Question 1

The downstream swimming rate of a woman is three times her upstream swimming rate. If she covers 12 km upstream in 2.5 hours, what distance (in km) will she cover in 5 hours downstream?

Accepted Answer

Introduction / Context: This problem links upstream and downstream swimming speeds through a given ratio. You are told how far the woman swims upstream in a given time and how her downstream rate compares to this upstream rate. From this, you must compute her downstream speed and then the distance she can cover in a specified time. Given Data / Assumptions: - Downstream swimming rate is three times the upstream swimming rate. - Upstream distance covered = 12 km. - Time upstream = 2.5 hours. - Required: distance swum in 5 hours downstream. - Let u be the upstream speed (km/h). - Then downstream speed = 3u (km/h). Concept / Approach: First compute the upstream speed using speed = distance / time. Once upstream speed u is known, downstream speed is 3u by the given relationship. The distance she swims downstream in 5 hours is then simply speed * time. No need to separate still water and current speeds because the ratio is given directly between upstream and downstream speeds. Step-by-Step Solution: Step 1: Upstream speed u = distance / time = 12 / 2.5 km/h. Step 2: 12 / 2.5 = 12 / (5 / 2) = 12 * 2 / 5 = 24 / 5 = 4.8 km/h. Step 3: Downstream speed = 3u = 3 * 4.8 = 14.4 km/h. Step 4: Time spent swimming downstream = 5 hours. Step 5: Distance = speed * time = 14.4 * 5 km. Step 6: 14.4 * 5 = 72 km. Verification / Alternative check: Check that the upstream rate and downstream rate are consistent with the ratio. Upstream speed is 4.8 km/h. Downstream speed is 14.4 km/h. Ratio downstream : upstream = 14.4 : 4.8 = 3 : 1, exactly as stated. Therefore, the calculations are internally consistent and the 72 km distance in 5 hours downstream is correct. Why Other Options Are Wrong: - 36 km would correspond to a downstream speed of only 7.2 km/h, which is not three times 4.8. - 42 km, 56 km, and 60 km likewise imply downstream speeds that are not equal to 3 times the upstream speed of 4.8 km/h. Common Pitfalls: Some learners misinterpret three times as adding instead of multiplying, using u + 3 for the downstream speed instead of 3u. Others make mistakes in converting 2.5 hours or in dividing 12 by 2.5. Keeping a clear step for computing the initial upstream speed and using precise arithmetic avoids these errors. Final Answer: In 5 hours downstream, the woman will cover 72 km.

Question 2

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

Accepted Answer

Introduction / Context: This round trip boats and streams problem asks you to determine the one way distance to a point on a river, given the boat speed in still water, the speed of the current, and the total time for the trip. The key here is to correctly compute upstream and downstream speeds and then use the total time for going and returning. Given Data / Assumptions: - Speed of boat in still water = 5 km/h. - Speed of river current = 1 km/h. - Total time for going to the place and coming back = 75 minutes. - 75 minutes = 75 / 60 = 1.25 hours. - Let one way distance be d km. - Downstream speed = 5 + 1 = 6 km/h. - Upstream speed = 5 - 1 = 4 km/h. Concept / Approach: Time downstream equals distance divided by downstream speed, and time upstream equals distance divided by upstream speed. The total round trip time is the sum of these two times. Setting this equal to 1.25 hours gives an equation in d that can be solved to find the distance to the place. This method is standard for such symmetric round trip problems. Step-by-Step Solution: Step 1: Time taken downstream = d / 6 hours. Step 2: Time taken upstream = d / 4 hours. Step 3: Total time = d / 6 + d / 4. Step 4: Given total time = 1.25 hours. Step 5: So d / 6 + d / 4 = 1.25. Step 6: Use common denominator 12: (2d / 12) + (3d / 12) = 1.25. Step 7: This simplifies to 5d / 12 = 1.25. Step 8: Multiply both sides by 12: 5d = 1.25 * 12. Step 9: 1.25 * 12 = 15, so 5d = 15. Step 10: Therefore d = 15 / 5 = 3 km. Verification / Alternative check: If the place is 3 km away, time downstream = 3 / 6 = 0.5 hour (30 minutes). Time upstream = 3 / 4 = 0.75 hour (45 minutes). Total time = 0.5 + 0.75 = 1.25 hours = 75 minutes, which matches the problem statement. Thus, the distance is confirmed as 3 km. Why Other Options Are Wrong: - 4 km or 5 km would give total times greater than 1.25 hours. - 2 km would give a total time less than 1.25 hours. - 6 km is far too large and would imply much longer travel times than given. Common Pitfalls: Learners may forget to convert minutes to hours, or incorrectly average the speeds instead of calculating separate upstream and downstream times. Another error is to assume 75 minutes is one way time instead of total round trip time. Carefully translating the language into a time equation is essential here. Final Answer: The place is 3 km away from the starting point.

Question 3

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

Accepted Answer

Introduction / Context: This is a direct application of the basic formulas for boats and streams. You are given upstream and downstream speeds, and you must find the speed of the current using simple averages and differences. It checks that you know the essential relationships between these quantities. Given Data / Assumptions: - Upstream speed = 16 km/h. - Downstream speed = 24 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). - Upstream speed = b - c, downstream speed = b + c. Concept / Approach: Because upstream speed is boat speed minus current speed and downstream speed is boat speed plus current speed, we can obtain b and c by solving the two equations. The current speed is half of the difference between downstream and upstream speeds, while the still water speed is their simple average. Step-by-Step Solution: Step 1: b + c = 24 (downstream). Step 2: b - c = 16 (upstream). Step 3: Subtract the upstream equation from the downstream equation: (b + c) - (b - c) = 24 - 16. Step 4: This gives 2c = 8. Step 5: Therefore c = 8 / 2 = 4 km/h. Verification / Alternative check: You can also use the shortcut formula for current speed: c = (downstream speed - upstream speed) / 2. Here that is c = (24 - 16) / 2 = 8 / 2 = 4 km/h. Using b = (24 + 16) / 2 = 20 km/h gives upstream speed = 20 - 4 = 16 km/h and downstream speed = 20 + 4 = 24 km/h, exactly as given. Why Other Options Are Wrong: - 6 km/h or 5 km/h would not give the correct upstream and downstream speeds when combined with the implied boat speed. - 3 km/h and 8 km/h are not equal to half the difference between 24 and 16. Common Pitfalls: Learners sometimes average the two speeds incorrectly or think that the current equals the smaller speed. Others forget that the current affects both directions and that the difference between downstream and upstream speeds is twice the current. Remember the simple formulas: b = (downstream + upstream) / 2 and c = (downstream - upstream) / 2. Final Answer: The speed of the current is 4 km/h.

Question 4

A motorboat takes exactly twice as long to cover a certain distance upstream as it takes to cover the same distance downstream. What is the ratio of the speed of the current to the speed of the boat in still water?

Accepted Answer

Introduction / Context: Here the key information is about the relative times to travel the same distance upstream and downstream. The motorboat takes twice the time going upstream, which tells us how its speed in still water compares to the speed of the current. The question asks specifically for the ratio of the current speed to the boat speed in still water. Given Data / Assumptions: - Time taken upstream is twice the time taken downstream for the same distance. - 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: Time equals distance divided by speed. For the same distance D, upstream time is D / (b - c) and downstream time is D / (b + c). Given that upstream time is twice downstream time, we have D / (b - c) = 2D / (b + c). Cancelling D gives a relation between b and c. From this relation we can form the ratio c : b that is required. Step-by-Step Solution: Step 1: Upstream time T up = D / (b - c). Step 2: Downstream time T down = D / (b + c). Step 3: Given T up = 2 * T down. Step 4: So D / (b - c) = 2D / (b + c). Step 5: Cancel D 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, which gives -b + 3c = 0. Step 9: So 3c = b, or c = b / 3. Step 10: Therefore the ratio c : b = 1 : 3. Verification / Alternative check: Assume a convenient value such as b = 3 units and c = 1 unit. Then downstream speed = 3 + 1 = 4 units, upstream speed = 3 - 1 = 2 units. For distance D, downstream time is D / 4 and upstream time is D / 2. The ratio of times is (D / 2) : (D / 4) = 2 : 1, so upstream time is indeed twice downstream time, confirming that c : b = 1 : 3 is correct. Why Other Options Are Wrong: - Ratios 3 : 2, 2 : 3, 3 : 1, and 1 : 2 all lead to different relationships between upstream and downstream speeds and do not produce a strict two to one ratio between upstream and downstream times. Common Pitfalls: Some learners accidentally reverse the ratio and give b : c instead of c : b, or they misread the condition as downstream taking twice as long. Others try to use numerical values for b and c without setting up the equation properly. Always translate the time condition precisely into a fraction equation before solving for the ratio. Final Answer: The ratio of the speed of the current to the speed of the boat in still water is 1 : 3.

Question 5

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

Accepted Answer

Introduction / Context: This problem uses the relationship between upstream and downstream times when the speed of a boat in still water is known. You are told that travelling against the current takes three times as long as travelling with the current over the same distance, and you must infer the stream speed from this ratio. Given Data / Assumptions: - Speed of the motorboat in still water b = 8 km/h. - Time taken upstream is three times the time taken downstream for the same distance. - Let c be the speed of the current (km/h). - Upstream speed = b - c, downstream speed = b + c. - Let the distance each way be L km. Concept / Approach: Time equals distance divided by speed. For distance L, upstream time is L / (b - c), downstream time is L / (b + c). The statement that upstream time is three times downstream time gives L / (b - c) = 3L / (b + c). Cancelling L and solving for c in terms of b leads to a simple linear equation. Substituting b = 8 gives the numerical value of c, which is the current speed. Step-by-Step Solution: Step 1: Upstream time T up = L / (8 - c). Step 2: Downstream time T down = L / (8 + c). Step 3: Given T up = 3 * T down. Step 4: So L / (8 - c) = 3L / (8 + c). Step 5: Cancel L: 1 / (8 - c) = 3 / (8 + c). Step 6: Cross multiply: 8 + c = 3(8 - c). Step 7: Expand right side: 8 + c = 24 - 3c. Step 8: Bring terms together: c + 3c = 24 - 8. Step 9: 4c = 16, so c = 16 / 4 = 4 km/h. Verification / Alternative check: Take b = 8 km/h and c = 4 km/h. Then upstream speed = 8 - 4 = 4 km/h, downstream speed = 8 + 4 = 12 km/h. For some distance such as 12 km, downstream time = 12 / 12 = 1 hour. Upstream time for the same distance = 12 / 4 = 3 hours, which is exactly three times the downstream time, confirming the correctness of c = 4 km/h. Why Other Options Are Wrong: - 6 km/h or 5 km/h would make upstream speed too low, leading to a time ratio more than three. - 3 km/h or 2 km/h would not produce a time ratio of exactly three when used with 8 km/h as the still water speed. Common Pitfalls: Learners often invert the ratio, using downstream time equal to three times upstream time, or they average speeds instead of setting up a time equation. Another mistake is to forget the minus sign for upstream speed. Rewriting the relationship clearly as T up = 3 T down before substituting formulas for time helps avoid these issues. Final Answer: The speed of the current is 4 km/h.

Question 6

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 12 km/h. What is the speed of the boat in still water (in km/h)?

Accepted Answer

Introduction / Context: You are given the upstream and downstream speeds for the same distance and asked for the boat speed in still water. This is one of the most fundamental types of boats and streams questions, reinforcing the formula that the still water speed is the average of upstream and downstream speeds. Given Data / Assumptions: - Upstream speed = 6 km/h. - Downstream speed = 12 km/h. - Distance covered each way = 29 km (though it is not directly needed for the still water speed). - Let b be the speed in still water and c be the current speed. - Upstream speed = b - c, downstream speed = b + c. Concept / Approach: Since upstream and downstream speeds are known, we can add them and divide by 2 to get the speed in still water. This is because (b - c + b + c) / 2 = (2b) / 2 = b. It is not necessary to use the distance or times in the calculation of b for this kind of question. Step-by-Step Solution: Step 1: Let upstream speed u = 6 km/h and downstream speed d = 12 km/h. Step 2: Speed in still water b = (u + d) / 2. Step 3: b = (6 + 12) / 2 = 18 / 2 = 9 km/h. Verification / Alternative check: Using b = 9 km/h, we can find the current speed c as (d - u) / 2 = (12 - 6) / 2 = 3 km/h. Then upstream speed should be b - c = 9 - 3 = 6 km/h and downstream speed should be 9 + 3 = 12 km/h. These match the given speeds, confirming that 9 km/h is correct for the still water speed. Why Other Options Are Wrong: - 6 km/h would correspond to a zero current, contradicting the different upstream and downstream speeds. - 8 km/h, 10 km/h, and 11 km/h are not equal to the average of 6 and 12 and would not reproduce the same upstream and downstream speeds when combined with a current. Common Pitfalls: Some learners incorrectly average distances or times, or think that the still water speed equals the higher of the two speeds. Others forget that current affects both directions equally and that the still water speed must lie exactly midway between upstream and downstream speeds. Remember the simple rule: boat in still water = (downstream + upstream) / 2. Final Answer: The speed of the boat in still water is 9 km/h.

Question 7

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

Accepted Answer

Introduction / Context: In this boats and streams question you must derive the boat speed in still water from an upstream journey and a known stream speed. This requires converting time properly, computing the effective upstream speed, and then adding the current speed to recover the still water speed. Given Data / Assumptions: - Upstream distance = 7 km. - Time taken upstream = 42 minutes. - Speed of the stream = 3 km/h. - Let b be the speed of the boat in still water (km/h). - Upstream speed = b - 3 km/h. Concept / Approach: First, convert the time from minutes to hours to keep units consistent. Then use speed = distance / time to find the upstream speed. Since upstream speed equals b minus the stream speed, we can solve for b by adding the current speed to the upstream speed. This is a direct application of the definition of upstream speed. Step-by-Step Solution: Step 1: Convert 42 minutes to hours: 42 / 60 hours = 0.7 hour. Step 2: Upstream speed = distance / time = 7 / 0.7 km/h. Step 3: 7 / 0.7 = 10 km/h (because 0.7 * 10 = 7). Step 4: Let b be the speed of the boat in still water. Step 5: Upstream speed = b - stream speed, so b - 3 = 10. Step 6: Solve for b: b = 10 + 3 = 13 km/h. Verification / Alternative check: If the boat speed in still water is 13 km/h and the stream is 3 km/h, the upstream speed is 13 - 3 = 10 km/h. For a 7 km journey, time taken upstream = 7 / 10 hours = 0.7 hour = 42 minutes, exactly matching the data. Therefore, 13 km/h is consistent with the given journey details. Why Other Options Are Wrong: - 14 km/h would give upstream speed 11 km/h and an upstream time of 7 / 11 hours, not equal to 42 minutes. - 12 km/h would give upstream speed 9 km/h and a time that is longer than 42 minutes. - 11 km/h and 10 km/h correspond to even lower still water speeds and do not satisfy the 7 km in 42 minutes condition when combined with 3 km/h current. Common Pitfalls: Learners sometimes forget to convert minutes into hours or miscalculate 7 / 0.7. Others mistakenly subtract the upstream speed from the current instead of adding the current to the upstream speed to get b. Carefully tracking units and using the formula upstream speed = b - c helps avoid such mistakes. Final Answer: The speed of the boat in still water is 13 km/h.

Question 8

A boat travels 4 km upstream and 4 km downstream in a total time of 1 hour. The same boat travels 5 km downstream and 3 km upstream in a total time of 55 minutes. Based on this information, what is the speed of the boat in still water (in km/h)?

Accepted Answer

Introduction / Context: This boats and streams problem asks you to determine the speed of a boat in still water when it makes two different trips involving both upstream and downstream motion. Because upstream and downstream speeds are affected by the current, we use the concept of relative speed to set up equations and solve for the unknown speed of the boat in still water. Given Data / Assumptions: The boat goes 4 km upstream and 4 km downstream in a total time of 1 hour. It then goes 5 km downstream and 3 km upstream in a total time of 55 minutes (which is 11/12 hour). Let the speed of the boat in still water be b km/h. Let the speed of the stream be s km/h. Upstream speed = (b - s) km/h, downstream speed = (b + s) km/h. Concept / Approach: We use the idea that time = distance / speed. For each mixed trip, we express the total time as the sum of times for upstream and downstream segments. This gives us two equations in terms of b and s. Solving these simultaneous equations gives the values of b and s. We are finally interested in b, the speed of the boat in still water. Step-by-Step Solution: Let b = speed of boat in still water, s = speed of stream. Upstream speed = b - s, downstream speed = b + s. From the first trip: 4 / (b - s) + 4 / (b + s) = 1. From the second trip: 5 / (b + s) + 3 / (b - s) = 11 / 12. Multiply the first equation by (b - s)(b + s): 4(b + s) + 4(b - s) = (b^2 - s^2). So 4b + 4s + 4b - 4s = b^2 - s^2 ⇒ 8b = b^2 - s^2. (1) For the second equation, multiply by (b - s)(b + s): 5(b - s) + 3(b + s) = (11 / 12)(b^2 - s^2). This gives 5b - 5s + 3b + 3s = (11 / 12)(b^2 - s^2). So 8b - 2s = (11 / 12)(b^2 - s^2). (2) From (1), b^2 - s^2 = 8b. Substitute into (2): 8b - 2s = (11 / 12) * 8b = (88 / 12)b = (22 / 3)b. Thus, 8b - 2s = (22 / 3)b ⇒ multiply by 3: 24b - 6s = 22b ⇒ 2b = 6s ⇒ b = 3s. Substitute b = 3s into (1): (3s)^2 - s^2 = 8b ⇒ 9s^2 - s^2 = 8 * 3s ⇒ 8s^2 = 24s. Since s ≠ 0, divide both sides by 8s: s = 3 km/h and b = 3 * 3 = 9 km/h. Verification / Alternative check: With b = 9 and s = 3, upstream speed is 6 km/h, downstream speed is 12 km/h. First trip time = 4/6 + 4/12 = 2/3 + 1/3 = 1 hour, correct. Second trip time = 5/12 + 3/6 = 5/12 + 1/2 = 5/12 + 6/12 = 11/12 hour, which matches 55 minutes. So the solution is consistent. Why Other Options Are Wrong: Speeds such as 6.5, 7.75, 8.25 or 10.5 km/h do not satisfy both time equations simultaneously. Substituting any of these values into the expressions for upstream and downstream times leads to totals different from 1 hour and 55 minutes. Only 9 km/h satisfies both conditions exactly. Common Pitfalls: Students often try to average distances or speeds without using proper equations, or they forget to convert 55 minutes into hours. Another frequent mistake is mixing upstream and downstream speeds. Always define variables clearly, write the time equations carefully and solve them systematically. Final Answer: The speed of the boat in still water is 9 km/h.

Question 9

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

Accepted Answer

Introduction / Context: This is a standard boats and streams question where you are given upstream and downstream journeys over known distances and times. From these, you can find the upstream and downstream speeds and then deduce the speeds of the boat in still water and of the current. Given Data / Assumptions: Upstream distance = 24 km, time taken = 6 hours. Downstream distance = 20 km, time taken = 4 hours. Upstream speed = speed of boat in still water minus speed of current. Downstream speed = speed of boat in still water plus speed of current. All speeds are in km/h. Concept / Approach: Upstream speed is found by dividing upstream distance by upstream time. Similarly, downstream speed is found from downstream data. Once these two effective speeds are known, we use the formulas: boat speed in still water = (upstream speed + downstream speed) / 2 current speed = (downstream speed - upstream speed) / 2 These come from solving the two basic equations for upstream and downstream speeds. Step-by-Step Solution: Step 1: Upstream speed = 24 km / 6 h = 4 km/h. Step 2: Downstream speed = 20 km / 4 h = 5 km/h. Step 3: Let b be the speed of the boat in still water and s be the speed of the current. Step 4: Then b - s = 4 and b + s = 5 from the upstream and downstream speeds. Step 5: Add the two equations: (b - s) + (b + s) = 4 + 5 ⇒ 2b = 9 ⇒ b = 4.5 km/h. Step 6: Subtract the first equation from the second: (b + s) - (b - s) = 5 - 4 ⇒ 2s = 1 ⇒ s = 0.5 km/h. Verification / Alternative check: Check the effective speeds. Upstream effective speed = b - s = 4.5 - 0.5 = 4 km/h, which matches 24 km in 6 hours. Downstream effective speed = b + s = 4.5 + 0.5 = 5 km/h, matching 20 km in 4 hours. This confirms that the calculated speeds are correct. Why Other Options Are Wrong: Pairs like 4.5 km/h and 3 km/h or 5 km/h and 2 km/h would give very different upstream and downstream speeds that do not fit the given times and distances. For example, with 5 km/h and 2 km/h, the upstream speed would be 3 km/h, not the required 4 km/h. Only the pair 4.5 km/h and 0.5 km/h satisfies both journeys. Common Pitfalls: Learners sometimes forget to compute separate upstream and downstream speeds before applying the formulas, or they accidentally swap them. Another common mistake is mixing up the addition and subtraction when finding the still water and current speeds. Always write the equations clearly and solve them step by step. 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 10

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

Accepted Answer

Introduction / Context: This boats and streams question combines information about total journey time with a key relationship between upstream and downstream travel times over different distances. Using this relationship, we find the ratio between upstream and downstream speeds, then combine it with the total time for a round trip to determine the speed of the boat in still water. Given Data / Assumptions: Distance from starting point to destination = 80 km. Total distance for the round trip = 80 km upstream + 80 km downstream = 160 km. Total time for the round trip = 20 hours. 8 km downstream takes the same time as 4 km upstream. Let b be the speed of the boat in still water (km/h). Let s be the speed of the stream (km/h). Upstream speed = b - s, downstream speed = b + s. Concept / Approach: First, use the time equality 8 km downstream = 4 km upstream to relate upstream and downstream speeds. This gives a ratio that leads to a simple connection between b and s. Next, express the total round trip time in terms of upstream and downstream speeds over 80 km each. Solving the resulting equation yields the value of the stream speed and then the boat speed in still water. Step-by-Step Solution: Step 1: From the time equality, 8 / (b + s) = 4 / (b - s). Step 2: Cross multiply: 8(b - s) = 4(b + s). Step 3: Expand: 8b - 8s = 4b + 4s. Step 4: Rearrange: 8b - 4b = 8s + 4s ⇒ 4b = 12s ⇒ b = 3s. Step 5: Upstream speed = b - s = 3s - s = 2s; downstream speed = b + s = 3s + s = 4s. Step 6: Time to go 80 km upstream = 80 / (2s) = 40 / s. Step 7: Time to come 80 km downstream = 80 / (4s) = 20 / s. Step 8: Total time for the round trip = 40 / s + 20 / s = 60 / s hours. Step 9: Given that total time = 20 hours, so 60 / s = 20 ⇒ s = 60 / 20 = 3 km/h. Step 10: Therefore b = 3s = 3 * 3 = 9 km/h. Verification / Alternative check: With b = 9 km/h and s = 3 km/h, upstream speed = 6 km/h, downstream speed = 12 km/h. Time for 80 km upstream = 80 / 6 ≈ 13.33 hours. Time for 80 km downstream = 80 / 12 ≈ 6.67 hours. Total time ≈ 20 hours, which matches the given total. Also, 8 km downstream takes 8 / 12 hours, and 4 km upstream takes 4 / 6 hours; both are 2/3 hour, confirming the condition. Why Other Options Are Wrong: If the boat speed were 7, 5, 2 or 11 km/h, the resulting upstream and downstream speeds could not satisfy both the total time of 20 hours and the specific timing relationship between 8 km downstream and 4 km upstream. Only 9 km/h accurately fits both conditions. Common Pitfalls: A common error is to treat 8 and 4 as directly giving a speed ratio without considering that times are equal, not distances. Another mistake is to assume a simple average speed for the round trip, which does not hold when speeds differ. Always derive equations from time relationships and solve them carefully. Final Answer: Mahesh's boat speed in still water is 9 km/h.

Question 11

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

Accepted Answer

Introduction / Context: This is a multi-step boats and streams problem that combines ideas of upstream and downstream speeds, percentage comparison between boat speed and stream speed, and total travel time for equal distances. You are also asked to choose an approximate distance, so rounding to a nearby option is required after calculation. Given Data / Assumptions: The boat travels an equal distance d km upstream and d km downstream. Total time for both journeys = 5 hours. Sum of upstream and downstream speeds = 40 km/h. The speed of the boat in still water is 600% more than the speed of the stream. Let b be the speed of the boat in still water (km/h). Let s be the speed of the stream (km/h). Upstream speed = b - s, downstream speed = b + s. Concept / Approach: First, use the given sum of upstream and downstream speeds to find the boat speed in still water. Then interpret “600% more” correctly to relate b and s. This allows us to find both b and s as numerical values. Finally, write the total time for equal distances upstream and downstream, use the known speeds and total time of 5 hours to solve for the distance d, and then choose the closest option. Step-by-Step Solution: Step 1: Upstream speed + downstream speed = (b - s) + (b + s) = 2b = 40 ⇒ b = 20 km/h. Step 2: “Speed of boat in still water is 600% more than speed of stream” means b = s + 600% of s = s + 6s = 7s. Step 3: From b = 7s and b = 20, we get 7s = 20 ⇒ s = 20 / 7 ≈ 2.857 km/h. Step 4: Upstream speed = b - s ≈ 20 - 2.857 ≈ 17.143 km/h. Step 5: Downstream speed = b + s ≈ 20 + 2.857 ≈ 22.857 km/h. Step 6: Time upstream = d / 17.143 hours; time downstream = d / 22.857 hours. Step 7: Total time = d / 17.143 + d / 22.857 = 5 hours. Step 8: Compute the sum of reciprocals: 1 / 17.143 + 1 / 22.857 ≈ 0.102083 per hour. Step 9: So d * 0.102083 ≈ 5 ⇒ d ≈ 5 / 0.102083 ≈ 48.96 km. Verification / Alternative check: If the downstream distance is about 49 km, the upstream time is roughly 49 / 17.143 ≈ 2.86 hours and the downstream time is about 49 / 22.857 ≈ 2.14 hours. The total is approximately 5 hours, agreeing with the question. Since the answer choices are rounded, 48.96 km is closest to 50 km. Why Other Options Are Wrong: Options 45 and 40 km would lead to total times noticeably less than 5 hours, while 55 and 60 km would give total times significantly greater than 5 hours. Only about 49–50 km keeps the total time near 5 hours with the calculated speeds, making 50 km the best approximate choice. Common Pitfalls: A frequent mistake is to misinterpret “600% more” as b = 6s instead of b = 7s. Another issue is forgetting that 2b equals the sum of upstream and downstream speeds. Finally, some learners try to avoid working with decimals and guess the distance without computing, which often leads to the wrong option. Working through the equations carefully and then rounding at the end is the safest method. Final Answer: The approximate distance covered by the boat in the downstream direction is 50 km.

Question 12

A boat takes 2 hours to travel from point A to point B in still water. You want to find the speed of the boat when it is travelling upstream. Which of the following pieces of information are required? A. Distance between points A and B. B. Time take…

Accepted Answer

Introduction / Context: This question is about data sufficiency in a boats and streams setting. You are told how long a boat takes to travel between two points in still water, and asked which additional pieces of information are needed to determine the upstream speed. Rather than calculating a numeric value, you must analyse which combinations of given data would make that computation possible. Given Data / Assumptions: Time taken from A to B in still water = 2 hours. Let distance between A and B be D km. Let speed of boat in still water be b km/h and speed of stream be s km/h. Upstream speed = b - s, downstream speed = b + s. Statements: A: Distance between A and B (value of D). B: Time taken to travel downstream from B to A. C: Speed of the stream s. Concept / Approach: From the still water information, we know that D = b * 2, so D = 2b. This relationship links distance and still water speed. We need to decide which combinations of A, B and C allow us to find b and s and hence compute b - s, the upstream speed. The key is to see that any two of A, B and C, together with the still water time, are sufficient to determine b and s uniquely. Step-by-Step Solution: Case 1 (A and B): If we know distance D and downstream time t_down from B to A, we can find downstream speed = D / t_down. From the still water time, D = 2b, so b = D / 2. Then downstream speed = b + s = D / t_down. This gives s = D / t_down - b, so we can find b - s, the upstream speed. Thus A and B are sufficient. Case 2 (B and C): Knowing downstream time t_down and stream speed s, we know that downstream speed = D / t_down = b + s. From the still water relation, D = 2b. So D / t_down = 2b / t_down = b + s. This is an equation in b because s and t_down are known, allowing us to solve for b and then compute b - s. Thus B and C are also sufficient. Case 3 (C and A): Knowing D and s, we have b = D / 2 from still water and can immediately compute upstream speed = b - s. Verification / Alternative check: In every case above, once b and s are known, upstream speed b - s is fully determined. There is no scenario where all three A, B and C are simultaneously necessary; any pair provides enough information. Therefore, the correct logical conclusion is that any one pair among (A, B), (B, C) or (C, A) is sufficient. Why Other Options Are Wrong: Option stating “Only A and B” ignores that B and C together or C and A together also work. Option “Only B and C” similarly neglects other valid pairs. Saying “All are required” is too strong, because we have already shown each pair is sufficient. “Only A and C” incorrectly rules out B and C as a sufficient combination. Common Pitfalls: Many learners mistakenly think distance is always essential, or they assume that without all three pieces of data the problem is unsolvable. The key insight is to express D in terms of b using the still water time and then see how each extra statement supplies enough information to determine the unknowns. Practising this type of logical reasoning is crucial for data sufficiency questions. Final Answer: Any one pair of A and B, B and C, or C and A is sufficient to determine the upstream speed.

Question 13

A boat travels upstream from point B to point A and then downstream from point A back to point B in a total time 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 …

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Introduction / Context: This boats and streams problem gives you the speed of a boat in still water, the speed of the current, and the total time for an upstream and downstream journey between two points. You are asked to find the distance between the points. Because the distance each way is the same, the time relation can be used directly with upstream and downstream speeds. Given Data / Assumptions: Speed of boat in still water = 9 km/h. Speed of current = 3 km/h. Let the distance between A and B be D km. Total time for upstream (B to A) and downstream (A to B) journeys = 3 hours. Upstream speed = 9 - 3 = 6 km/h; downstream speed = 9 + 3 = 12 km/h. Concept / Approach: When a boat goes a certain distance upstream and the same distance downstream, the total time is the sum of upstream time and downstream time. We use the formula time = distance / speed for each leg, express the total time in terms of D, and then solve for D. Since speeds are already known, the algebra is straightforward. Step-by-Step Solution: Step 1: Upstream speed = 9 - 3 = 6 km/h. Step 2: Downstream speed = 9 + 3 = 12 km/h. Step 3: Time taken to go upstream from B to A = D / 6 hours. Step 4: Time taken to go downstream from A to B = D / 12 hours. Step 5: Total time for both journeys = D / 6 + D / 12 = 3 hours. Step 6: Combine terms: D / 6 + D / 12 = (2D / 12) + (D / 12) = 3D / 12 = D / 4. Step 7: So D / 4 = 3 ⇒ D = 3 * 4 = 12 km. Verification / Alternative check: Check the times using D = 12 km. Upstream time = 12 / 6 = 2 hours. Downstream time = 12 / 12 = 1 hour. Total time = 2 + 1 = 3 hours, which exactly matches the given total time. Therefore, the distance of 12 km is consistent with all data. Why Other Options Are Wrong: If the distance were 9, 10 or 11 km, the combined upstream and downstream times would not total 3 hours when calculated with speeds 6 km/h and 12 km/h. For example, with 9 km, upstream time is 1.5 hours and downstream time is 0.75 hours, totaling 2.25 hours, not 3. Only 12 km gives a total of exactly 3 hours. Common Pitfalls: Many learners forget that the distance upstream and downstream is the same and mistakenly use different distances. Others neglect to simplify the fractions correctly when summing times. Remember that upstream speed is lower, so most of the total time is spent going against the current, and the final answer must reflect that balance. Final Answer: The distance between A and B is 12 km.

Question 14

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

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Introduction / Context: This aptitude question focuses on the concept of average speed over a round trip when the distances are the same but the speeds differ. It is important to remember that the average speed is not the simple average of the two speeds. Instead, for equal distances, the correct formula is based on the harmonic mean. Given Data / Assumptions: The boat travels from point 1 to point 2 at 30 km/h. It returns from point 2 to point 1 at 60 km/h. The distance one way is the same in both directions (say d km). We are asked to find the average speed for the entire journey. Concept / Approach: Average speed is defined as total distance travelled divided by total time taken. For a round trip with equal distances d at two different speeds v1 and v2, the formula simplifies to: average speed = (2 * v1 * v2) / (v1 + v2) This is the harmonic mean of the two speeds. We can derive this quickly by computing times for each leg and then using the definition of average speed. Step-by-Step Solution: Step 1: Let the one-way distance be d km. Step 2: Time taken for the onward journey at 30 km/h = d / 30 hours. Step 3: Time taken for the return journey at 60 km/h = d / 60 hours. Step 4: Total distance for the round trip = d + d = 2d km. Step 5: Total time = d / 30 + d / 60. Step 6: Compute total time: d / 30 + d / 60 = (2d / 60) + (d / 60) = 3d / 60 = d / 20 hours. Step 7: Average speed = total distance / total time = (2d) / (d / 20) = 2d * (20 / d) = 40 km/h. Verification / Alternative check: Using the harmonic mean formula directly: average speed = (2 * 30 * 60) / (30 + 60) = (3600) / 90 = 40 km/h. This matches the value obtained from the full calculation, confirming the result is correct. Why Other Options Are Wrong: A common incorrect guess is 45 km/h, the arithmetic mean of 30 and 60, but this ignores the longer time spent at the lower speed. Options 50, 35 and 36 km/h also do not match the ratio of total distance to total time. Only 40 km/h correctly balances the slower and faster legs of the trip. Common Pitfalls: Many students mistakenly average the speeds directly, which is valid only if the times at each speed are equal, not the distances. In this problem, more time is spent travelling at 30 km/h, which pulls the overall average down below 45 km/h. Always base average speed on total distance divided by total time. Final Answer: The average speed for the entire journey is 40 km/h.

Question 15

A steamer moves with a speed of 4.5 km/h in still water. It travels to a certain point upstream and then returns to the starting point downstream in a river whose current flows at 1.5 km/h. What is the average speed (in km/h) of the steamer for the …

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Introduction / Context: This question tests your understanding of average speed when travelling equal distances upstream and downstream in a river. The steamer has a known speed in still water, and the river has a known current speed. We must use relative speed to get effective upstream and downstream speeds, and then calculate the overall average speed for a round trip. Given Data / Assumptions: Speed of steamer in still water = 4.5 km/h. Speed of river current = 1.5 km/h. Let the one-way distance to the upstream point be d km. Upstream speed = 4.5 - 1.5 = 3 km/h. Downstream speed = 4.5 + 1.5 = 6 km/h. The steamer covers distance d upstream and distance d downstream. Concept / Approach: For a round trip with equal distances travelled at two different speeds, average speed is total distance divided by total time. As in earlier questions, this leads to the harmonic mean of the two speeds. Here the two speeds are the effective upstream and downstream speeds. We can compute average speed using either the full distance-and-time method or the harmonic mean formula. Step-by-Step Solution: Step 1: Upstream speed = 4.5 - 1.5 = 3 km/h. Step 2: Downstream speed = 4.5 + 1.5 = 6 km/h. Step 3: Time taken upstream = d / 3 hours. Step 4: Time taken downstream = d / 6 hours. Step 5: Total distance for the complete journey = d + d = 2d km. Step 6: Total time = d / 3 + d / 6 = (2d / 6) + (d / 6) = 3d / 6 = d / 2 hours. Step 7: Average speed = total distance / total time = (2d) / (d / 2) = 2d * (2 / d) = 4 km/h. Verification / Alternative check: Using the harmonic mean formula directly: average speed = (2 * upstream speed * downstream speed) / (upstream speed + downstream speed) = (2 * 3 * 6) / (3 + 6) = 36 / 9 = 4 km/h. Both methods give the same value, so the result is confirmed. Why Other Options Are Wrong: Options such as 6, 10 or 12 km/h ignore the fact that the steamer spends more time moving slowly upstream than quickly downstream. The still water speed of 4.5 km/h is not itself the average because the current affects the effective speeds. Only 4 km/h matches the total distance divided by total time. Common Pitfalls: A common error is to take the arithmetic mean of 3 and 6 (which is 4.5) or mistakenly use the still water speed directly. Another mistake is to ignore that the upstream leg takes longer, which lowers the average. Always compute total distance and total time explicitly or apply the harmonic mean formula correctly. Final Answer: The average speed of the steamer for the entire journey is 4 km/h.

Question 16

A boat goes a certain distance at a speed of 40 km/h and returns the same distance at a speed of 24 km/h. What is the average speed (in km/h) for the entire two-way journey?

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Introduction / Context: This is another average speed question involving a round trip with different speeds in each direction. Because the distances are the same in both directions, the correct average speed is the harmonic mean of the two speeds, not the simple arithmetic mean. Understanding this concept is essential in many aptitude tests. Given Data / Assumptions: Onward journey speed = 40 km/h. Return journey speed = 24 km/h. Let the one-way distance be d km (same both ways). We seek the overall average speed for the round trip. Concept / Approach: Average speed is total distance divided by total time. For a round trip with equal distances at speeds v1 and v2, the average speed formula becomes: average speed = (2 * v1 * v2) / (v1 + v2) This is quicker than repeatedly reinventing the distance-and-time calculation but can always be derived from first principles by computing times for each leg. Step-by-Step Solution: Step 1: Let the one-way distance be d km. Step 2: Time taken for the first leg at 40 km/h = d / 40 hours. Step 3: Time taken for the return leg at 24 km/h = d / 24 hours. Step 4: Total distance travelled = d + d = 2d km. Step 5: Total time taken = d / 40 + d / 24. Step 6: Take the LCM of 40 and 24: 120. So d / 40 = 3d / 120 and d / 24 = 5d / 120. Step 7: Total time = (3d / 120) + (5d / 120) = 8d / 120 = d / 15 hours. Step 8: Average speed = total distance / total time = (2d) / (d / 15) = 2d * (15 / d) = 30 km/h. Verification / Alternative check: Using the harmonic mean formula: average speed = (2 * 40 * 24) / (40 + 24) = (1920) / 64 = 30 km/h. This matches the result above, confirming that 30 km/h is correct. Why Other Options Are Wrong: The arithmetic mean of 40 and 24 is 32, but this ignores that more time is spent at the lower speed. Options 28, 26 or 34 km/h do not equal total distance divided by total time when the correct speeds are used. Only 30 km/h is consistent with the travel scenario described. Common Pitfalls: Students frequently take the simple average of speeds, which is only valid if the times at each speed are equal, not the distances. Here, the slower speed of 24 km/h increases the time spent on that leg, pulling the average speed below 32. Always derive average speed from total distance and total time or apply the harmonic mean formula directly. Final Answer: The average speed for the entire journey is 30 km/h.

Question 17

The speeds of a boat along the current and against the current are 14 km/h and 7 km/h respectively. What is the speed of the boat in still water (in km/h)?

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Introduction / Context: This question directly uses the basic formulas for boats and streams. You are given the effective speeds of a boat downstream (along the current) and upstream (against the current). From these, you can easily find the speed of the boat in still water and the speed of the current. Given Data / Assumptions: Downstream speed (along the current) = 14 km/h. Upstream speed (against the current) = 7 km/h. Let b be the speed of the boat in still water. Let s be the speed of the stream. Then downstream speed = b + s, upstream speed = b - s. Concept / Approach: When downstream and upstream speeds are given, the speed of the boat in still water is simply the average of these two speeds. The current speed is half of their difference. This follows from solving the two simultaneous equations: b + s = downstream speed b - s = upstream speed Adding and subtracting these equations isolates b and s. Step-by-Step Solution: Step 1: Write the equations: b + s = 14 and b - s = 7. Step 2: Add the two equations: (b + s) + (b - s) = 14 + 7. Step 3: This gives 2b = 21, so b = 21 / 2 = 10.5 km/h. Step 4: If desired, we can also find the stream speed by subtracting: (b + s) - (b - s) = 14 - 7 ⇒ 2s = 7 ⇒ s = 3.5 km/h. Verification / Alternative check: Check that b = 10.5 and s = 3.5 fit both given speeds. Downstream speed = b + s = 10.5 + 3.5 = 14 km/h, which matches. Upstream speed = b - s = 10.5 - 3.5 = 7 km/h, which also matches the given value. Thus, the speed in still water is confirmed as 10.5 km/h. Why Other Options Are Wrong: Values like 3.5 or 14 km/h are actually the stream speed and downstream speed respectively, not the still water speed. Options 7.5 and 9.5 km/h do not satisfy the equations b + s = 14 and b - s = 7 for any real s. Only 10.5 km/h gives consistent upstream and downstream speeds. Common Pitfalls: Some learners mistakenly think the still water speed equals the downstream speed or confuse it with the stream speed. Others forget to divide by 2 after adding or subtracting the equations. Remember that still water speed is the average of upstream and downstream speeds, and always check your answer by substituting back into the original conditions. Final Answer: The speed of the boat in still water is 10.5 km/h.

Question 18

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

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Introduction / Context: This question combines a ratio relationship involving the downstream speed and the current speed with a simple distance = speed * time calculation for upstream travel. Once the actual speeds are determined, computing the upstream distance over a given time is straightforward. Given Data / Assumptions: Ratio of downstream boat speed to stream speed = 9 : 1. Speed of the current (stream) = 3 km/h. Let b be the speed of the boat in still water. Downstream speed = b + s, where s = 3 km/h. Upstream speed = b - s. We need the distance travelled upstream in 5 hours. Concept / Approach: First, interpret the ratio correctly: downstream speed : stream speed = 9 : 1. Using the actual stream speed, we find the downstream speed and then deduce the boat's still water speed. With the still water speed known, we easily calculate upstream speed and from that compute distance travelled in 5 hours using distance = speed * time. Step-by-Step Solution: Step 1: Let downstream speed = 9k and stream speed = 1k for some constant k. Step 2: Given the stream speed is 3 km/h, we have 1k = 3 ⇒ k = 3. Step 3: Therefore downstream speed = 9k = 9 * 3 = 27 km/h. Step 4: Let b be the speed of the boat in still water and s = 3 km/h. Step 5: Downstream speed = b + s ⇒ b + 3 = 27 ⇒ b = 24 km/h. Step 6: Upstream speed = b - s = 24 - 3 = 21 km/h. Step 7: Distance travelled upstream in 5 hours = upstream speed * time = 21 * 5 = 105 km. Verification / Alternative check: Check the ratio: downstream speed : stream speed = 27 : 3 = 9 : 1, as given. Also, with still water speed 24 km/h and stream speed 3 km/h, upstream speed 21 km/h is reasonable. Multiplying 21 by 5 gives 105 km, which is consistent and within a realistic range for the given speeds and time. Why Other Options Are Wrong: Distances like 100 km, 98 km or 90 km would require smaller upstream speeds than 21 km/h or less time. A distance like 109 km would require an upstream speed greater than 21 km/h over 5 hours, which does not come from the derived still water and stream speeds. Only 105 km matches the product of the correctly calculated upstream speed and the given time. Common Pitfalls: A common mistake is to treat the ratio as relating still water speed to stream speed instead of downstream speed to stream speed. Another error is to forget that the stream speed is given directly, which allows you to find the constant k. Always decode ratios carefully and check which actual speeds they refer to before performing final distance calculations. Final Answer: The boat will travel 105 km upstream in 5 hours.

Question 19

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

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Introduction / Context: This is a straightforward boats and streams question where the effective upstream and downstream speeds are given. From these, we can directly calculate the speed of the stream and, if needed, the speed of the boat in still water. Given Data / Assumptions: Upstream speed (against the current) = 8 km/h. Downstream speed (with the current) = 12 km/h. Let b be the speed of the boat in still water. Let s be the speed of the stream. Then upstream speed = b - s, downstream speed = b + s. Concept / Approach: When upstream and downstream speeds are known, we can find both the still water speed and the current speed by solving two simple linear equations. Adding the equations gives twice the boat speed; subtracting them gives twice the stream speed. This method is standard and very quick once memorised. Step-by-Step Solution: Step 1: Write the equations based on definitions: b - s = 8 and b + s = 12. Step 2: Add the two equations: (b - s) + (b + s) = 8 + 12. Step 3: Left side simplifies to 2b; right side gives 20, so 2b = 20 ⇒ b = 10 km/h. Step 4: Subtract the first equation from the second: (b + s) - (b - s) = 12 - 8. Step 5: Left side simplifies to 2s; right side is 4, so 2s = 4 ⇒ s = 2 km/h. Verification / Alternative check: With b = 10 km/h and s = 2 km/h, upstream speed should be b - s = 10 - 2 = 8 km/h, which matches the given upstream speed. Downstream speed should be b + s = 10 + 2 = 12 km/h, matching the given downstream speed. Therefore, the stream speed of 2 km/h is fully consistent with the problem data. Why Other Options Are Wrong: If the stream speed were 3, 4.5, 5 or 6 km/h, then the implied still water speed would not simultaneously satisfy both given effective speeds. For example, if s = 3 km/h, then b would have to be 11 km/h to get 8 km/h upstream, but b + s would then be 14 km/h, not 12 km/h. Only s = 2 km/h works correctly in both directions. Common Pitfalls: A typical mistake is to confuse which equation to add and which to subtract, sometimes leading to b = 2 or s = 10, which are clearly unreasonable. Another issue is to misinterpret the values as still water and stream speeds instead of effective speeds. Always remember that upstream is slower (b - s) and downstream is faster (b + s), and solve the pair of equations systematically. Final Answer: The speed of the stream is 2 km/h.

Question 20

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

Accepted Answer

Upstream speed = B-S Downstream speed = B+s B-S = 15/5 = 3 km/h Again B= 4S Therefore B-S = 3= 3S => S = 1 and B= 4 km/h Therefore B+S = 5km/h Therefore, Time during downstream = 15/5 = 3h

Boats and Streams Questions