to stay in miles per hour. Using the given times and common distance, we find the effective downstream and upstream speeds. These give us two equations in b and c. Adding and subtracting the equations allows us to solve for the speed of the current c. Step-by-Step Solution: Step 1: Convert 20 minutes to hours: 20 / 60 = 1/3 hour. Step 2: Downstream speed = distance / time = 10 / (1/3) = 30 mph. Step 3: Convert 30 minutes to hours: 30 / 60 = 1/2 hour. Step 4: Upstream speed = distance / time = 10 / (1/2) = 20 mph. Step 5: Therefore, b + c = 30 and b - c = 20. Step 6: Add the two equations: (b + c) + (b - c) = 30 + 20 gives 2b = 50. Step 7: So b = 50 / 2 = 25 mph. Step 8: Substitute b into b + c = 30 to get 25 + c = 30, so c = 5 mph. Verification / Alternative check: Using b = 25 mph and c = 5 mph, downstream speed = 25 + 5 = 30 mph. Time downstream = 10 / 30 = 1/3 hour, which is 20 minutes, as given. Upstream speed = 25 - 5 = 20 mph. Time upstream = 10 / 20 = 1/2 hour, which is 30 minutes, also matching the data. So the calculated current speed is fully consistent with the problem conditions. Why Other Options Are Wrong: - 8 mph or 7 mph would give upstream or downstream times different from 20 and 30 minutes. - 6 mph or 4 mph also fail when checked back into the equations b + c = 30 and b - c = 20. - Only 5 mph satisfies both the downstream and upstream conditions exactly. Common Pitfalls: Many learners forget to convert minutes into hours, which leads to wrong speeds. Others mistakenly average the two times rather than working with speeds. A further mistake is to assume the current speed is half of one of the given speeds without using equations. Always use distance / time carefully and set up equations for downstream and upstream speeds. Final Answer: The speed of the river current is 5 mph.

Question 9

In one hour, a boat travels 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 question uses basic boats and streams concepts to find the speed of a boat in still water. When a boat travels with the stream and against the stream, the current either helps or opposes the motion, creating two different effective speeds. The middle value of these speeds gives the speed in still water. Given Data / Assumptions: - Speed along the stream (downstream speed) = 12 km/h. - Speed against the stream (upstream speed) = 6 km/h. - Let b be the speed of the boat in still water (km/h). - Let c be the speed of the current (km/h). - Downstream speed = b + c, upstream speed = b - c. Concept / Approach: Downstream the current adds to the boat speed, and upstream it subtracts. Therefore we can write two simple equations and then find the average of the downstream and upstream speeds to get the still water speed: b = (downstream + upstream) / 2. This is a standard shortcut result in boats and streams problems. Step-by-Step Solution: Step 1: b + c = 12 (downstream speed). Step 2: b - c = 6 (upstream speed). Step 3: Add the equations: (b + c) + (b - c) = 12 + 6. Step 4: This gives 2b = 18. Step 5: Divide by 2 to find b: b = 18 / 2 = 9 km/h. Step 6: So the speed of the boat in still water is 9 km/h. Verification / Alternative check: Use the average formula directly: speed in still water = (downstream speed + upstream speed) / 2 = (12 + 6) / 2 = 18 / 2 = 9 km/h. This quick method matches the algebraic solution, confirming the result. Why Other Options Are Wrong: - 8 km/h and 7 km/h do not satisfy both equations b + c = 12 and b - c = 6 together. - 7.5 km/h is simply the plain average of 9 and 6, not of 12 and 6, and does not work when checked in the equations. - 6 km/h would imply zero current, which contradicts the problem information. Common Pitfalls: Learners sometimes incorrectly average 12 and 6 and then subtract something for the current, or they confuse which speed is upstream and which is downstream. Remember that the boat speed in still water is always the simple average of upstream and downstream speeds, while the current is half of their difference. Final Answer: The speed of the boat in still water is 9 km/h.

Question 10

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

Accepted Answer

Introduction / Context: This question combines knowledge of speed, distance, and time with boats and streams concepts. You are given the sum of upstream and downstream speeds and one actual upstream journey. From this, you must work out the downstream speed and then determine the time taken to travel a different downstream distance. Given Data / Assumptions: - Sum of upstream and downstream speeds = 82 km/h. - Upstream distance covered = 105 km. - Time taken upstream = 3 hours. - Required: time to cover 126 km downstream. - Let upstream speed be u km/h and downstream speed be d km/h. Concept / Approach: First, find the upstream speed using speed = distance / time. Then use the given sum u + d = 82 to find downstream speed d. Finally, use the standard formula time = distance / speed to find the time for the downstream trip of 126 km. An approximate decimal answer is acceptable when the division is not exact. Step-by-Step Solution: Step 1: Upstream speed u = distance / time = 105 / 3 = 35 km/h. Step 2: We know u + d = 82, so 35 + d = 82. Step 3: Solve for d: d = 82 - 35 = 47 km/h. Step 4: Downstream distance required = 126 km. Step 5: Time taken downstream = distance / downstream speed. Step 6: Time = 126 / 47 hours. Step 7: 47 * 2 = 94 and 126 - 94 = 32, so 126 / 47 is a little more than 2.6. Step 8: Actual value 126 / 47 is approximately 2.68 hours, which rounds to about 2.7 hours. Verification / Alternative check: If we take 2.7 hours as an approximate time, the implied distance at 47 km/h is 47 * 2.7 = 126.9 km, which is very close to 126 km, allowing for rounding. Using 2.6 hours would give only 122.2 km, which is too low. Thus 2.7 hours is the best approximation among the options. Why Other Options Are Wrong: - 2.5 hours and 2.6 hours imply speeds that do not match 47 km/h when combined with 126 km. - 2.8 hours would suggest a distance greater than 126 km at 47 km/h. - 3 hours is clearly too long and would correspond to 141 km at 47 km/h. Common Pitfalls: A common error is to average 82 and 35 instead of using their relationship, or to miscalculate the upstream speed. Another frequent mistake is to round too early and then get a significantly different answer. Always find the exact division first, then round to the nearest option provided. Final Answer: The boat will take approximately 2.7 hours to cover 126 km downstream.

Question 11

A boat travels 72 km downstream in 8 hours and 84 km upstream in 12 hours. What are the speeds (in km/h) of the boat in still water and of the water current respectively?

Accepted Answer

Introduction / Context: Here we have a classic boats and streams problem where we are given separate downstream and upstream journeys with different distances and times. From these we can compute effective speeds in each direction and then deduce both the speed of the boat in still water and the speed of 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 (km/h). - Let c be the speed of the current (km/h). - Downstream speed = b + c, upstream speed = b - c. Concept / Approach: First we compute downstream and upstream speeds by distance / time. Then we use the standard relations b + c = downstream speed and b - c = upstream speed. Solving these two simple equations gives values for b and c. This method is central to most boats and streams aptitude questions. Step-by-Step Solution: Step 1: Downstream speed = 72 / 8 = 9 km/h. Step 2: Upstream speed = 84 / 12 = 7 km/h. Step 3: So, b + c = 9 and b - c = 7. Step 4: Add the equations: (b + c) + (b - c) = 9 + 7 gives 2b = 16. Step 5: Therefore b = 16 / 2 = 8 km/h. Step 6: Substitute b into b + c = 9: 8 + c = 9, so c = 1 km/h. Verification / Alternative check: With b = 8 km/h and c = 1 km/h, downstream speed = 8 + 1 = 9 km/h, giving time downstream = 72 / 9 = 8 hours, which matches the question. Upstream speed = 8 - 1 = 7 km/h, giving time upstream = 84 / 7 = 12 hours, also matching. The values are therefore correct and consistent with both journeys. Why Other Options Are Wrong: - 9 and 3 km/h give downstream and upstream speeds that do not match 9 and 7 km/h. - 6 and 7 km/h or 7 and 2 km/h would lead to huge differences between calculated and given times. - 10 and 2 km/h would produce different effective speeds than those derived from the distances and times in the problem. Common Pitfalls: Some learners mistakenly divide a wrong distance or misread upstream and downstream data. Others average the distances instead of speeds. Always compute the actual speeds first before forming equations, and be careful with which direction is upstream and which is downstream. Final Answer: The speed of the boat in still water is 8 km/h and the speed of the current is 1 km/h.

Question 12

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

Accepted Answer

Introduction / Context: This question checks your understanding of how to use downstream information to find the speed of a boat in still water and then use that to compute an upstream journey time. It connects distance, speed, and time with the influence of river current on rowing speed. Given Data / Assumptions: - Downstream distance = 96 km. - Time taken downstream = 8 hours. - Speed of current = 4 km/h. - Required: time to cover 8 km upstream. - Let b be the speed of the boat in still water (km/h). Concept / Approach: First find the downstream speed using distance / time. Since downstream speed equals b + c, where c is current speed, you can find b. Then upstream speed equals b - c. Finally, use time = distance / speed to calculate how long it takes to go 8 km upstream. This sequence is typical in many boat and stream problems. Step-by-Step Solution: Step 1: Downstream speed = 96 / 8 = 12 km/h. Step 2: Let b be the boat speed in still water and c = 4 km/h be the current speed. Step 3: b + c = downstream speed, so b + 4 = 12. Step 4: Solve for b: b = 12 - 4 = 8 km/h. Step 5: Upstream speed = b - c = 8 - 4 = 4 km/h. Step 6: Required upstream distance = 8 km. Step 7: Time upstream = distance / upstream speed = 8 / 4 = 2 hours. Verification / Alternative check: Check consistency: with b = 8 km/h and c = 4 km/h, downstream speed is 8 + 4 = 12 km/h, giving time 96 / 12 = 8 hours as in the question. Upstream speed is 8 - 4 = 4 km/h; at this speed, in 2 hours the boat covers 2 * 4 = 8 km. The calculations match the problem statement perfectly. Why Other Options Are Wrong: - 1 hour would correspond to an unrealistic upstream speed of 8 km/h, which would not match the still water speed found from downstream data. - 1.5 hours or 2.5 hours both represent incorrect divisions of 8 by the upstream speed. - 3 hours is much too long and would imply a very low upstream speed that conflicts with the given scenario. Common Pitfalls: Learners may mistakenly use the downstream speed directly for upstream travel or subtract the current from the distance instead of the speed. Others forget to separate the steps: first find boat speed in still water, then find upstream speed, and only then calculate time. Keeping a clear step order avoids confusion. Final Answer: The boatman will take 2 hours to cover 8 km upstream.

Question 13

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

Accepted Answer

Introduction / Context: Average speed in a round trip with different speeds in each direction is a common concept tested in aptitude exams. This boat and stream version requires you to compute effective downstream and upstream speeds first and then use total distance and total time to find the true average speed for the whole journey. Given Data / Assumptions: - Speed of Rajesh in still water = 4.5 km/h. - Speed of river current = 1.5 km/h. - Let distance to the place (one way) be d km. - Downstream speed = 4.5 + 1.5 km/h. - Upstream speed = 4.5 - 1.5 km/h. Concept / Approach: Average speed for a trip is defined as total distance divided by total time. Because the speeds downstream and upstream differ, we cannot simply average the two speeds. Instead, we compute the time taken for downstream and upstream separately using time = distance / speed, then add them, and finally divide the total distance 2d by the total time to get the average speed. Step-by-Step Solution: Step 1: Downstream speed = 4.5 + 1.5 = 6 km/h. Step 2: Upstream speed = 4.5 - 1.5 = 3 km/h. Step 3: Time taken downstream = d / 6 hours. Step 4: Time taken upstream = d / 3 hours. Step 5: Total distance for the round trip = d (going) + d (returning) = 2d. Step 6: Total time T = d / 6 + d / 3. Step 7: Simplify T: d / 6 + d / 3 = d / 6 + 2d / 6 = 3d / 6 = d / 2 hours. Step 8: Average speed = total distance / total time = 2d / (d / 2). Step 9: 2d divided by (d / 2) = 2d * (2 / d) = 4 km/h. Verification / Alternative check: Choose a convenient distance, say d = 6 km. Then downstream time = 6 / 6 = 1 hour; upstream time = 6 / 3 = 2 hours. Total distance = 12 km, total time = 3 hours. Average speed = 12 / 3 = 4 km/h. This specific example confirms the general calculation. Why Other Options Are Wrong: - 2 km/h and 8 km/h are far from the speeds involved and do not reflect correct averaging. - 6 km/h simply repeats the downstream speed and ignores the slower upstream leg. - 4.5 km/h is the still water speed, not the effective average when current affects both directions differently. Common Pitfalls: Many learners incorrectly average the downstream and upstream speeds, giving (6 + 3) / 2 = 4.5 km/h. This is wrong because average speed must always be computed as total distance divided by total time, especially when distances are same but speeds differ. Always calculate times first to avoid this mistake. Final Answer: Rajesh average speed for the whole journey is 4 km/h.

Question 14

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

Accepted Answer

Introduction / Context: Here you are asked to infer the speed of the stream using information about how long the same distance takes downstream and upstream. This is a standard boats and streams problem where times and distances are given instead of speeds directly, and you must work backwards to the current speed. Given Data / Assumptions: - Distance downstream and upstream = 36 km. - Downstream time = 6 hours. - Upstream time is 3 hours more, so upstream time = 9 hours. - Let b be the speed in still water, c be the speed of the stream. - Downstream speed = b + c, upstream speed = b - c. Concept / Approach: First find effective downstream and upstream speeds from distance / time. Then use the two equations b + c = downstream speed and b - c = upstream speed. Solving these simultaneous equations yields both the boat speed in still water and the stream speed. Our interest here is the stream speed c. Step-by-Step Solution: Step 1: Downstream speed = 36 / 6 = 6 km/h. Step 2: Upstream time = 6 + 3 = 9 hours. Step 3: Upstream speed = 36 / 9 = 4 km/h. Step 4: So b + c = 6 and b - c = 4. Step 5: Add the equations: (b + c) + (b - c) = 6 + 4, giving 2b = 10. Step 6: Hence b = 10 / 2 = 5 km/h. Step 7: Substitute into b + c = 6: 5 + c = 6, so c = 1 km/h. Verification / Alternative check: With b = 5 km/h and c = 1 km/h, downstream speed = 6 km/h and upstream speed = 4 km/h. Time downstream = 36 / 6 = 6 hours. Time upstream = 36 / 4 = 9 hours, which is exactly 3 hours more than the downstream time. So the stream speed of 1 km/h is consistent with all given information. Why Other Options Are Wrong: - 1.5, 0.75, and 0.5 km/h would all produce upstream and downstream speeds that do not result in times of 6 and 9 hours for 36 km. - 2 km/h would give upstream speed 3 km/h, leading to 12 hours upstream, not 9 hours. Common Pitfalls: Some learners mistakenly subtract the times or distances directly to get the stream speed or confuse upstream and downstream speeds. Others incorrectly average times instead of working with speeds. Always convert time and distance into speeds first, then use the b + c and b - c system of equations. Final Answer: The speed of the stream is 1 km/h.

Question 15

A man can row three-fourths of a kilometre against the current in 15 minutes and returns the same distance with the current in 10 minutes. What is the ratio of his speed in still water to the speed of the current?

Accepted Answer

Introduction / Context: This problem focuses on finding the ratio of the rowing speed in still water to the speed of the current using upstream and downstream times for the same distance. It tests your ability to convert minutes to hours, compute speeds, and then combine them using standard boats and streams formulas. Given Data / Assumptions: - Distance each way = three fourth of a kilometre = 3/4 km = 0.75 km. - Time taken upstream = 15 minutes. - Time taken downstream = 10 minutes. - Let b be speed in still water (km/h), c speed of current (km/h). - Upstream speed = b - c, downstream speed = b + c. Concept / Approach: Speed equals distance divided by time, so we first compute the upstream and downstream speeds from the 0.75 km journeys. Then we use the relations b - c = upstream speed and b + c = downstream speed. Solving these equations gives both b and c, from which the ratio b : c can be simplified to a whole number ratio. Step-by-Step Solution: Step 1: Convert times to hours. Upstream: 15 minutes = 15 / 60 = 0.25 hour. Step 2: Upstream speed = distance / time = 0.75 / 0.25 = 3 km/h. Step 3: Downstream: 10 minutes = 10 / 60 hour = 1/6 hour. Step 4: Downstream speed = 0.75 / (1/6) = 0.75 * 6 = 4.5 km/h. Step 5: So b - c = 3 and b + c = 4.5. Step 6: Add equations: (b - c) + (b + c) = 3 + 4.5 gives 2b = 7.5. Step 7: Therefore b = 7.5 / 2 = 3.75 km/h. Step 8: Substitute into b + c = 4.5: 3.75 + c = 4.5, so c = 0.75 km/h. Step 9: Ratio b : c = 3.75 : 0.75. Step 10: Divide both by 0.75 to get 5 : 1. Verification / Alternative check: Check with the found speeds. Upstream speed = b - c = 3.75 - 0.75 = 3 km/h. Time for 0.75 km upstream = 0.75 / 3 = 0.25 hour = 15 minutes. Downstream speed = 3.75 + 0.75 = 4.5 km/h, time = 0.75 / 4.5 = 1/6 hour = 10 minutes. Both match the given data, so the ratio 5 : 1 is correct. Why Other Options Are Wrong: - Ratios 3 : 1, 4 : 1, 2 : 1, and 6 : 1 do not produce upstream and downstream speeds that match 3 km/h and 4.5 km/h for the given distance and times. - Any incorrect ratio would change either the upstream or downstream time away from 15 and 10 minutes. Common Pitfalls: Some learners forget to convert minutes into hours, leading to incorrect speeds. Others may try to directly ratio the times rather than deriving speeds. Always use distance / time to find correct upstream and downstream speeds before forming equations and ratios. Final Answer: The ratio of the man speed in still water to the speed of the current is 5 : 1.

Question 16

A man swims 72 km downstream in 9 hours and 45 km upstream in 9 hours. What is the speed of the current (in km/h)?

Accepted Answer

Introduction / Context: This swimming problem is another boats and streams style question. You are given downstream and upstream journeys over different distances but equal times. From these journeys you can determine downstream and upstream speeds and then compute the speed of the current itself. Given Data / Assumptions: - Downstream distance = 72 km, time = 9 hours. - Upstream distance = 45 km, time = 9 hours. - Let b be the swimmer speed in still water (km/h). - Let c be the speed of the current (km/h). - Downstream speed = b + c, upstream speed = b - c. Concept / Approach: Compute the effective speeds in both directions by dividing distance by time. These are equal to b + c and b - c respectively. Once we have these values, adding and subtracting the two equations allows us to solve for b and c. The question specifically asks for the value of c, the current speed. Step-by-Step Solution: Step 1: Downstream speed = 72 / 9 = 8 km/h. Step 2: Upstream speed = 45 / 9 = 5 km/h. Step 3: So b + c = 8 and b - c = 5. Step 4: Add the equations: (b + c) + (b - c) = 8 + 5 gives 2b = 13. Step 5: Thus b = 13 / 2 = 6.5 km/h. Step 6: Substitute into b + c = 8: 6.5 + c = 8, so c = 8 - 6.5 = 1.5 km/h. Verification / Alternative check: Check the downstream and upstream speeds using b = 6.5 and c = 1.5. Downstream speed = 6.5 + 1.5 = 8 km/h, so time for 72 km is 72 / 8 = 9 hours, as given. Upstream speed = 6.5 - 1.5 = 5 km/h, so time for 45 km is 45 / 5 = 9 hours, also matching the question. Therefore, the current speed 1.5 km/h is correct. Why Other Options Are Wrong: - 1 km/h or 2 km/h would give different downstream and upstream speeds that do not match 8 km/h and 5 km/h. - 3.2 km/h and 2.5 km/h would cause large mismatches in times when checked back with the given distances. Common Pitfalls: Sometimes learners average the distances or the times instead of calculating speeds separately. Another error is mixing up downstream and upstream formulas. Always start with speed = distance / time, then identify which speed corresponds to b + c and which corresponds to b - c, and finally solve for c. Final Answer: The speed of the current is 1.5 km/h.

Question 17

A person can row at 8.5 km/h in still water. He finds that 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 (in km/h)?

Accepted Answer

Introduction / Context: This question explores the relationship between upstream and downstream times when the boat speed in still water is known. If the time taken upstream is double the time taken downstream for the same distance, there is a specific relationship between the boat speed and stream speed that can be derived using the time formula. Given Data / Assumptions: - Speed of rowing in still water b = 8.5 km/h. - Time taken upstream is twice the time taken downstream for the same distance. - Let c be the speed of the stream (km/h). - Upstream speed = b - c, downstream speed = b + c. - Let the distance each way be L km. Concept / Approach: Time = distance / speed. For the same distance L, upstream time is L / (b - c) and downstream time is L / (b + c). The problem states that upstream time is twice downstream time, which gives an equation relating b and c. Substituting b = 8.5 into this relation allows us to solve for c and thus find the stream speed. Step-by-Step Solution: Step 1: Upstream time T up = L / (b - c). Step 2: Downstream time T down = L / (b + c). Step 3: Given T up = 2 * T down. Step 4: So L / (b - c) = 2L / (b + c). Step 5: Cancel L from both sides: 1 / (b - c) = 2 / (b + c). Step 6: Cross multiply: b + c = 2(b - c). Step 7: Expand right side: b + c = 2b - 2c. Step 8: Rearrange: b + c - 2b + 2c = 0 gives -b + 3c = 0. Step 9: So 3c = b and c = b / 3. Step 10: Substitute b = 8.5 km/h: c = 8.5 / 3 km/h. Step 11: 8.5 / 3 ≈ 2.833 km/h, which is about 2.83 km/h. Verification / Alternative check: With b = 8.5 and c ≈ 2.83, downstream speed ≈ 11.33 km/h and upstream speed ≈ 5.67 km/h. For a distance of, say, 11.33 km, downstream time ≈ 1 hour. Upstream time for the same distance is 11.33 / 5.67 ≈ 2 hours, which is twice the downstream time as required. This confirms that the computed stream speed is correct. Why Other Options Are Wrong: - 1.78, 2.35, and 3.15 km/h lead to time ratios that are not exactly 2:1 between upstream and downstream. - 4 km/h would make upstream speed very low and the ratio between times much larger than 2. Common Pitfalls: A common mistake is to try to average speeds or to assume that the stream speed is some simple fraction of 8.5 without using the time relationship. Others may set up the equation with the ratio inverted. Always be clear which time is longer and by what factor, and translate that precisely into a mathematical equation. Final Answer: The speed of the stream is approximately 2.83 km/h.

Question 18

Amith can row a boat a distance d km upstream and the same distance d km downstream in a total time of 5 hours 15 minutes. He can also row the boat 2d km upstream in 7 hours. How long will it take Amith to row the distance 2d km downstream?

Accepted Answer

Introduction / Context: This is a multi step boats and streams problem involving an unknown distance and two different journey scenarios. It requires careful algebraic modelling of upstream and downstream times and is a good test of problem solving with averages and relative speeds on water. Given Data / Assumptions: - Amith rows a distance d km upstream and d km downstream in 5 hours 15 minutes. - 5 hours 15 minutes = 5.25 hours. - He rows 2d km upstream in 7 hours. - Let b be the speed of the boat in still water (km/h). - Let c be the speed of the current (km/h). - Upstream speed = b - c, downstream speed = b + c. Concept / Approach: Time equals distance divided by speed. The given information can be translated into two equations involving d, b, and c. However, we can work smartly by focusing on times per unit distance instead of finding b and c explicitly. By letting t up denote time to travel d km upstream and t down denote time to travel d km downstream, we can first compute t up from the 2d upstream journey, then use the total 5.25 hours for d up and d down to find t down. From that, the time for 2d downstream is simply 2 * t down. Step-by-Step Solution: Step 1: For upstream, speed is b - c, so time for distance d upstream is d / (b - c). Step 2: For downstream, speed is b + c, so time for distance d downstream is d / (b + c). Step 3: From the first scenario, total time for d upstream and d downstream is 5.25 hours. Step 4: So d / (b - c) + d / (b + c) = 5.25. Call this Equation (1). Step 5: From the second scenario, 2d km upstream takes 7 hours, so 2d / (b - c) = 7. Step 6: Divide both sides by 2 to get d / (b - c) = 3.5 hours. Step 7: Substitute d / (b - c) = 3.5 into Equation (1): 3.5 + d / (b + c) = 5.25. Step 8: Hence d / (b + c) = 5.25 - 3.5 = 1.75 hours. Step 9: Time to row 2d km downstream = 2 * [d / (b + c)] = 2 * 1.75 = 3.5 hours. Step 10: 3.5 hours = 3 hours 30 minutes. Verification / Alternative check: Notice that upstream time per d is 3.5 hours and downstream time per d is 1.75 hours. For the 5.25 hour combined trip, 3.5 + 1.75 = 5.25, which matches the given total. For the 2d upstream trip, 2 * 3.5 = 7 hours, which matches the given 7 hours. Therefore the derived values are consistent, and 2d downstream must indeed take 2 * 1.75 hours = 3.5 hours. Why Other Options Are Wrong: - 4 hours 10 minutes and 4 hours 1 minute are larger than 3.5 hours and would not align with the ratio between upstream and downstream times. - 3 hours 15 minutes and 3 hours 45 minutes give downstream times per d that do not combine correctly to produce the 5.25 and 7 hour totals given in the question. Common Pitfalls: Learners often try to find b and c directly and get lost in algebra, or they misuse the distances d and 2d. Another common mistake is to convert 5 hours 15 minutes incorrectly, or to mix hours and minutes during calculations. Working in decimal hours and focusing on time per d is a clean, efficient approach here. Final Answer: Amith will take 3 hours 30 minutes to row 2d km downstream.

Question 19

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

Accepted Answer

Introduction / Context: This question involves comparing upstream and downstream times for the same distance to find the ratio of the speed of the motorboat in still water to the speed of the current. It is a good example of how relative speed in water is affected by current and how timing data can be converted into speed ratios. Given Data / Assumptions: - Time taken upstream = 8 hours 48 minutes. - Time taken downstream = 4 hours. - 8 hours 48 minutes = 8 + 48/60 = 8.8 hours. - Let the distance be D km. - Let b be the speed of the boat in still water (km/h). - Let c be the speed of the current (km/h). - Upstream speed = b - c, downstream speed = b + c. Concept / Approach: Speed equals distance divided by time. Using the same distance D, we can express upstream and downstream speeds in terms of D and times. These two speeds must match b - c and b + c. We then add and subtract the resulting equations to isolate b and c, and finally form the ratio b : c. The distance D cancels out, so its specific value is not needed. Step-by-Step Solution: Step 1: Upstream speed = D / 8.8 km/h. Step 2: Downstream speed = D / 4 km/h. Step 3: So b - c = D / 8.8 and b + c = D / 4. Step 4: Write 8.8 as 44/5 so that 1 / 8.8 = 5 / 44. Step 5: Hence b - c = D * 5 / 44 and b + c = D / 4. Step 6: Add equations: (b - c) + (b + c) = D * 5 / 44 + D / 4. Step 7: This gives 2b = D * (5 / 44 + 11 / 44) = D * 16 / 44 = D * 4 / 11. Step 8: So b = D * 2 / 11. Step 9: Subtract equations: (b + c) - (b - c) = D / 4 - D * 5 / 44. Step 10: This gives 2c = D * (11 / 44 - 5 / 44) = D * 6 / 44 = D * 3 / 22. Step 11: Therefore c = D * 3 / 44. Step 12: Ratio b : c = (D * 2 / 11) : (D * 3 / 44). Step 13: Cancel D and simplify: (2 / 11) : (3 / 44) = (2 / 11) * (44 / 3) = 88 / 33 = 8 / 3. Verification / Alternative check: The ratio 8 : 3 indicates that the boat speed in still water is more than twice the current speed, which is reasonable given that upstream time is more than double downstream time. Any alternative ratio must be tested against the time data, and 8 : 3 is the only one consistent with the observed 8.8 versus 4 hours for the same distance. Why Other Options Are Wrong: - Ratios 3 : 7, 7 : 9, 5 : 4, and 4 : 1 do not match the precise relationship between upstream and downstream times shown by the 8.8 and 4 hour durations. - Using these incorrect ratios would give very different predicted times for either upstream or downstream travel. Common Pitfalls: Mistakes often occur when converting 8 hours 48 minutes to hours, or when trying to directly write ratios from times without going through speeds. Another common error is to try averaging times instead of speeds. Always reduce mixed times to hours, express speeds as D divided by time, and then work systematically with equations. Final Answer: The ratio of the speed of the motorboat in still water to the speed of the current is 8 : 3.

Question 20

Sravan drove from home to a neighbouring town at a speed of 50 km/h. On his return journey he drove at 45 km/h and took 1 hour longer to reach home. What distance (in km) did he cover one way?

Accepted Answer

Introduction / Context: This is a classic time, speed, and distance problem in which the distances are the same in both directions but speeds differ, causing different travel times. The key idea is to express both times in terms of distance and use the given difference in time to solve for the distance. Given Data / Assumptions: - Speed from home to town = 50 km/h. - Speed from town back to home = 45 km/h. - Return journey takes 1 hour longer than the onward journey. - Let one way distance be d km. - Distances going and coming back are equal. Concept / Approach: Time equals distance divided by speed. Let t1 be the time from home to town and t2 be the time from town to home. Then t1 = d / 50 and t2 = d / 45. According to the problem, t2 is 1 hour longer than t1, so t2 - t1 = 1. This equation in d can be solved using simple fraction manipulation to obtain the distance. Step-by-Step Solution: Step 1: Time for onward journey: t1 = d / 50 hours. Step 2: Time for return journey: t2 = d / 45 hours. Step 3: Given that t2 = t1 + 1. Step 4: So d / 45 = d / 50 + 1. Step 5: Bring terms together: d / 45 - d / 50 = 1. Step 6: Use common denominator 2250: (50d - 45d) / 2250 = 1. Step 7: This gives 5d / 2250 = 1. Step 8: Multiply both sides by 2250 to get 5d = 2250. Step 9: Therefore d = 2250 / 5 = 450 km. Verification / Alternative check: If d = 450 km, onward time t1 = 450 / 50 = 9 hours. Return time t2 = 450 / 45 = 10 hours. Difference t2 - t1 = 10 - 9 = 1 hour, which matches the given condition. Therefore, 450 km is consistent with the problem statement. Why Other Options Are Wrong: - 350 km or 500 km would lead to time differences smaller or larger than 1 hour. - 700 km and 900 km give onward and return times whose differences do not match the required 1 hour. - Only 450 km satisfies the exact time difference required. Common Pitfalls: A typical mistake is to set up the equation as t1 - t2 = 1 instead of t2 - t1 = 1 or to average the two speeds incorrectly. Another error is to forget that the distance both ways is the same. Always start with d / speed expressions for each leg and carefully follow the condition about which time is longer. Final Answer: Sravan covered a one way distance of 450 km.

Boats and Streams Questions