Question 1

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. The speed of the boat in still water is (I.I.F.T., 2008)

Accepted Answer

Concept & Linear Systems: This classic boat problem presents two journey scenarios, forming a linear system of two equations. We solve for the upstream speed (U) and downstream speed (D) to find the still water speed. Still Water Speed = (D + U/2) Step-by-Step Solution: • Let the upstream speed be U and the downstream speed be D. • Scenario 1: (30/U) + (44/D) = 10 \quad --- (Equation 1) • Scenario 2: (40/U) + (55/D) = 13 \quad --- (Equation 2) • To eliminate U, multiply Equation 1 by 4 and Equation 2 by 3: (120/U) + (176/D) = 40 \quad --- (Equation 3) (120/U) + (165/D) = 39 \quad --- (Equation 4) • Subtract Equation 4 from Equation 3: (176/D) - (165/D) = 40 - 39 (11/D) = 1 \implies D = 11 km/hr • Substitute D = 11 into Equation 1: (30/U) + (44/11) = 10 (30/U) + 4 = 10 (30/U) = 6 \implies U = 5 km/hr • The speed of the boat in still water is: (D + U/2) = (11 + 5/2) = (16/2) = 8 km/hr Exam Strategy & Shortcut: Look for common factors in the downstream distances. 44 and 55 are both multiples of 11. Guessing D = 11 is a highly educated shortcut. If D=11, then (44/11) = 4 hours. Leaving 10 - 4 = 6 hours for 30 km upstream, so U = 5. Test this on the second equation: (40/5) + (55/11) = 8 + 5 = 13. It works perfectly, bypassing all complex algebra. Common Pitfall: Solving for the stream speed instead of the boat's speed in still water. The stream speed would be (11 - 5/2) = 3, which is prominently listed as an incorrect trap option (a). Final Answer: Therefore, the correct answer is 8 km/hr.

Question 2

A man can row upstream at $10$ kmph and downstream at $18$ kmph. Find the man's rate in still water? [Indian Railways Gr. 'D' Exam, 2014]

Accepted Answer

Concept & Basic Averages: The speed of a boat or person in still water is the arithmetic mean of their upstream and downstream speeds. Still Water Speed = \frac{Downstream Speed + Upstream Speed}{2} Step-by-Step Solution: • Identify the given speeds: Downstream speed (D) = 18 kmph, Upstream speed (U) = 10 kmph. • Apply the formula for rate in still water: (18 + 10/2). • Calculate the sum: 18 + 10 = 28. • Divide by 2: (28/2) = 14 kmph. Exam Strategy & Shortcut: This is a direct formula application. Mentally add the two speeds and halve the result for an instant answer. Common Pitfall: Subtracting the values instead of adding them, which would mistakenly give the speed of the current (4 kmph) instead of the still water speed. Final Answer: Therefore, the correct answer is (a) 14 kmph.

Question 3

Boat A travels downstream from Point X to Point Y in $3$ hours less than the time taken by Boat B to travel upstream from Point Y to Point Z. The distance between X and Y is $20$ km, which is half of the distance between Y and Z. The speed of Boat B…

Accepted Answer

Concept & Algebraic Formulation: Break complex relational problems down by expressing all unknown speeds in terms of a single variable, typically the speed of the current. Time = \frac{Distance}{Speed} Step-by-Step Solution: • Let the speed of the current be c. • Analyze Distances: Distance X to Y = 20 km. Distance Y to Z = 20 × 2 = 40 km. • Analyze Boat B: Speed in still water = 10 km/h. Upstream speed = 10 - c. Time for B to travel Y to Z upstream = (40/10 - c). • Analyze Boat A: Speed in still water equals Boat B's upstream speed, so it is 10 - c. Downstream speed of A = (10 - c) + c = 10 km/h. Time for A to travel X to Y downstream = (20/10) = 2 hours. • Set up the equation based on the time relationship: Time of A = Time of B - 3. 2 = (40/10 - c) - 3 • Solve for c: 5 = (40/10 - c) 10 - c = 8 c = 2 km/h. • Calculate Boat A's still water speed: 10 - c = 10 - 2 = 8 km/h. Exam Strategy & Shortcut: Observe that Boat A's downstream speed perfectly cancels out the current variable (10 - c + c = 10), immediately giving Boat A's time as 2 hours. This drastically simplifies the equation setup right from the start. Common Pitfall: Losing track of the variables and forgetting that Boat A's still water speed already incorporates a subtraction of c, leading to double-counting the current when calculating downstream speed. Final Answer: Therefore, the correct answer is (d) 8 km/h.

Question 4

The speed of the boat in still water is $5$ times that of the current, it takes $1.1$ hours to row to point B form point A downstream. The distance between point A and point B is $13.2$km. How much distance (in km) will it cover in $312$ minutes ups…

Accepted Answer

Concept & Relative Speeds: Establish a ratio between the boat's speed and the current's speed to express downstream and upstream velocities in terms of a single variable. Distance = Speed × Time Step-by-Step Solution: • Establish variables: Let current speed = c. The boat's still water speed is 5c. • Determine relative speeds: Downstream Speed (D) = 5c + c = 6c. Upstream Speed (U) = 5c - c = 4c. • Find the actual value of c: The boat covers 13.2 km in 1.1 hours downstream. D = (13.2/1.1) = 12 km/h. Since 6c = 12, we find c = 2 km/h. • Calculate Upstream Speed (U): U = 4c = 4 × 2 = 8 km/h. • Calculate target distance: Time is 312 minutes. Convert to hours: (312/60) = 5.2 hours. Distance = U × Time = 8 × 5.2 = 41.6 km. Exam Strategy & Shortcut: The speed ratio of Downstream (6c) to Upstream (4c) is 6:4 or 3:2. If downstream speed is 12 km/h, upstream speed is instantly (2/3) × 12 = 8 km/h. Multiply by the hour equivalent of 312 mins (5.2) to get 41.6. Common Pitfall: Forgetting to convert minutes to hours before multiplying with the calculated upstream speed in km/h. Final Answer: Therefore, the correct answer is (c) 41.6.

Question 5

A boat can travel $36$ km upstream in $5$ hours. If the speed of the stream is $2.4$ kmph, how much time will the boat take to cover a distance of $78$ km downstream? (in hours) [United India Insurance Co. Ltd. (UIICL) Assistant (Online) Exam, 2015]

Accepted Answer

Concept & Speed Conversions: Find the boat's still water speed using the upstream speed and stream speed, then use that to calculate the downstream time. Upstream Speed = Boat Speed - Stream Speed Downstream Speed = Boat Speed + Stream Speed Step-by-Step Solution: • Calculate upstream speed (U): Distance = 36 km, Time = 5 hours. U = (36/5) = 7.2 kmph. • Given stream speed (V) = 2.4 kmph. • Find boat speed in still water (B): Since U = B - V, we have 7.2 = B - 2.4. B = 7.2 + 2.4 = 9.6 kmph. • Calculate downstream speed (D): D = B + V = 9.6 + 2.4 = 12 kmph. • Calculate time for 78 km downstream: Time = \frac{Distance}{D} = (78/12) = 6.5 hours. Exam Strategy & Shortcut: Notice the relationship D = U + 2V. This allows you to skip finding B entirely. D = 7.2 + 2(2.4) = 7.2 + 4.8 = 12 kmph. The time is immediately 78 / 12 = 6.5 hours. Common Pitfall: Adding the stream speed only once to the upstream speed, mistaking the result for the downstream speed rather than just finding the still water speed first. Final Answer: Therefore, the correct answer is (b) 6.5.

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