Question 1

A man rows 3/4 km 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: Here we are asked to find the ratio of a man’s rowing speed in still water to the speed of the current. The man rows a short distance upstream and downstream with different times for each leg. By using distance over time we find effective speeds and then use the boats and streams relationships to derive the required ratio. Ratios are often tested to avoid computing exact decimal values. Given Data / Assumptions: Distance each way = 3/4 km. Upstream time = 15 minutes. Downstream time = 10 minutes. Speeds are constant in both directions. We want the ratio speed in still water : stream speed. Concept / Approach: If b is the man’s speed in still water and c is the speed of the current, then: Upstream speed = b - c. Downstream speed = b + c. We first compute the numerical values of upstream and downstream speeds from distance and time. Then we solve for b and c and finally express their ratio b : c in simplest form. Step-by-Step Solution: Convert times to hours: 15 minutes = 15 / 60 = 1/4 hour, 10 minutes = 10 / 60 = 1/6 hour. Upstream speed = distance / time = (3/4) / (1/4) = 3 km/h. Downstream speed = (3/4) / (1/6) = (3/4) * 6 = 18 / 4 = 4.5 km/h. Let b be speed in still water and c be stream speed. Then b - c = 3 and b + c = 4.5. Add the equations: 2b = 7.5 so b = 7.5 / 2 = 3.75 km/h. Subtract the equations: 2c = 1.5 so c = 1.5 / 2 = 0.75 km/h. Thus the ratio b : c = 3.75 : 0.75. Divide both numbers by 0.75 to obtain 5 : 1. Verification / Alternative check: Using b = 3.75 km/h and c = 0.75 km/h, upstream speed is 3.75 - 0.75 = 3 km/h, downstream speed is 3.75 + 0.75 = 4.5 km/h. For a distance of 3/4 km, upstream time is (3/4) / 3 = 1/4 hour or 15 minutes, and downstream time is (3/4) / 4.5 = 1/6 hour or 10 minutes. These match the problem statement, confirming that the speeds and ratio are correct. Why Other Options Are Wrong: Ratios 3:1, 4:1, and 2:1 would correspond to different pairs of b and c that do not produce the observed upstream and downstream times. For example, a 3:1 ratio might lead to speeds that yield equal or different time differences. Only the 5:1 ratio reconstructs the correct upstream and downstream times for the given distance. Common Pitfalls: One common mistake is to forget to convert minutes to hours, causing incorrect speeds. Another is to try to directly ratio the given times without first computing speeds. Also, some learners might mistakenly compute stream speed as the difference of times instead of the difference of speeds. Clear steps and consistent use of units prevent these errors. Final Answer: The ratio of the man’s speed in still water to the speed of the current is 5:1.

Question 2

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

Accepted Answer

Introduction / Context: This swimming problem is a boats and streams type question with different distances downstream and upstream but equal times. We can use the distances and times to find effective downstream and upstream speeds. Once we know those speeds, we can find the swimmer’s speed in still water and the current speed by solving simple equations. The focus is on manipulating distance time relations and using the symmetric structure of the problem. Given Data / Assumptions: Downstream distance = 72 km. Downstream time = 9 hours. Upstream distance = 45 km. Upstream time = 9 hours. Speeds are assumed constant, and current is uniform. Concept / Approach: First we compute: Downstream speed = distance / time. Upstream speed = distance / time. Then we let b be the swimmer’s speed in still water and c be the current speed. We use: b + c = downstream speed. b - c = upstream speed. Solving the system gives c, the required speed of the current. Step-by-Step Solution: Downstream speed = 72 / 9 = 8 km/h. Upstream speed = 45 / 9 = 5 km/h. Let b be swimming speed in still water and c be stream speed. Then b + c = 8 and b - c = 5. Add the equations: 2b = 13 so b = 6.5 km/h. Subtract the equations: 2c = 3 so c = 1.5 km/h. Therefore, the speed of the current is 1.5 km/h. Verification / Alternative check: Using b = 6.5 km/h and c = 1.5 km/h, downstream speed is 8 km/h, giving time 72 / 8 = 9 hours for the downstream journey. Upstream speed is 5 km/h, giving time 45 / 5 = 9 hours for the upstream journey. Both match the given times, confirming that the calculation of the current speed is accurate. Why Other Options Are Wrong: A current speed of 1 km/h would produce different upstream and downstream speeds and would not match the given distances for 9 hours. Values such as 2 km/h or 3.2 km/h would change the speeds even more and would not yield the exact distances within 9 hours. Only 1.5 km/h leads to effective speeds of 8 km/h and 5 km/h that satisfy the problem conditions. Common Pitfalls: Some students average the distances or times incorrectly, while others forget that the swimmer’s own speed is different from the effective speeds. Another mistake is to divide the sum or difference of distances by total time instead of computing each speed separately. Always calculate upstream and downstream speeds first, then use the system of equations to obtain still water and current speeds. Final Answer: The speed of the current is 1.5 km/h.

Question 3

A person can row at 8.5 km/h in still water and finds that it takes twice as long to row upstream as it does to row downstream over the same distance. What is the speed of the stream (in km/h)?

Accepted Answer

Introduction / Context: This question relates the ratio of times taken upstream and downstream to the speeds of a boat in still water and the stream. The person’s speed in still water is given, and we know that rowing upstream takes twice as long as rowing downstream for the same distance. Using the relationship between time and speed, we can set up an equation and solve for the stream speed. Given Data / Assumptions: Speed 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. Upstream speed = b - c, downstream speed = b + c. Distances for upstream and downstream are equal. Concept / Approach: If the distance is D, then: Time upstream = D / (b - c). Time downstream = D / (b + c). We are told that upstream time is twice downstream time, so D / (b - c) = 2 * D / (b + c). Distance cancels, and we get an equation only in b and c. Solving this equation gives c. The stream speed can then be compared with the options. Step-by-Step Solution: Let b = 8.5 km/h and c be the stream speed. Upstream time = D / (b - c), downstream time = D / (b + c). Given that upstream time is twice downstream time: D / (b - c) = 2 * D / (b + c). Cancel D from both sides to get 1 / (b - c) = 2 / (b + c). Cross multiply: b + c = 2(b - c). So b + c = 2b - 2c. Rearrange: b + c - 2b + 2c = 0 which simplifies to -b + 3c = 0. Thus 3c = b so c = b / 3. Substitute b = 8.5 to get c = 8.5 / 3 ≈ 2.833 km/h. Verification / Alternative check: If c ≈ 2.83 km/h, then downstream speed is about 8.5 + 2.83 = 11.33 km/h and upstream speed is about 8.5 - 2.83 = 5.67 km/h. For any fixed distance, say D = 11.33 km, downstream time is 11.33 / 11.33 = 1 hour and upstream time is 11.33 / 5.67 ≈ 2 hours, which is twice as long. This confirms the relationship described in the question. Why Other Options Are Wrong: If the stream speed were 1.78 km/h, the time ratio would not be exactly 2:1. Similar mismatches occur with 2.35 km/h and 3.15 km/h. Only a stream speed close to 2.83 km/h makes the upstream time exactly double the downstream time for the same distance when the still water speed is 8.5 km/h. Common Pitfalls: A common misunderstanding is to invert the given ratio and set downstream time to be double instead of upstream. Another error is to relate speed ratio directly to time ratio without considering the reciprocal relationship. The correct approach is to express times as distance divided by speed and only then apply the time ratio condition. Final Answer: The speed of the stream is approximately 2.83 km/h.

Question 4

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

Accepted Answer

Introduction / Context: This is a slightly more algebraic boats and streams problem that uses symbolic distance d rather than concrete numbers. We are told the total time for an upstream and downstream trip of distance d each, and the time for an upstream trip of distance 2d. Using these conditions, we can determine the upstream and downstream speeds and then compute the time needed for a downstream trip of length 2d. The problem tests comfort with symbolic manipulation and relative speed concepts. Given Data / Assumptions: Time to row d km upstream and d km downstream = 5 hours 15 minutes = 5.25 hours. Time to row 2d km upstream = 7 hours. Let upstream speed be u km/h and downstream speed be v km/h. Speeds are constant and d is a positive distance. Concept / Approach: We can express times in terms of distance and speed: Time upstream for distance d = d / u. Time downstream for distance d = d / v. Time upstream for distance 2d = 2d / u. From the second condition, 2d / u = 7, which gives u in terms of d. We substitute that into the first condition d / u + d / v = 5.25 and solve for v. Once v is known, time for 2d downstream is 2d / v. Step-by-Step Solution: Let u be upstream speed and v be downstream speed. Given 2d / u = 7, so u = 2d / 7. Total time for one upstream d and one downstream d is d / u + d / v = 5.25 hours. Substitute u: d / (2d / 7) + d / v = 5.25. Simplify d / (2d / 7) = 7 / 2 = 3.5 hours. So 3.5 + d / v = 5.25. Then d / v = 5.25 - 3.5 = 1.75 hours. Thus v = d / 1.75. Time for 2d downstream = 2d / v = 2d / (d / 1.75) = 2 * 1.75 = 3.5 hours. 3.5 hours = 3 hours 30 minutes. Verification / Alternative check: We can verify by choosing a convenient value for d, for example d = 7 km. Then u = 2d / 7 = 2 km/h. Time for 2d upstream is 14 / 2 = 7 hours, which matches the given condition. From d / v = 1.75, with d = 7 we get v = 7 / 1.75 = 4 km/h. Then time for one upstream 7 km is 7 / 2 = 3.5 hours, and one downstream 7 km is 7 / 4 = 1.75 hours. Total is 3.5 + 1.75 = 5.25 hours, as required. Time for 2d downstream is 14 / 4 = 3.5 hours or 3 hours 30 minutes, confirming the result. Why Other Options Are Wrong: Options such as 4 hours 10 minutes or 4 hours 1 minute would correspond to different downstream speeds that conflict with the earlier conditions about 5.25 hours and 7 hours. Similarly, 3 hours 15 minutes is less than the calculated required time. Only 3 hours 30 minutes is consistent with all given time relations. Common Pitfalls: Students sometimes attempt to guess d rather than treating it symbolically, and may choose inconsistent values. Another common mistake is to confuse the upstream and downstream times or forget to convert 5 hours 15 minutes into 5.25 hours. Working carefully with algebra and converting mixed time formats into pure hours helps avoid such misunderstandings. Final Answer: Amith will take 3 hours 30 minutes to row the distance 2d km downstream.

Question 5

A motorboat running upstream takes 8 hours 48 minutes to cover a certain distance and 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 problem is structurally identical to a typical boats and streams question in which we are given upstream and downstream times for the same distance. The aim is to find the ratio of the speed of the boat in still water to the speed of the current. It illustrates a useful shortcut that expresses this ratio directly in terms of the upstream and downstream times. Given Data / Assumptions: Upstream time = 8 hours 48 minutes. Downstream time = 4 hours. Distance upstream and downstream is the same. Speeds are constant and current is uniform. Concept / Approach: For a fixed distance D: Upstream speed = D / T_up. Downstream speed = D / T_down. Let b be the speed of the boat in still water and c be the speed of the current. Then b = (downstream speed + upstream speed) / 2 and c = (downstream speed - upstream speed) / 2. There is a known result that b : c equals (T_up + T_down) : (T_up - T_down). We can use this time based ratio directly to avoid computing D. Step-by-Step Solution: Convert 8 hours 48 minutes into hours: 48 minutes = 48 / 60 = 0.8 hours. So T_up = 8.8 hours. T_down = 4 hours. Ratio 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 numbers by 0.8 to simplify: 12.8 / 0.8 = 16 and 4.8 / 0.8 = 6, giving 16 : 6. Divide by 2 to get 8 : 3. Verification / Alternative check: We can also confirm by assuming a convenient distance, say D = 1 unit, computing upstream and downstream speeds as 1 / 8.8 and 1 / 4, then finding b and c from (v_up + v_down) / 2 and (v_down - v_up) / 2. Simplifying b : c from those values leads to the same ratio 8 : 3. This double checks the shortcut formula and validates the answer. Why Other Options Are Wrong: Ratios like 3:7 or 7:9 do not match the relationship between times, and if we try to back compute speeds from them we do not obtain the given upstream and downstream times. The ratio 5:4 is also inconsistent with the time difference. Only 8:3 is consistent with the given travel times for the same distance. Common Pitfalls: A frequent mistake is to use T_up : T_down directly as the ratio of speeds, which is incorrect because speed is inversely proportional to time. Another issue is failing to convert 8 hours 48 minutes correctly to 8.8 hours, which leads to wrong numbers in the ratio formula. Proper unit conversion and careful use of the shortcut formula are essential. Final Answer: The ratio of the speed of the motorboat in still water to the speed of the current is 8:3.

Question 6

Sravan drives from home to a neighbouring town at 50 km/h and returns at 45 km/h. The return journey takes one hour longer than the onward journey. What distance (in km) does he cover one way?

Accepted Answer

Introduction / Context: This is a classic average speed type question on a road journey, structurally similar to boats and streams problems. Sravan drives to a town at one speed and returns at a different speed, with the return trip taking one hour longer than the outward trip. The distance one way is the same in both directions, so we can formulate an equation in terms of distance and travel times to find the unknown distance. Given Data / Assumptions: Speed from home to town = 50 km/h. Speed from town back home = 45 km/h. Return trip takes 1 hour longer than the onward trip. Let D be the one way distance in km. Speeds are constant and there are no other delays. Concept / Approach: Time is distance divided by speed. If T1 is the time from home to town and T2 is the time back, then: T1 = D / 50. T2 = D / 45. Given that T2 = T1 + 1, we form an equation and solve for D. This algebraic approach is straightforward and robust. Step-by-Step Solution: Let D be the distance (in km) from home to the town. Time going out T1 = D / 50 hours. Time coming back T2 = D / 45 hours. Given that the return journey takes 1 hour longer: D / 45 = D / 50 + 1. Multiply both sides by 450 to clear denominators. This gives 10D = 9D + 450. So 10D - 9D = 450 which implies D = 450 km. Verification / Alternative check: With D = 450 km, time from home to town is 450 / 50 = 9 hours. Time for the return trip is 450 / 45 = 10 hours. The difference between these times is 1 hour, exactly as stated in the problem. This confirms that the distance 450 km is correct. Why Other Options Are Wrong: If D were 350 km, times would be 7 hours and about 7.78 hours, which differ by less than an hour. If D were 700 km, times would be 14 hours and about 15.56 hours, a difference of 1.56 hours. For 900 km, the difference in times would be even larger. Only 450 km results in a difference of exactly 1 hour between onward and return journey times. Common Pitfalls: A common mistake is to think that distance is proportional to the difference in speeds alone, without using the time relation. Another error is to set up the equation backwards, for example writing D / 50 = D / 45 + 1, which reverses the problem’s condition. Carefully translating the statement "return journey takes one hour longer" into the equation is essential for success. Final Answer: Sravan covers a one way distance of 450 km between his home and the town.

Question 7

A woman’s downstream swimming speed is three times her upstream swimming speed. If she covers 12 km upstream in 2.5 hours, how much distance (in km) will she cover in 5 hours downstream?

Accepted Answer

Introduction / Context: This swimming problem describes a relationship between upstream and downstream speeds. The downstream speed is three times the upstream speed, and we are given the upstream distance and time. From this, we can compute the upstream speed, derive the downstream speed, and then calculate how far the woman can swim downstream in a given time. It illustrates proportional reasoning in the context of relative motion in water. Given Data / Assumptions: Upstream distance = 12 km. Upstream time = 2.5 hours. Downstream speed = 3 times upstream speed. We need downstream distance in 5 hours. Speeds are constant, and water flow is uniform. Concept / Approach: First we compute upstream speed from distance and time. Then downstream speed is given as three times that upstream speed. Once downstream speed is known, we simply multiply it by the given downstream time to find the required distance. This problem does not require separate still water and stream speeds; it only needs the effective speeds. Step-by-Step Solution: Upstream distance = 12 km, upstream time = 2.5 hours. Upstream speed = distance / time = 12 / 2.5 km/h. Compute 12 / 2.5 = 4.8 km/h. Given that downstream speed = 3 times upstream speed. So downstream speed = 3 * 4.8 = 14.4 km/h. Time available downstream = 5 hours. Downstream distance = speed * time = 14.4 * 5 = 72 km. Verification / Alternative check: We can double check the calculations. Upstream speed of 4.8 km/h over 2.5 hours gives 4.8 * 2.5 = 12 km, which matches the given upstream distance. Downstream speed of 14.4 km/h for 5 hours yields 14.4 * 5 = 72 km, which is consistent with the proportional relation provided. The internal arithmetic is straightforward and fully supports the answer. Why Other Options Are Wrong: If we mistakenly used a different multiple or computed upstream speed incorrectly, we might get values like 36 km, 56 km, or 42 km. However, those values do not match a downstream speed that is exactly three times the correctly computed upstream speed over 5 hours. Only 72 km corresponds to 14.4 km/h sustained for 5 hours. Common Pitfalls: Some learners confuse the relation and assume upstream speed is three times downstream speed. Others miscalculate 12 / 2.5 or incorrectly double instead of tripling the upstream speed. It is important to carefully read the phrase "downstream swimming rate is thrice of her upstream swimming rate" and perform precise arithmetic when converting between speed, time, and distance. Final Answer: The woman will cover 72 km in 5 hours while swimming downstream.

Question 8

A boat can move at 5 km/h in still water. If the river is flowing at 1 km/h, it takes the boat 75 minutes to go to a place and return. How far (in km) is the place from the starting point?

Accepted Answer

Introduction / Context: This problem involves a round trip in a river with a known boat speed in still water and known stream speed. We are given the total time for the trip to a certain place and back. From these data, we must determine the one way distance. By computing upstream and downstream speeds, expressing the round trip time in terms of distance, and equating it to the given total time, we can solve for the distance. Given Data / Assumptions: Boat speed in still water = 5 km/h. Stream speed = 1 km/h. Total time for the journey to the place and back = 75 minutes. Let the one way distance be x km. Speeds are constant and water current is uniform. Concept / Approach: The downstream speed is boat speed plus stream speed, and the upstream speed is boat speed minus stream speed. Time taken downstream is distance divided by downstream speed, and time taken upstream is distance divided by upstream speed. The sum of these two times equals the given total time converted into hours. Solving this equation yields x, the required distance. Step-by-Step Solution: Let x be the one way distance in km. Downstream speed = 5 + 1 = 6 km/h. Upstream speed = 5 - 1 = 4 km/h. Total time = 75 minutes = 75 / 60 hours = 1.25 hours. Time downstream = x / 6 hours. Time upstream = x / 4 hours. Total time equation: x / 6 + x / 4 = 1.25. Find common denominator 12: (2x + 3x) / 12 = 1.25. So 5x / 12 = 1.25. Multiply both sides by 12: 5x = 1.25 * 12 = 15. Thus x = 15 / 5 = 3 km. Verification / Alternative check: With x = 3 km, downstream time is 3 / 6 = 0.5 hours (30 minutes). Upstream time is 3 / 4 = 0.75 hours (45 minutes). Total time is 0.5 + 0.75 = 1.25 hours, which equals 75 minutes as given. This confirms that the distance 3 km is consistent with all data in the problem. Why Other Options Are Wrong: If x were 4 km, downstream and upstream times would be 4 / 6 and 4 / 4, summing to more than 75 minutes. A distance of 5 km or 6 km would further increase times and contradict the given total time. Therefore, only 3 km produces the exact 75 minute round trip duration. Common Pitfalls: Many learners forget to convert minutes into hours, or they mistakenly add or subtract the stream speed incorrectly when computing upstream and downstream speeds. Others may think that total time should be divided equally between upstream and downstream legs, which is not true due to different speeds. Writing out the full equation x / 6 + x / 4 = 1.25 and solving carefully avoids these issues. Final Answer: The place is 3 km away from the starting point.

Question 9

A motorboat takes only half as much time to cover a certain distance downstream as it takes to cover the same distance upstream. What is the ratio between the rate of the current and the rate of the boat in still water?

Accepted Answer

Introduction / Context: This question is about motion in a stream where a motorboat travels the same distance both upstream and downstream. We are told that the time taken to travel downstream is exactly half of the time taken to travel upstream. From this information, we must determine the ratio of the speed of the current to the speed of the boat in still water. This is a standard boats and streams problem that relies on the relationship between speed, distance and time. Given Data / Assumptions: The boat travels the same distance downstream and upstream. Time taken downstream is half of the time taken upstream. Let the speed of the boat in still water be b km/h. Let the speed of the current be c km/h. Downstream speed is b + c and upstream speed is b - c. We must find the ratio c : b, that is the rate of current to the rate of boat in still water. Concept / Approach: For any journey, time = distance / speed. If the distance is the same in both directions, then the ratio of the times is the inverse of the ratio of the speeds. Here, the time taken downstream is half the upstream time, so the downstream speed must be twice the upstream speed. We use this idea with b + c and b - c to form an equation, solve for the relation between b and c, and then express the required ratio c : b in its simplest form. Step-by-Step Solution: Step 1: Let the one way distance be d km. Step 2: Time taken upstream = d / (b - c). Time taken downstream = d / (b + c). Step 3: Given that downstream time is half of upstream time. d / (b + c) = (1 / 2) * d / (b - c). Step 4: Cancel d (since d is non zero). 1 / (b + c) = (1 / 2) * 1 / (b - c). Step 5: Cross multiply to remove fractions. 2 = (b + c) / (b - c). Step 6: Multiply both sides by (b - c). 2(b - c) = b + c. 2b - 2c = b + c. Step 7: Collect like terms. 2b - b = 2c + c. b = 3c. Step 8: Rearrange to find the ratio of current speed to boat speed. c / b = 1 / 3, so c : b = 1 : 3. Verification / Alternative check: Assume c = 1 and b = 3 (which matches b = 3c). Then downstream speed = 3 + 1 = 4 km/h, upstream speed = 3 - 1 = 2 km/h. For a distance of 4 km, time downstream = 4 / 4 = 1 hour and time upstream = 4 / 2 = 2 hours. Downstream time is exactly half of upstream time, so the ratio works correctly. Why Other Options Are Wrong: Options like 3 : 2 or 2 : 3 do not satisfy the condition that downstream time is half of upstream time when used to form b and c values. If the ratio were 3 : 1 or 1 : 4, you would not get a downstream speed that is exactly double the upstream speed for any positive b, c that respect those ratios. Common Pitfalls: A common mistake is to set up the equation with times directly but forget that downstream time must be smaller when speed is higher, leading to inverted ratios. Another error is to confuse the ratio of times with the ratio of speeds and write 2 = (b - c) / (b + c) instead of 2 = (b + c) / (b - c). Final Answer: The ratio between the rate of current and the rate of the boat in still water is 1 : 3.

Question 10

A man can row a boat upstream at 16 km/h and downstream at 24 km/h in a river. What is the speed of the current (in km/h)?

Accepted Answer

Introduction / Context: This boats and streams question gives the rowing speeds of a man upstream and downstream and asks for the speed of the current. When a boat moves upstream, the current opposes the motion, and when it moves downstream, the current helps the motion. Using the two given effective speeds, we can extract the speed of the boat in still water and the speed of the current using simple algebraic relationships. Given Data / Assumptions: Upstream speed = 16 km/h. Downstream speed = 24 km/h. Let the speed of the boat in still water be b km/h. Let the speed of the current be c km/h. Upstream speed = b - c and downstream speed = b + c. We must find c, the speed of the current. Concept / Approach: In problems where both upstream and downstream speeds are known, we use the standard formulas: b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 These formulas come from solving the two linear equations b + c = downstream speed and b - c = upstream speed. Once we compute b and c, the answer is simply the value of c in km/h. Step-by-Step Solution: Step 1: Write the equations for upstream and downstream speeds. b - c = 16. b + c = 24. Step 2: Add the two equations to solve for b. (b - c) + (b + c) = 16 + 24. 2b = 40. b = 40 / 2 = 20 km/h. Step 3: Substitute b into either equation to find c. Using b + c = 24 gives 20 + c = 24. c = 24 - 20 = 4 km/h. Verification / Alternative check: We can check using the upstream equation. If b = 20 and c = 4, then b - c = 20 - 4 = 16 km/h, which matches the given upstream speed. Downstream speed with these values is b + c = 20 + 4 = 24 km/h, which also matches the given downstream speed. Why Other Options Are Wrong: If c were 6 km/h, then b would be 21 km/h to keep downstream at 27 km/h, which does not match the given data. If c were 5, 3 or 8 km/h, substituting into b - c and b + c would give upstream and downstream speeds that do not equal 16 km/h and 24 km/h respectively. Common Pitfalls: A common error is to average the two speeds incorrectly or forget to divide the difference by 2 when computing the current speed. Another pitfall is mixing up which expression gives the boat speed and which gives the current speed, leading to swapped values. Final Answer: The speed of the current is 4 kmph.

Question 11

A motorboat can go at 8 km/h in still water. It takes three times as long to go a fixed distance against the current as it takes to go the same distance with the current. What is the speed (in km/h) of the current?

Accepted Answer

Introduction / Context: This problem describes a motorboat whose speed in still water is known. The boat takes three times as long to go a certain distance against the current as it takes to go the same distance with the current. We are asked to find the speed of the current. This is a typical algebraic boats and streams question that uses the relationship between time, distance and effective speed in upstream and downstream directions. Given Data / Assumptions: Speed of the boat in still water b = 8 km/h. Let the speed of the current be c km/h. Downstream speed = b + c = 8 + c km/h. Upstream speed = b - c = 8 - c km/h. The time taken upstream is three times the time taken downstream for the same distance. We must find c. Concept / Approach: If a fixed distance d is covered in time t1 upstream and t2 downstream, then: t1 = d / (b - c) t2 = d / (b + c) We are told that the upstream time is three times the downstream time, so t1 = 3t2. After substituting the formulas for t1 and t2 and cancelling the common distance d, we get an equation involving only b and c. Since b is known, we can solve this equation to obtain c, the speed of the current. Step-by-Step Solution: Step 1: Let the one way distance be d km. Upstream time = d / (8 - c). Downstream time = d / (8 + c). Step 2: Use the condition that upstream time is three times downstream time. d / (8 - c) = 3 * d / (8 + c). Step 3: Cancel d from both sides. 1 / (8 - c) = 3 / (8 + c). Step 4: Cross multiply to remove fractions. 8 + c = 3(8 - c). 8 + c = 24 - 3c. Step 5: Collect like terms. c + 3c = 24 - 8. 4c = 16. c = 16 / 4 = 4 km/h. Verification / Alternative check: With c = 4 km/h, downstream speed = 8 + 4 = 12 km/h and upstream speed = 8 - 4 = 4 km/h. For a distance of 12 km, downstream time = 12 / 12 = 1 hour. Upstream time = 12 / 4 = 3 hours, which is exactly three times the downstream time. This confirms that c = 4 km/h is consistent with the conditions. Why Other Options Are Wrong: If c were 6 km/h, upstream speed would be 2 km/h and downstream speed 14 km/h, which would not give an upstream time exactly three times the downstream time for the same distance. Similarly, c values of 3, 2 or 1 km/h lead to different time ratios that do not equal 3. Common Pitfalls: One common mistake is to reverse the time ratio and write downstream time as three times the upstream time, which leads to the wrong equation. Another error is not cancelling the distance d properly and trying to keep it as an extra variable, which complicates the algebra unnecessarily. Final Answer: The speed of the current is 4 kmph.

Question 12

A boat travelling upstream covers a distance of 29 km at a speed of 6 km/h, and while travelling downstream it covers the same distance at a speed of 12 km/h. What is the speed (in km/h) of the boat in still water?

Accepted Answer

Introduction / Context: This question gives the effective speeds of a boat while travelling upstream and downstream over the same distance. The objective is to find the speed of the boat in still water. Because the distance is identical in both directions, we can ignore it when applying the standard relationships between upstream speed, downstream speed, boat speed and current speed. Given Data / Assumptions: Upstream distance = 29 km. Upstream speed = 6 km/h. Downstream distance = 29 km. Downstream speed = 12 km/h. Let boat speed in still water be b km/h. Let current speed be c km/h. Upstream speed = b - c and downstream speed = b + c. We need to find b. Concept / Approach: When we know the upstream and downstream speeds, the boat speed and current speed can be found using the formulas: b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 These come from solving the two linear equations b + c = downstream speed and b - c = upstream speed. Here, the distances are not needed to compute these speeds because the given speeds already incorporate the effect of the current. Step-by-Step Solution: Step 1: Write equations from the given speeds. b - c = 6. b + c = 12. Step 2: Add these equations to eliminate c. (b - c) + (b + c) = 6 + 12. 2b = 18. b = 18 / 2 = 9 km/h. Step 3: If required, we can also compute current speed. Substitute b = 9 into b + c = 12 to get 9 + c = 12, so c = 3 km/h. Verification / Alternative check: Check upstream speed with b = 9 and c = 3. Upstream speed = b - c = 9 - 3 = 6 km/h, which matches the given upstream speed. Downstream speed = b + c = 9 + 3 = 12 km/h, matching the given downstream speed. Thus the computed boat speed in still water is consistent. Why Other Options Are Wrong: If b were 6, 8, 10 or 11 km/h, there would be no non negative current speed c that makes both upstream and downstream speeds equal to 6 km/h and 12 km/h. Each such choice would lead to contradictions in at least one of the equations b - c = 6 or b + c = 12. Common Pitfalls: A frequent error is to try to calculate average speed from 6 km/h and 12 km/h as if they are simple averages, instead of using the correct formulas for b and c. Another mistake is to mix up b and c and mistakenly report the current speed instead of the boat speed in still water. Final Answer: The speed of the boat in still water is 9 kmph.

Question 13

A boat goes 7 km upstream in 42 minutes. If the speed of the stream is 3 km/h, what is the speed (in km/h) of the boat in still water?

Accepted Answer

Introduction / Context: Here we are given the time taken by a boat to travel a known distance upstream and the speed of the stream. From this information, we can compute the upstream speed and then determine the speed of the boat in still water. This is a straightforward application of the relationships between upstream speed, boat speed, and current speed. Given Data / Assumptions: Upstream distance = 7 km. Time taken upstream = 42 minutes. Speed of the stream c = 3 km/h. Let boat speed in still water be b km/h. Upstream speed = b - c. We must find b. Concept / Approach: First we convert the time from minutes to hours, because speeds are given in km/h. Using distance and time, we compute the upstream speed. Since upstream speed equals b - c, where c is known, we can rearrange to get b. This is a direct and common pattern in boats and streams questions. Step-by-Step Solution: Step 1: Convert time from minutes to hours. Time upstream = 42 minutes = 42 / 60 hours = 0.7 hours. Step 2: Compute the upstream speed using speed = distance / time. Upstream speed = 7 / 0.7 = 10 km/h. Step 3: Use the upstream relationship b - c = upstream speed. b - 3 = 10. Step 4: Solve for b. b = 10 + 3 = 13 km/h. Verification / Alternative check: With b = 13 km/h and c = 3 km/h, upstream speed becomes 13 - 3 = 10 km/h. Time to cover 7 km upstream = 7 / 10 hours = 0.7 hours = 42 minutes, which matches the original information. This confirms that b = 13 km/h is correct. Why Other Options Are Wrong: If b were 12 or 11 km/h, upstream speeds would be 9 or 8 km/h respectively, leading to times of 7 / 9 or 7 / 8 hours, which do not equal 42 minutes. Values such as 14 or 10 km/h would produce upstream speeds that also do not lead to the given time of 42 minutes. Common Pitfalls: A common mistake is to forget to convert minutes into hours, resulting in an incorrect upstream speed. Another is to misinterpret the 3 km/h as the upstream speed rather than the current speed, which leads to an incorrect equation. Final Answer: The speed of the boat in still water is 13 kmph.

Question 14

A boat goes 4 km upstream and 4 km downstream in a total time of 1 hour. The same boat goes 5 km downstream and 3 km upstream in 55 minutes. What is the speed (in km/h) of the boat in still water?

Accepted Answer

Introduction / Context: This is a more involved boats and streams problem. We are told about two different trips made by the same boat, each involving a mix of upstream and downstream travel over different distances and times. From this combined information, we must determine the speed of the boat in still water. This requires setting up a pair of equations in two unknowns and solving them systematically. Given Data / Assumptions: Let the speed of the boat in still water be b km/h. Let the speed of the current be c km/h. Upstream speed = b - c. Downstream speed = b + c. First trip: 4 km upstream + 4 km downstream in 1 hour. Second trip: 5 km downstream + 3 km upstream in 55 minutes (which is 11/12 hour). We must find b. Concept / Approach: For each trip, total time equals the sum of individual times. Time is given by distance divided by speed. We write one time equation for the first trip and another for the second trip, each involving b and c. To simplify, we introduce variables x = b - c and y = b + c so that the equations involve x and y directly. After solving for x and y, we recover b as (x + y) / 2. This method turns a boats and streams word problem into a clean algebra exercise. Step-by-Step Solution: Step 1: Define x and y for convenience. Let x = b - c (upstream speed) and y = b + c (downstream speed). Step 2: Form the equation for the first trip. Time upstream = 4 / x hours. Time downstream = 4 / y hours. Total time = 1 hour, so 4 / x + 4 / y = 1. Step 3: Form the equation for the second trip. Time downstream = 5 / y hours. Time upstream = 3 / x hours. Total time = 55 minutes = 11 / 12 hours. So 5 / y + 3 / x = 11 / 12. Step 4: Multiply the first equation by xy. 4y + 4x = xy. Rearrange as xy - 4x - 4y = 0. Step 5: Multiply the second equation by 12xy to clear denominators. 12(5 / y + 3 / x) = 11. This becomes 60x + 36y = 11xy. Rearrange to 11xy - 60x - 36y = 0. Step 6: Use the first equation to express xy. From xy - 4x - 4y = 0, we get xy = 4x + 4y. Substitute into 11xy - 60x - 36y = 0. 11(4x + 4y) - 60x - 36y = 0. 44x + 44y - 60x - 36y = 0. (-16x) + 8y = 0, so 8y = 16x and y = 2x. Step 7: Substitute y = 2x back into xy = 4x + 4y. x(2x) = 4x + 4(2x). 2x^2 = 4x + 8x = 12x. If x is not zero, divide by 2x: x = 6. Then y = 2x = 12. Step 8: Recover the boat speed in still water. b = (x + y) / 2 = (6 + 12) / 2 = 18 / 2 = 9 km/h. Verification / Alternative check: With b = 9 and c = (y - x) / 2 = (12 - 6) / 2 = 3 km/h, upstream speed is 6 km/h and downstream speed is 12 km/h. First trip: 4 / 6 + 4 / 12 = 2/3 + 1/3 = 1 hour, which matches. Second trip: 5 / 12 + 3 / 6 = 5/12 + 1/2 = 5/12 + 6/12 = 11/12 hour = 55 minutes, also correct. Why Other Options Are Wrong: If b were 6.5, 7.75, 10.5 or 8.5 km/h, it would be impossible to choose a single current speed c that satisfies both time equations simultaneously. Quick substitution of these values leads to inconsistencies between the calculated trip times and the given total times. Common Pitfalls: A common pitfall is to try to treat each trip independently as if they give direct upstream and downstream speeds, without setting up proper equations. Another is algebraic error when working with fractions, especially when clearing denominators; careful multiplication is essential to avoid mistakes. Final Answer: The speed of the boat in still water is 9 km/h.

Question 15

A boat travels 24 km upstream in 6 hours and 20 km downstream in 4 hours. What are the speeds (in km/h) of the boat in still water and of the current, respectively?

Accepted Answer

Introduction / Context: This question provides separate upstream and downstream journeys of a boat over different distances and times. From this information we can compute the effective speeds in each direction and then determine the speed of the boat in still water and the speed of the current. This is a direct application of the relationships between upstream speed, downstream speed, boat speed and current speed. Given Data / Assumptions: Upstream distance = 24 km, time = 6 hours. Downstream distance = 20 km, time = 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. We must find b and c. Concept / Approach: First we compute the effective upstream and downstream speeds by dividing distance by time. Once upstream and downstream speeds are known, we use: b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 These formulas are obtained by solving the linear system b - c = upstream speed and b + c = downstream speed. The calculations here are simple and rely only on accurate arithmetic. Step-by-Step Solution: Step 1: Compute upstream speed. Upstream speed = distance / time = 24 / 6 = 4 km/h. Step 2: Compute downstream speed. Downstream speed = 20 / 4 = 5 km/h. Step 3: Use the formulas for boat and current speeds. b = (downstream speed + upstream speed) / 2 = (5 + 4) / 2 = 9 / 2 = 4.5 km/h. c = (downstream speed - upstream speed) / 2 = (5 - 4) / 2 = 1 / 2 = 0.5 km/h. Verification / Alternative check: Check upstream with these values: b - c = 4.5 - 0.5 = 4 km/h, which matches the calculated upstream speed. Check downstream: b + c = 4.5 + 0.5 = 5 km/h, matching the calculated downstream speed. Therefore the pair 4.5 km/h and 0.5 km/h satisfies all conditions. Why Other Options Are Wrong: Pairs such as 4.5 km/h and 3 km/h or 5 km/h and 2 km/h imply upstream and downstream speeds that are far from 4 and 5 km/h respectively. Similarly, 4 km/h and 2 km/h or 6 km/h and 1 km/h do not yield upstream and downstream speeds equal to 4 km/h and 5 km/h when plugged into b - c and b + c. Common Pitfalls: A common error is miscomputing the initial upstream or downstream speeds from distance and time, for example mixing up the distances or times. Another pitfall is to average the two effective speeds but forget to halve the difference to find the current speed, which leads to reporting only b and not c. Final Answer: The speed of the boat in still water and the speed of the current are 4.5 km/h and 0.5 km/h respectively.

Question 16

Mahesh rows to a place 80 km away and back in a total of 20 hours. He observes that he can row 8 km downstream in the same time as 4 km upstream. What is the speed (in km/h) of the boat in still water?

Accepted Answer

Introduction / Context: In this question, Mahesh rows to a destination and back, with a given total time for the round trip. Additionally, he notes a relationship between the time taken to cover 8 km downstream and 4 km upstream. We are asked to find the speed of the boat in still water. This is a classic boats and streams problem that combines time comparisons and round trip travel in a current. Given Data / Assumptions: Distance from starting point to destination = 80 km. Total distance for round trip = 160 km. Total time for round trip = 20 hours. He can row 8 km downstream in the same time as 4 km upstream. 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: We use two key ideas. First, the equality of times for 8 km downstream and 4 km upstream gives a direct relationship between b and c. Second, the total time of 20 hours for rowing 80 km downstream and 80 km upstream provides another equation. By combining these, we can solve for b, the speed of the boat in still water. The algebra is straightforward once the equations are set up correctly. Step-by-Step Solution: Step 1: Use the equal time condition. Time for 8 km downstream = 8 / (b + c). Time for 4 km upstream = 4 / (b - c). Given that these times are equal: 8 / (b + c) = 4 / (b - c). Cross multiply: 8(b - c) = 4(b + c). 8b - 8c = 4b + 4c. Rearrange: 8b - 4b = 8c + 4c. 4b = 12c, so b = 3c. Step 2: Use the round trip time. Downstream distance = 80 km; time downstream = 80 / (b + c). Upstream distance = 80 km; time upstream = 80 / (b - c). Total time = 20 hours, so: 80 / (b + c) + 80 / (b - c) = 20. Step 3: Substitute b = 3c. Downstream speed = 3c + c = 4c. Upstream speed = 3c - c = 2c. So 80 / (4c) + 80 / (2c) = 20. Simplify: 20 / c + 40 / c = 60 / c. Thus 60 / c = 20, so c = 60 / 20 = 3 km/h. Step 4: Find the boat speed in still water. b = 3c = 3 * 3 = 9 km/h. Verification / Alternative check: With b = 9 and c = 3, downstream speed = 12 km/h and upstream speed = 6 km/h. Downstream time for 80 km = 80 / 12 hours and upstream time = 80 / 6 hours. 80 / 12 ≈ 6.67 hours, 80 / 6 ≈ 13.33 hours, sum ≈ 20 hours, matching the given total. Time for 8 km downstream = 8 / 12 ≈ 0.67 hours and time for 4 km upstream = 4 / 6 ≈ 0.67 hours, confirming the equality. Why Other Options Are Wrong: If the boat speed were 7, 5, 11 or 2 km/h, it would be impossible to choose a current speed that satisfies both the equal time condition and the total round trip time of 20 hours. Quick checks show that those speeds lead to serious mismatches in the times for the given distances. Common Pitfalls: A common pitfall is to misinterpret the statement about 8 km downstream and 4 km upstream, treating the distances as equal when they are not. Another frequent mistake is to plug in numerical approximations too early, which can make the algebra harder to follow. Final Answer: The speed of the boat in still water is 9 kmph.

Question 17

A boat covers equal distances upstream and downstream in a total of 5 hours. The sum of its upstream and downstream speeds is 40 km/h. The speed of the boat in still water is 600% more than the speed of the stream. What is the approximate distance (…

Accepted Answer

Introduction / Context: This question combines several pieces of information about a boat moving in a river: total time for equal upstream and downstream journeys, the sum of the two effective speeds, and a percentage relationship between the boat speed in still water and the stream speed. Our goal is to use these conditions to determine the approximate downstream distance covered by the boat. This involves working with ratios, percentages and time expressions together. Given Data / Assumptions: The boat covers equal distances upstream and downstream in a total of 5 hours. Let the one way distance be s km. Sum of upstream and downstream speeds = 40 km/h. Let b be the speed of the boat in still water (km/h). Let c be the speed of the stream (km/h). Upstream speed = b - c; downstream speed = b + c. b is 600% more than c, meaning b = c + 6c = 7c. We must find s, the downstream distance, approximately. Concept / Approach: First we translate the 600% condition into the relation b = 7c. Then we express upstream and downstream speeds in terms of c. Using the given sum of speeds, we solve for c and hence b. Next, we write the equation for total time as the sum of the upstream and downstream times for distance s, equated to 5 hours. Solving this time equation yields s in terms of c, which we then evaluate numerically. Finally, we compare the result to the given options, picking the closest integer value. Step-by-Step Solution: Step 1: Use the 600% condition. b is 600% more than c, so b = c + 6c = 7c. Step 2: Express upstream and downstream speeds in terms of c. Upstream speed u = b - c = 7c - c = 6c. Downstream speed d = b + c = 7c + c = 8c. Step 3: Use the sum of speeds. u + d = 6c + 8c = 14c. Given that u + d = 40 km/h, we have 14c = 40. c = 40 / 14 = 20 / 7 km/h. Then b = 7c = 7 * (20 / 7) = 20 km/h. Thus u = 6c = 6 * (20 / 7) = 120 / 7 km/h. And d = 8c = 8 * (20 / 7) = 160 / 7 km/h. Step 4: Write the total time equation with equal distances. Time upstream = s / u = s / (120 / 7) = 7s / 120. Time downstream = s / d = s / (160 / 7) = 7s / 160. Total time = 7s / 120 + 7s / 160 = 5 hours. Step 5: Solve for s. Take 7s common: 7s (1 / 120 + 1 / 160) = 5. Compute 1 / 120 + 1 / 160 = (160 + 120) / (120 * 160) = 280 / 19200. Simplify 280 / 19200 = 28 / 1920 = 7 / 480. So 7s * (7 / 480) = 5, meaning 49s / 480 = 5. Therefore s = 5 * 480 / 49 = 2400 / 49 km. Step 6: Approximate the value of s. 49 * 49 = 2401, so 2400 / 49 is just under 49. Thus s ≈ 48.98 km, which is approximately 49 km. Verification / Alternative check: With s ≈ 49 km, upstream time = 49 / (120 / 7) ≈ 49 * 7 / 120 ≈ 343 / 120 ≈ 2.86 hours. Downstream time = 49 / (160 / 7) ≈ 49 * 7 / 160 ≈ 343 / 160 ≈ 2.14 hours. Total ≈ 2.86 + 2.14 = 5 hours, matching the given total time. Why Other Options Are Wrong: Distances like 45, 55 or 60 km would produce total times significantly different from 5 hours when combined with the calculated upstream and downstream speeds. A distance of 50 km is closest to the computed value of about 49 km, whereas 40 km would be too small to account for 5 hours of travel at the given speeds. Common Pitfalls: One common misconception is to interpret 600% more as 6 times instead of 7 times the original speed, leading to b = 6c instead of b = 7c. Another frequent error is mishandling fractions when adding 1 / 120 and 1 / 160, which can lead to an incorrect total time expression and hence a wrong distance. Final Answer: The approximate distance covered downstream is 50 km.

Question 18

A boat takes 2 hours to travel from point A to point B in still water. To determine its speed upstream, which of the following pieces of information are sufficient? A: Distance between A and B. B: Time taken to travel downstream from B to A. C: Spee…

Accepted Answer

Introduction / Context: This is a conceptual data sufficiency style question about boats and streams. We know how long the boat takes to travel from A to B in still water, and we are given three possible additional pieces of information. The task is not to calculate a numerical value, but to decide which combinations of information are sufficient to determine the upstream speed of the boat. Understanding how each piece of data relates to the unknown speeds is the key. Given Data / Assumptions: The boat takes 2 hours to go from A to B in still water. Let the distance between A and B be d km. Let the speed of the boat in still water be b km/h. Let the speed of the stream be c km/h. Upstream speed = b - c; downstream speed = b + c. Information A: the distance d between A and B. Information B: the time taken to travel downstream from B to A. Information C: the speed c of the stream of water. Question: Which combinations of A, B, C are sufficient to find b - c? Concept / Approach: From the still water travel, we know the boat covers distance d in 2 hours at speed b, so d = 2b. To get the upstream speed b - c we need enough information to calculate b and c or to deduce b - c directly. We analyse each pair of data: A with B, B with C, and C with A. If any such pair allows us to form equations that lead to a single value for b - c, that pair is sufficient. If the same holds for all three pairs, then any one pair is enough. Step-by-Step Solution: Step 1: Use the still water information. From travelling from A to B in still water: distance d = b * 2, so d = 2b. This relationship holds in all cases. Step 2: Consider A and B together. With A, we know d exactly. With B, we know the downstream time t_down. Downstream speed = d / t_down = b + c. Using d = 2b, we get b + c = 2b / t_down. Thus c = 2b / t_down - b. Upstream speed b - c = b - (2b / t_down - b) = 2b - 2b / t_down = 2b(1 - 1 / t_down). Since d and t_down are known and d = 2b, we can solve for b and hence compute b - c. So A and B together are sufficient. Step 3: Consider B and C together. With B, we know t_down, the downstream time. With C, we know c, the stream speed. Downstream speed b + c = d / t_down. From still water travel, d = 2b. So 2b / t_down = b + c. Rearrange: 2b = t_down(b + c) = t_down b + t_down c. Thus b(2 - t_down) = t_down c, giving b = (t_down c) / (2 - t_down). Knowing b and c now allows us to find upstream speed b - c. So B and C together are sufficient. Step 4: Consider C and A together. With A, we know d, the distance. From still water travel, b = d / 2. With C, we know c directly. Upstream speed b - c = d / 2 - c, which is now completely determined. Thus A and C together are also sufficient. Verification / Alternative check: We see that with each pair (A,B), (B,C) or (C,A), we end up with enough equations to uniquely determine b - c, the upstream speed. No single item A, B or C on its own is enough, but any pair gives sufficient information. Why Other Options Are Wrong: Option a states that only A and B suffice, but we have shown B and C or C and A also suffice. Option b restricts sufficiency to B and C only, which is too narrow. Option c says all are required, which is incorrect because any one pair already gives enough information. Option e limits sufficiency to A and C only, again ignoring the fact that the other pairs work. Common Pitfalls: A common mistake is to think that distance must always be known numerically to find speed, ignoring that time and stream speed can also determine the boat speed when combined. Another pitfall is to focus on finding b and c separately, rather than realising that any method that yields b - c directly is also valid and sufficient. Final Answer: Any one pair among A and B, B and C, or C and A is sufficient to determine the upstream speed, so the correct choice is Any one pair of A and B, B and C or C and A is sufficient.

Question 19

A boat travels upstream from B to A and then downstream from A to B in a total of 3 hours. The speed of the boat in still water is 9 km/h and the speed of the current is 3 km/h. What is the distance (in km) between A and B?

Accepted Answer

Introduction / Context: This problem describes a boat travelling from B to A upstream and from A to B downstream in a total of 3 hours. The boat's speed in still water and the speed of the current are known. We must determine the distance between A and B. This is a direct application of time = distance / speed for two legs of a journey with different effective speeds. Given Data / Assumptions: Speed of the boat in still water b = 9 km/h. Speed of the current c = 3 km/h. Upstream speed = b - c = 9 - 3 = 6 km/h. Downstream speed = b + c = 9 + 3 = 12 km/h. Total time for upstream plus downstream journey = 3 hours. Let the distance between A and B be d km. We must find d. Concept / Approach: For each leg of the journey, we write time as distance / speed. The upstream leg takes d / 6 hours, and the downstream leg takes d / 12 hours. Since the total time is 3 hours, we set the sum of these times equal to 3 and solve for d. The arithmetic is simple if we combine the fractions correctly. Step-by-Step Solution: Step 1: Express upstream and downstream times. Upstream time = d / 6 hours. Downstream time = d / 12 hours. Step 2: Use the total time condition. d / 6 + d / 12 = 3. Step 3: Combine the fractions on the left. d / 6 + d / 12 = (2d + d) / 12 = 3d / 12 = d / 4. So d / 4 = 3. Step 4: Solve for d. d = 3 * 4 = 12 km. Verification / Alternative check: With d = 12 km, upstream time = 12 / 6 = 2 hours. Downstream time = 12 / 12 = 1 hour. Total time = 2 + 1 = 3 hours, which matches the given total. Why Other Options Are Wrong: If d were 9, 10 or 11 km, the sum of upstream and downstream times would be less than 3 hours. If d were 8 km, total time would be 8 / 6 + 8 / 12, which also does not equal 3 hours. Common Pitfalls: A common error is to average the upstream and downstream speeds to get 9 km/h and then divide the total distance by this average, which does not correctly represent the time for two different speeds over equal distances. Another pitfall is misadding the fractions d / 6 and d / 12 without finding a common denominator. Final Answer: The distance between A and B is 12 km.

Question 20

A boat goes a certain distance at 30 km/h and returns the same distance at 60 km/h. What is the average speed (in km/h) for the entire journey?

Accepted Answer

Introduction / Context: This question is about average speed for a round trip where the distances in both directions are equal but the speeds are different. The boat goes a certain distance at 30 km/h and comes back the same distance at 60 km/h. We must find the average speed for the entire journey. This is a classic example of when the average speed is not the simple arithmetic mean of the two speeds, but instead the harmonic mean due to equal distances. Given Data / Assumptions: Speed from starting point to destination = 30 km/h. Speed on the way back = 60 km/h. The distance one way is the same in both directions. Let the one way distance be d km. We must find the average speed over the whole trip. Concept / Approach: For two equal distances covered at speeds v1 and v2, the average speed is given by: Average speed = 2 * v1 * v2 / (v1 + v2). This formula comes from using total distance divided by total time. Here v1 = 30 and v2 = 60. Alternatively, we can compute the total distance as 2d and the total time as d / 30 + d / 60 and then divide. Both methods give the same result. Step-by-Step Solution: Step 1: Let the one way distance be d km. Total distance = 2d. Step 2: Compute the time for each leg. Time forward = d / 30 hours. Time return = d / 60 hours. Total time = d / 30 + d / 60. Step 3: Combine the times. d / 30 + d / 60 = (2d + d) / 60 = 3d / 60 = d / 20. Step 4: Use average speed = total distance / total time. Average speed = (2d) / (d / 20) = 2d * (20 / d) = 40 km/h. Verification / Alternative check: Use the harmonic mean formula directly: Average speed = 2 * 30 * 60 / (30 + 60) = 3600 / 90 = 40 km/h. Both methods yield the same average speed, confirming the result. Why Other Options Are Wrong: The arithmetic mean (30 + 60) / 2 = 45 km/h is not correct because the boat spends more time travelling at the slower speed. Speeds 50, 35 or 36 km/h do not match the total distance divided by the total time for the given legs. Common Pitfalls: A frequent error is to simply average the two speeds, forgetting that average speed is based on total distance and total time, not on the arithmetic mean of speeds. Another pitfall is mishandling the addition of the time fractions, leading to an incorrect total time and hence a wrong average speed. Final Answer: The average speed for the total journey is 40 km/h.

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