📘 Quiz

Solution: Step 1: Let the speed of the boat in still water be 'B' km/hr and the speed of the current be 'W' km/hr. Step 2: Define upstream speed (U) = (B - W) km/hr and downstream speed (D) = (B + W) km/hr. Step 3: Formulate equations based on the given conditions. * For the first journey: 24/U + 36/D = 6 (Equation 1) * For the second journey: 36/U + 24/D = 6.5 (or 13/2) (Equation 2) Step 4: To simplify, let x = 1/U and y = 1/D. * 24x + 36y = 6 (Equation 1') * 36x + 24y = 13/2 (Equation 2') Step 5: Multiply Equation 1' by 3 and Equation 2' by 2: * 72x + 108y = 18 * 72x + 48y = 13 Step 6: Subtract the second new equation from the first to eliminate 'x': * (72x + 108y) - (72x + 48y) = 18 - 13 * 60y = 5 * y = 5/60 = 1/12 Step 7: Since y = 1/D, D = 12 km/hr. Step 8: Substitute y = 1/12 into Equation 1': * 24x + 36(1/12) = 6 * 24x + 3 = 6 * 24x = 3 * x = 3/24 = 1/8 Step 9: Since x = 1/U, U = 8 km/hr. Step 10: Calculate the speed of the current (W): * W = (D - U) / 2 * W = (12 - 8) / 2 = 4 / 2 = 2 km/hr. Step 11: The speed of the current is 2 km/hr.

Solution: Step 1: Define variables for the boat's speeds. * Let U be the speed of the boat upstream (km/h). * Let D be the speed of the boat downstream (km/h). Step 2: Formulate two equations based on the given time-distance conditions: * Equation 1: (48 / U) + (72 / D) = 12 * Equation 2: (72 / U) + (48 / D) = 13 Step 3: Solve these simultaneous equations. Multiply Equation 1 by 2 and Equation 2 by 3/2 (or equivalent to eliminate one variable easily). Let's aim to eliminate 1/U: * Multiply Equation 1 by 3: 144/U + 216/D = 36 (Equation 3) * Multiply Equation 2 by 2: 144/U + 96/D = 26 (Equation 4) Step 4: Subtract Equation 4 from Equation 3: * (144/U + 216/D) - (144/U + 96/D) = 36 - 26 * (216/D) - (96/D) = 10 * 120/D = 10 * D = 120 / 10 = 12 km/h (Downstream speed). Step 5: Substitute D = 12 km/h into Equation 1: * 48/U + 72/12 = 12 * 48/U + 6 = 12 * 48/U = 12 - 6 * 48/U = 6 * U = 48 / 6 = 8 km/h (Upstream speed). Step 6: Calculate the speed of the stream. * Speed of stream = (Downstream speed - Upstream speed) / 2 * Speed of stream = (D - U) / 2 = (12 - 8) / 2 = 4 / 2 = 2 km/h.

Solution: Step 1: Let the length of the train be L meters and its speed be S m/sec. Step 2: When the train passes a pole, the distance covered is its own length. Using Distance = Speed × Time: L = S × 15. From this, we can express speed S = L/15. Step 3: When the train passes a platform, the total distance covered is the sum of the train's length and the platform's length. Total distance = (L + 100) meters. Step 4: The time taken to cross the platform is 25 seconds. Using Distance = Speed × Time: L + 100 = S × 25. Step 5: Substitute S = L/15 into the second equation: L + 100 = (L/15) × 25. Step 6: Simplify the equation: L + 100 = (5L)/3. Step 7: Multiply the entire equation by 3 to eliminate the denominator: 3(L + 100) = 5L. Step 8: Distribute: 3L + 300 = 5L. Step 9: Rearrange terms to solve for L: 300 = 5L - 3L. Step 10: Simplify: 300 = 2L. Step 11: Calculate L: L = 300 / 2 = 150 meters. Step 12: The length of the train is 150 meters.

Solution: Step 1: Calculate the relative speed of the train with respect to the man, as they are moving in the same direction. Step 2: Relative Speed = Speed of train - Speed of man = (63 - 3) km/hr = 60 km/hr. Step 3: Convert the relative speed from km/hr to m/sec: 60 km/hr = 60 * (5/18) m/sec = 50/3 m/sec. Step 4: When a train crosses a man, the distance covered is the length of the train. Step 5: Distance = 500 meters. Step 6: Time = Distance / Relative Speed. Step 7: Time = 500 / (50/3) = 500 * (3/50) = 30 seconds.

Solution: Step 1: Define the speeds of the ship and the sea plane. - Let the speed of the ship (S_ship) = S miles/hour. - The speed of the sea plane (S_plane) = 10S miles/hour. Step 2: Determine the initial relative position and direction of movement. - The ship is initially 18 miles from the shore. - The sea plane starts from the shore (0 miles) and chases the ship. - Both are moving away from the shore (in the same direction). Step 3: Calculate the relative speed of the plane with respect to the ship. - Since they are moving in the same direction, Relative Speed (S_rel) = S_plane - S_ship = 10S - S = 9S miles/hour. Step 4: Calculate the time taken for the plane to catch up with the ship. - The plane needs to cover the initial 18-mile gap at its relative speed. - Time (T) = Initial Distance / Relative Speed = 18 miles / (9S miles/hour) = 2/S hours. Step 5: Calculate the distance from the shore where the plane catches the ship. - The plane starts from the shore and travels for T hours. - Distance from shore = S_plane * T = (10S miles/hour) * (2/S hours) = 20 miles. - (Alternatively, the ship also travels for T hours. Ship's travel = S_ship * T = S * (2/S) = 2 miles. So, ship's final position = initial 18 miles + 2 miles = 20 miles from shore). Step 6: The sea plane catches up with the ship 20 miles from the shore.

Solution: Step 1: Identify the given speeds. * Downstream speed (D) = 15 km/hr. * Speed of current (W) = 3 km/hr. Step 2: Calculate the speed of the boat in still water (B). * We know D = B + W. * 15 = B + 3 * B = 15 - 3 = 12 km/hr. Step 3: Calculate the upstream speed (U). * U = B - W = 12 - 3 = 9 km/hr. Step 4: Calculate the time taken to travel 15 km upstream. * Distance = 15 km * Speed = U = 9 km/hr * Time_us = 15 / 9 = 5/3 hours. Step 5: Calculate the time taken to travel 15 km downstream. * Distance = 15 km * Speed = D = 15 km/hr * Time_ds = 15 / 15 = 1 hour. Step 6: Calculate the total time. * Total Time = Time_us + Time_ds = 5/3 + 1 = 8/3 hours. Step 7: Convert total time to hours and minutes. * 8/3 hours = 2 and 2/3 hours. * (2/3) hours * 60 minutes/hour = 40 minutes. Step 8: Total time = 2 hours 40 minutes.

Solution: Step 1: Let the distance from Rohan's office to Shikha's home be D. Step 2: Rohan's speed (S_R) = 60 km/hr. Step 3: If Rohan drove the full distance D to pick Shikha, it would take him 48 minutes. Step 4: So, D = S_R * (48/60) hours = 60 km/hr * (4/5) hours = 48 km. Step 5: Rohan was initially going to be 16 minutes late. By Shikha moving towards him, they managed to arrive on time, meaning they saved 16 minutes of travel time. Step 6: The time Rohan saved is due to him not having to travel the distance Shikha covered twice (once to go pick her up, and once to cover the same segment back with her). So, the distance saved by Rohan is 2 * (distance Shikha traveled). Step 7: Let x_S be the distance Shikha traveled in the auto-rickshaw. The time Rohan saved is (2 * x_S) / S_R. Step 8: (2 * x_S) / 60 kmph = 16 minutes = 16/60 hours. Step 9: From the equation: 2 * x_S = 16 => x_S = 8 km. This is the distance Shikha covered. Step 10: Rohan and Shikha traveled simultaneously until they met. The time taken for them to meet (T) is the same for both. Step 11: Distance covered by Rohan until meeting = D - x_S = 48 km - 8 km = 40 km. Step 12: Time T = (Distance covered by Rohan) / S_R = 40 km / 60 km/hr = 2/3 hours. Step 13: Now, calculate Shikha's speed (S_S). Shikha covered x_S = 8 km in T = 2/3 hours. Step 14: S_S = x_S / T = 8 km / (2/3) hours = 8 * (3/2) km/hr = 12 km/hr. Step 15: The speed of the auto-rickshaw is 12 km/hr.

Solution: Method 1: Using Algebraic Equations Step 1: Let the time spent traveling by car be T_c hours and by train be T_t hours. Step 2: The total time is 16 hours, so T_c + T_t = 16 ... (Equation 1). Step 3: The distance traveled by car is Speed_car * T_c = 40 * T_c km. Step 4: The distance traveled by train is Speed_train * T_t = 80 * T_t km. Step 5: The total distance is 1200 km, so 40 * T_c + 80 * T_t = 1200 ... (Equation 2). Step 6: From Equation 1, express T_t in terms of T_c: T_t = 16 - T_c. Step 7: Substitute this into Equation 2: 40 * T_c + 80 * (16 - T_c) = 1200. Step 8: Simplify the equation: 40T_c + 1280 - 80T_c = 1200. Step 9: Combine like terms: -40T_c + 1280 = 1200. Step 10: Isolate T_c: 1280 - 1200 = 40T_c => 80 = 40T_c. Step 11: Solve for T_c: T_c = 80 / 40 = 2 hours. Step 12: The distance traveled by car = Speed_car * T_c = 40 km/h * 2 hours = 80 km. Method 2: Using Alligation Step 1: Calculate the overall average speed for the journey: Average Speed = Total Distance / Total Time = 1200 km / 16 hours = 75 km/h. Step 2: Apply the alligation rule for speeds and average speed. The ratio of the differences will give the ratio of times spent. - Speed by Car: 40 km/h - Speed by Train: 80 km/h - Average Speed: 75 km/h - Difference (Train - Avg): 80 - 75 = 5 - Difference (Avg - Car): 75 - 40 = 35 Step 3: The ratio of time spent by car to time spent by train is 5 : 35, which simplifies to 1 : 7. Step 4: The total time is 16 hours. Distribute this time according to the ratio 1:7. Step 5: Time spent by car = (1 / (1 + 7)) * 16 hours = (1/8) * 16 = 2 hours. Step 6: Distance traveled by car = Speed_car * Time_car = 40 km/h * 2 hours = 80 km.

Solution: Step 1: Calculate the relative speed of the train with respect to the man. Since they are moving in opposite directions, Relative Speed = Speed of train + Speed of man. Relative Speed = 60 kmph + 6 kmph = 66 kmph. Step 2: Convert the relative speed from kmph to m/sec. Relative Speed = 66 * (5/18) m/sec = 55/3 m/sec. Step 3: When a train passes a man, the distance covered is the length of the train. Distance = Length of the train = 110 metres. Step 4: Use the formula: Time = Distance / Speed. Time = 110 m / (55/3 m/sec) Time = 110 * (3/55) sec Time = 2 * 3 sec Time = 6 seconds.

Solution: Step 1: Let A's speed = a m/s, B's speed = b m/s Step 2: When A finishes 300m, B covers (300 - 22.5) = 277.5m Step 3: Time taken by A = 300/a seconds Step 4: Time taken by B = 277.5/b seconds Step 5: Given time difference = 6 seconds: 300/a - 277.5/b = 6 Step 6: Relative speed difference = 22.5/6 = 3.75 m/s Step 7: Set up equation for B's time: 300/a = 277.5/(a - 3.75) + 6 Step 8: Solve for a and b: a = 7.5 m/s, b = 3.75 m/s Step 9: Total time for B = 300/3.75 = 80 seconds

Solution: Step 1: To compare the speeds, convert all given speeds to a common unit, such as kilometres per hour (km/hr). Step 2: Convert the first speed: 25 m/sec. To convert m/sec to km/hr, multiply by (18/5). Speed = 25 * (18/5) km/hr = 5 * 18 km/hr = 90 km/hr. Step 3: Convert the second speed: 1500 m/min. Convert metres to kilometres: 1500 m = 1.5 km. Convert minutes to hours: 1 min = 1/60 hr. Speed = 1.5 km / (1/60) hr = 1.5 * 60 km/hr = 90 km/hr. Step 4: The third speed is already in km/hr: 90 km/hr. Step 5: Compare all converted speeds. Speed 1 = 90 km/hr. Speed 2 = 90 km/hr. Speed 3 = 90 km/hr. Step 6: Since all three speeds are equal, none is faster than the others. They are all the same speed.

Solution: Step 1: Distance traveled by Vehicle A in 40 sec = 40 × 20 = 800 m Step 2: Length of Vehicle A = Total distance - Platform length = 800 - 400 = 400 m Step 3: Length of Vehicle B = (2/5) × Length of Vehicle A = (2/5) × 400 = 160 m

Solution: Step 1: Let the original speed be S km/h and the original time be T hours. Step 2: From the first condition: Distance = 240 km. So, T = 240 / S. (Equation 1) Step 3: From the second condition: If speed is (S - 8) km/h, time taken is (T + 1) hours. So, T + 1 = 240 / (S - 8). (Equation 2) Step 4: Substitute T from Equation 1 into Equation 2: (240 / S) + 1 = 240 / (S - 8). (240 + S) / S = 240 / (S - 8). Step 5: Cross-multiply and solve for S: (240 + S)(S - 8) = 240S. 240S - 1920 + S^2 - 8S = 240S. S^2 - 8S - 1920 = 0. Step 6: Solve the quadratic equation. Factors of -1920 that sum to -8 are -48 and 40. (S - 48)(S + 40) = 0. Since speed cannot be negative, S = 48 km/h. Step 7: Now, calculate the time required to cover 480 km at the original speed (48 km/h). Time = Distance / Speed = 480 km / 48 km/h = 10 hours.

Solution: Step 1: Downstream speed = 20 km / (24/60) hr = 50 km/hr Step 2: Let speed of vessel in still water = 4x, speed of stream = x Step 3: Downstream speed = 4x + x = 5x = 50 km/hr → x = 10 km/hr Step 4: Upstream speed = 4x - x = 3x = 3*10 = 30 km/hr Step 5: Time upstream = 15 km / 30 km/hr = 0.5 hr = 30 mins

Solution: Step 1: Let Vehicle Y's length = L units and speed = S units/sec Step 2: Vehicle X's length = 1.25L, speed = 1.2S Step 3: Distance covered by Vehicle X to pass stationary point = 1.2S * 12 = 14.4S Step 4: Relative speed when overtaking = 1.2S - S = 0.2S Step 5: Total distance when overtaking = 1.25L + L = 2.25L Step 6: Time to overtake = 162 seconds Step 7: Set up equation: 2.25L / 0.2S = 162 Step 8: Solve for L: L = (162 * 0.2S) / 2.25 = 14.4S Step 9: Substitute L in relative distance: 2.25 * 14.4S / 0.2S = 162 Step 10: Simplify: 162 = 162, consistent Step 11: Convert S to kmph: S = 60 kmph

Solution: Step 1: Let the speed of the passenger train be S_p km/h and the speed of the express train be S_e km/h. Step 2: Let the time taken by the passenger train be T_p hours and by the express train be T_e hours. Step 3: From the problem statement: Distance (D) = 1176 km. T_p - T_e = 5 hours. S_e - S_p = 70 km/h => S_p = S_e - 70. Step 4: Using the formula Time = Distance / Speed: T_p = 1176 / S_p and T_e = 1176 / S_e. Step 5: Substitute these into the time difference equation: (1176 / S_p) - (1176 / S_e) = 5. Step 6: Substitute S_p = S_e - 70: (1176 / (S_e - 70)) - (1176 / S_e) = 5. Step 7: Simplify the equation: 1176 * [S_e - (S_e - 70)] / [S_e * (S_e - 70)] = 5. 1176 * 70 / (S_e^2 - 70 * S_e) = 5. 82320 = 5 * (S_e^2 - 70 * S_e). 16464 = S_e^2 - 70 * S_e. S_e^2 - 70 * S_e - 16464 = 0. Step 8: Solve the quadratic equation for S_e. By testing factors or using options, we find S_e = 168 km/h. (S_e - 168)(S_e + 98) = 0. Since speed cannot be negative, S_e = 168 km/h. Step 9: Calculate S_p: S_p = S_e - 70 = 168 - 70 = 98 km/h. Step 10: Calculate the time taken by the passenger train: T_p = 1176 km / 98 km/h = 12 hours.

Solution: Step 1: Let swimmer's speed in still water = x km/hr, river's speed = y km/hr Step 2: Downstream speed = x + y, Upstream speed = x - y Step 3: Given: 20 = (x + y) * 1 and 20 = (x - y) * 2 Step 4: Simplify: x + y = 20 and x - y = 10 Step 5: Add equations: 2x = 30 => x = 15 Step 6: Substitute x = 15 into x + y = 20 => y = 5 Step 7: Swimmer's speed = 15 km/hr, River's speed = 5 km/hr

Solution: Step 1: Use the formula for average speed when distances traveled at different speeds are the same: Average Speed = (2 * speed1 * speed2) / (speed1 + speed2) Step 2: Plug in the given values: speed1 = 60 km/hr, speed2 = 90 km/hr Step 3: Calculate: Average Speed = (2 * 60 * 90) / (60 + 90) Step 4: Simplify: Average Speed = (10800) / (150) = 72 km/hr

Solution: Step 1: Let the speed downstream be 'Sd' km/hr and the speed upstream be 'Su' km/hr. Given, the time taken to row 4 km downstream is the same as 3 km upstream. Time = Distance / Speed. So, 4 / Sd = 3 / Su. This implies Sd = (4/3)Su. (Equation 1) Step 2: The total time for the 48 km round trip (downstream and upstream) is 14 hours. Time_downstream + Time_upstream = 14. 48 / Sd + 48 / Su = 14. (Equation 2) Step 3: Substitute Equation 1 into Equation 2. 48 / ((4/3)Su) + 48 / Su = 14. (48 * 3) / (4 * Su) + 48 / Su = 14. 36 / Su + 48 / Su = 14. (36 + 48) / Su = 14. 84 / Su = 14. Step 4: Solve for Su. Su = 84 / 14 = 6 km/hr. Step 5: Solve for Sd using Equation 1. Sd = (4/3) * 6 = 8 km/hr. Step 6: Let the speed of the man in still water be 'B' and the speed of the stream be 'S'. Speed_downstream (Sd) = B + S = 8. Speed_upstream (Su) = B - S = 6. Step 7: To find the rate of the stream (S), subtract the upstream speed from the downstream speed and divide by 2. S = (Sd - Su) / 2 = (8 - 6) / 2 = 2 / 2 = 1 km/hr.