Question 1

In covering a certain distance, the speeds of $A$ and $B$ are in the ratio of 3 : 4. $A$ takes 30 minutes more than $B$ to reach the destination. The time taken by $A$ to reach the destination is

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Concept & Inverse Proportion of Speed and Time: For a constant distance, the time taken is inversely proportional to the speed. If the ratio of speeds of two moving bodies is a : b, the ratio of the time taken by them to cover the same distance is b : a. Step-by-Step Solution: 1. We are given the ratio of speeds of A and B as S_A : S_B = 3 : 4. 2. Therefore, the ratio of time taken by them will be T_A : T_B = 4 : 3. 3. Let the time taken by A be 4x minutes and the time taken by B be 3x minutes. 4. The problem states that A takes 30 minutes more than B: 4x - 3x = 30 x = 30 minutes 5. We need to find the time taken by A, which is 4x: T_A = 4 × 30 = 120 minutes 6. Convert minutes to hours: 120 minutes = 2 hours Exam Strategy & Shortcut: The time ratio is 4 : 3. The difference in ratio units is 4 - 3 = 1 unit. This 1 unit difference corresponds to the 30 minutes given in the problem. A takes 4 units of time, so A's time is 4 × 30 = 120 minutes = 2 hours. Very fast and requires no complex algebra! Common Pitfall: A common mistake is applying the ratio directly without taking its inverse, incorrectly assuming the faster person (B) takes more time. Final Answer: Therefore, the correct answer is 2 hours.

Question 2

A journey of 192 km between two cities takes 2 hours less by a fast train than by a slow train. If the average speed of the slow train is 16 km/hr less than that of the fast train, then the average speed of the fast train is

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Concept & Quadratic Equation for Speed Difference: When comparing the time taken by two objects travelling the same distance with a known difference in their speeds and times, we construct an algebraic time difference equation resulting in a quadratic equation. (Distance/S_{slow)} - (Distance/S_{fast)} = \Delta Time Step-by-Step Solution: 1. Let the average speed of the fast train be x km/hr. 2. The average speed of the slow train is (x - 16) km/hr. 3. The time taken by the fast train to cover 192 km is (192/x) hours. 4. The time taken by the slow train to cover 192 km is (192/x - 16) hours. 5. The problem states the time difference is 2 hours: (192/x - 16) - (192/x) = 2 6. Factor out 192 and find a common denominator: 192 [ (x - (x - 16)/x(x - 16)) ] = 2 192 [ (16/x^2 - 16x) ] = 2 7. Divide both sides by 2 to simplify: 96 × 16 = x^2 - 16x 1536 = x^2 - 16x x^2 - 16x - 1536 = 0 8. Factor the quadratic equation. Look for two numbers that multiply to -1536 and add to -16. These numbers are -48 and +32. (x - 48)(x + 32) = 0 9. Since speed cannot be negative, x = 48 km/hr. Exam Strategy & Shortcut: Use Option Elimination based on the time difference. The calculation (192/x - 16) - (192/x) must equal 2. Test Option (c) 48 km/hr: Fast speed = 48. Slow speed = 48 - 16 = 32. Fast time = (192/48) = 4 hours. Slow time = (192/32) = 6 hours. Difference = 6 - 4 = 2 hours. This matches perfectly and skips solving the quadratic equation! Common Pitfall: Setting up the subtraction backwards (e.g., (192/x) - (192/x - 16) = 2) is a major pitfall. It produces a negative time difference and invalidates the setup. Always subtract the smaller time (faster speed) from the larger time (slower speed). Final Answer: Therefore, the correct answer is 48 km/hr.

Question 3

There are 8 equidistant points $A, B, C, D, E, F, G$ and $H$ in the clockwise direction on the periphery of a circle. In a time interval $t$, a person reaches from $A$ to $C$ with uniform motion while another person reaches the point $E$ from the po…

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Concept & Relative Distance on Circular Tracks: When two objects move on a circular track in the same direction starting from different positions, the time to meet depends on the initial gap and their relative speed. The faster object must close the "rear" gap to lap the slower one. Time = (Distance\ to\ Gain/Relative\ Speed) Step-by-Step Solution: 1. Define units: Let the distance between any two adjacent points be 1 unit. Total circumference = 8 units. 2. Determine speeds: Person 1 (P1) travels from A to C (2 units) in time t. Speed of P1 = 2/t units/time. Person 2 (P2) travels from B to E (3 units) in time t. Speed of P2 = 3/t units/time. 3. Analyze initial positions: P1 starts at A. P2 starts at B. Since points are clockwise (A, B, C...), P2 is ahead of P1 by 1 unit in the direction of motion. 4. Calculate relative speed: Since both move clockwise (same direction), relative speed = Speed of P2 - Speed of P1 = (3/t) - (2/t) = 1/t units/time. 5. Determine distance to gain: P2 is faster and is ahead. For P2 to meet P1, P2 must lap P1. This means P2 has to cover the entire circle minus the initial 1-unit lead to catch up from behind. Distance to gain = Total Circumference - Initial Lead = 8 - 1 = 7 units. 6. Calculate meeting time: Time = (Distance to gain) / Relative Speed = 7 / (1/t) = 7t. Exam Strategy & Shortcut: Think in terms of "gaps" instead of speed formulas. In time t, P1 covers 2 gaps, P2 covers 3 gaps. P2 gains 1 gap every t time. P2 is initially 1 gap ahead, so P2 needs to gain 7 gaps to catch P1 from behind. 7 gaps × (t per gap) = 7t. Common Pitfall: Assuming the faster person just needs to close the 1-unit gap. Since P2 is faster AND starts ahead, P2 is pulling away. P2 must go all the way around the circle to catch P1 from behind, which requires gaining 8 - 1 = 7 units. Final Answer: Therefore, the correct answer is 7t.

Question 4

Two guns were fired from the same place at an interval of 8 minutes, A person approaching the place observes that 5 minutes 52 seconds have elapsed between the hearing of the sound of the two guns. If the velocity of the sound is 330 m/sec, the man …

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Concept & Logic: This is a classic relative speed problem involving sound. The distance that sound would have traveled in the remaining time of the firing interval is equal to the total distance the approaching man traveled during his observed elapsed time. *(Note: There is a typographical error in the original printed question where "8 minutes" is written instead of the mathematically intended "6 minutes". The solution below assumes 6 minutes to align with standard logic and the provided options).* Step-by-Step Solution: • Given: Intended time interval of guns = 6 minutes. Time elapsed for the moving observer = 5 min 52 sec. Speed of sound = 330 m/s. • Calculation: Convert the time values into seconds. • Intended firing interval = 6 × 60 = 360 seconds. • Time elapsed heard by the observer = (5 × 60) + 52 = 352 seconds. • Difference in time = 360 - 352 = 8 seconds. • The distance the sound wave would have traveled in those "missing" 8 seconds = 330 m/s × 8 s = 2640 m. • The approaching man must have traveled this exact same distance (2640 m) over his entire traveling duration between hearing the shots (352 seconds). • Speed of the man in m/s = (2640/352) = 7.5 m/s. • To convert m/s to km/hr, multiply by (18/5): 7.5 × (18/5) = 27 km/hr. Exam Strategy & Shortcut: Use the direct proportionality formula for these problems: Speed of Man = Speed of Sound × ( \frac{Actual Interval - Heard Interval}{Heard Interval} ). Assuming the intended 6 minutes: Speed = 330 × ((360 - 352/352)) = 330 × (8/352) = 7.5 m/s \implies 27 km/hr. Common Pitfall: Using the typo value of "8 minutes" blindly will yield 120 m/s (432 km/hr), which is completely absent from the options. Recognizing standard typographical errors in competitive exam questions is an essential troubleshooting skill. Final Answer: Therefore, the correct answer is 27.

Question 5

$A$ and $B$ are two stations $10$ km apart. A man, $P$ starts from $A$ and travels towards $B$ at the rate of $3$ km/hr, whereas another man $Q$ starts from $B$ and travels to wards A at the rate of $2$ km/hr. When and where do they meet?

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Concept & Meeting Point Calculation: When two objects travel towards each other simultaneously, their relative speed is the sum of their speeds. The meeting time is the total initial distance divided by this relative speed. Once time is known, distance from a specific point is found using the respective traveler's speed. Meeting Time = \frac{Total Distance}{Relative Speed} Step-by-Step Solution: • Given: Distance between A and B = 10 km. Speed of P (from A) = 3 km/hr. Speed of Q (from B) = 2 km/hr. • Calculation: Relative speed of P and Q = 3 + 2 = 5 km/hr. • Time taken to meet = \frac{Total Distance}{Relative Speed} = (10/5) = 2 hours. • The distance from station A where they meet is the distance traveled by P in those 2 hours. • Distance from A = Speed of P × Meeting Time = 3 × 2 = 6 km. Exam Strategy & Shortcut: Find relative speed (5 km/hr). Total distance (10 km) divided by 5 gives 2 hours. Since the question asks for distance from A, multiply the meeting time (2 hrs) by the speed of the person leaving from A (3 km/hr) to get 6 km. Match these two values to the options. Common Pitfall: Calculating the correct meeting time (2 hours) but mistakenly calculating the meeting distance from station B (2 × 2 = 4 km) and selecting option (d). Final Answer: Therefore, the correct answer is After 2 hours, 6 km from A.

Question 6

Directions: These questions are based on the following information:  $P$ and $Q$ are $120$ km apart. $A$ starts from $P$ towards $Q$ at 6 a.m. $B$ starts from $Q$ towards $P$ at 11 a.m. on the same day. $A$ is $50$ % faster than $B$. They cross each…

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Concept & Total Journey Time: Once individual speeds are determined using distance proportionality from the meeting point, the total time for each person to complete the full journey is simply the total distance divided by their respective speeds. Step-by-Step Solution: • Given: Total distance = 120 km. A travels 14 hours and B travels 9 hours until meeting. A's speed is 1.5 times B's speed. • Calculation: From earlier derivations, A's speed is 6 km/hr and B's speed is 4 km/hr. • Total time taken by A to travel the entire 120 km = (120/6) = 20 hours. • Total time taken by B to travel the entire 120 km = (120/4) = 30 hours. • The question asks for the difference in their total travel times to reach their respective destinations. • Time difference = 30 - 20 = 10 hours. Exam Strategy & Shortcut: Since you already know A takes 20 hours and B travels at (2/3) the speed of A, it stands to reason that B will take (3/2) the time of A. So, B takes 1.5 × 20 = 30 hours. The difference is 30 - 20 = 10 hours, avoiding the need to fully calculate B's speed. Common Pitfall: Misreading the question and answering with the time difference it took them just to meet (which is 14 - 9 = 5 hours) instead of the time difference to complete the entire destination journey. Final Answer: Therefore, the correct answer is 10.

Question 7

Two planes move along a circle of circumference 1.2 kms with constant speeds. When they move in different directions, they meet every 15 seconds and when they move in the same direction one plane overtakes the other every 60 seconds. The speed of th…

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Concept & Relative Speed on a Circle: When two bodies move along a circular path, their relative speed depends on their direction. If moving in opposite directions, their relative speed is the sum of their speeds. If moving in the same direction, it is the difference. The distance covered to meet or overtake is the circumference of the circle. Distance = Relative\ Speed × Time Step-by-Step Solution: 1. Let the speed of the faster plane be u km/s and the slower plane be v km/s. 2. Opposite directions: They meet every 15 seconds. Distance = 1.2 km. Relative speed = (u + v) (u + v) = (1.2/15) = 0.08 km/s 3. Same direction: The faster plane overtakes the slower one every 60 seconds. Relative speed = (u - v) (u - v) = (1.2/60) = 0.02 km/s 4. Solve for v: We need the speed of the slower plane. Subtract the second equation from the first. (u + v) - (u - v) = 0.08 - 0.02 2v = 0.06 v = 0.03 km/s Exam Strategy & Shortcut: Once you establish the relative speeds (0.08 for sum, 0.02 for difference), simply subtract the difference from the sum and divide by 2 to find the smaller value directly: (0.08 - 0.02/2) = 0.03. Common Pitfall: Mixing up units. Ensure you notice that circumference is in "kms" and time is in "seconds", aligning perfectly with the options in "km/s". Do not needlessly convert to km/hr. Final Answer: Therefore, the correct answer is 0.03 km/s.

Question 8

Two trains start from stations $A$ and $B$ and travel towards each other at a speed of 50 kmph and 60 kmph respectively. A the time of their meeting, the second train had travelled 120 km more than the first. The distance between $A$ and $B$ is

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Concept & Speed Difference: When two objects move towards each other for the same duration, the difference in the distance they cover is simply their relative speed difference multiplied by the total time traveled. Distance\ Difference = (Speed_2 - Speed_1) × Time Step-by-Step Solution: 1. Identify the speeds: Train 1 speed = 50 km/hr, Train 2 speed = 60 km/hr. 2. Find the difference in speed: Speed\ Difference = 60 - 50 = 10 km/hr This means for every hour they travel, the second train covers 10 km more than the first. 3. Find the time to meet: The total extra distance covered by the second train is 120 km. Time = (Extra\ Distance/Speed\ Difference) Time = (120/10) = 12 hours They traveled for 12 hours before meeting. 4. Calculate total distance: The total distance is the sum of their individual speeds (relative speed towards each other) multiplied by the total time. Total\ Distance = (50 + 60) × 12 Total\ Distance = 110 × 12 = 1320 km Exam Strategy & Shortcut: Think of it logically: The faster train gains 10 km on the slower train every hour. To gain 120 km, it must have been traveling for 12 hours. In 1 hour, together they cover 110 km. In 12 hours, they cover 110 × 12 = 1320 km. Common Pitfall: Assuming 120 is the relative distance to solve for time using the sum of speeds. 120 is the *difference* in their individual distances, so it must be equated to the *difference* in their speeds. Final Answer: Therefore, the correct answer is 1320 km.

Question 9

Train A leaves Ludhiana for Delhi at 11 a.m, running at the speed of 60 km/hr. Train B leaves Ludhiana for Delhi by the same route at 2 p.m. on the same day, running at the speed of 72 km/hr. At what time will the two trains meet each other?

Accepted Answer

Concept & Relative Speed: When two bodies move in the same direction, their relative speed is the difference of their individual speeds. The time taken to meet is the initial distance separating them divided by this relative speed. Relative Speed = V_2 - V_1 Time to Catch Up = \frac{Initial Distance}{Relative Speed} Step-by-Step Solution: 1. Calculate Initial Distance: Train A leaves at 11 a.m. and Train B leaves at 2 p.m. Train A travels alone for 3 hours. 2. Distance covered by Train A in 3 hours = 60 km/hr × 3 hrs = 180 km. 3. Calculate Relative Speed: Both trains are moving in the same direction. Relative Speed = 72 km/hr - 60 km/hr = 12 km/hr. 4. Calculate Time to Catch Up: Time = \frac{Distance}{Relative Speed} = \frac{180 km}{12 km/hr} = 15 hours. 5. Calculate Meeting Time: Train B leaves at 2 p.m. Adding 15 hours to 2 p.m.: 2 p.m. + 10 hours = 12 a.m. (Midnight) 12 a.m. + 5 hours = 5 a.m. on the next day. Exam Strategy & Shortcut: Instead of forming algebraic equations 60(t+3) = 72t, directly think in terms of the "gap". Train A creates a 180 km gap. Train B closes this gap at a rate of 12 km per hour. The mental math (180/12) = 15 hours is significantly faster. Common Pitfall: A frequent error is adding the 15 hours to Train A's departure time (11 a.m.) instead of Train B's departure time (2 p.m.), which would lead to an incorrect answer of 2 a.m. Final Answer: Therefore, the correct answer is 5 a.m. on the next day.

Question 10

A car overtakes a bus travelling from Delhi to Jaipur at 4 : 30 p.m. The car reaches Jaipur at 6 : 00 p.m. After stopping there for 1 hour, it starts back towards Delhi and meets the same bus at 7 : 30 p.m. which was moving towards Jaipur at that ti…

Accepted Answer

Concept & Logic: This problem requires tracking the positions and times of two vehicles to determine their speeds relative to the total distance. Speed = \frac{Distance}{Time} Step-by-Step Solution: 1. Let the point where the car overtakes the bus (at 4:30 p.m.) be point A, and Jaipur be point B. Let the distance AB be D. 2. Car's Journey to Jaipur: The car travels from A to B between 4:30 p.m. and 6:00 p.m. Time taken by car = 1.5 hours. Speed of car (v_c) = (D/1.5) = (2D/3). 3. Car's Return Journey: The car stops at Jaipur for 1 hour (leaves at 7:00 p.m.) and meets the bus at 7:30 p.m. Let this meeting point be C. Time traveled by car from B to C = 0.5 hours. Distance BC = v_c × 0.5 = (2D/3) × (1/2) = (D/3). 4. Bus's Journey: The bus has been traveling from A since 4:30 p.m. At 7:30 p.m., it is at point C. Distance traveled by bus (AC) = AB - BC = D - (D/3) = (2D/3). Time taken by bus = 7:30 p.m. - 4:30 p.m. = 3 hours. Speed of bus (v_b) = \frac{Distance}{Time} = (\frac{2D/3)}{3} = (2D/9). 5. Bus's Total Time: Total time for the bus to travel distance D (AB) = (D/v_b) = (D/\frac{2D){9}} = (9/2) = 4.5 hours. 6. Arrival Time: The bus was at point A at 4:30 p.m. Arrival time at Jaipur = 4:30 p.m. + 4.5 hours = 9:00 p.m. Exam Strategy & Shortcut: Visualize the timeline. The car travels the whole distance in 1.5 hours. On the way back, it travels for 0.5 hrs, covering 1/3 of the distance. This means the bus has covered 2/3 of the distance in 3 hours. To cover the full distance (3/3), the bus will take 3 × (3/2) = 4.5 hours. 4:30 pm + 4.5 hrs = 9:00 pm. Common Pitfall: A common error is forgetting to account for the 1-hour stop the car makes in Jaipur when calculating the time it takes to travel back and meet the bus. Final Answer: Therefore, the correct answer is 9 : 00 p.m.

Question 11

The speed of a bus is 72 kmph. The distance covered by the bus in 5 sec is [Indian Railway Gr. 'D' Exam, 2014]

Accepted Answer

Concept & Formula: To find distance in meters given time in seconds, the speed must first be converted from kilometers per hour (kmph) to meters per second (m/s). Speed in m/s = Speed in kmph × (5/18) Distance = Speed × Time Step-by-Step Solution: 1. Convert Speed: Convert 72 kmph to m/s. 72 × (5/18) = 4 × 5 = 20 m/s 2. Calculate Distance: Use the converted speed and the given time (5 seconds). Distance = 20 m/s × 5 sec = 100 meters Exam Strategy & Shortcut: Memorize common conversions: 18 kmph = 5 m/s. Since 72 kmph = 4 × 18 kmph, the speed is 4 × 5 m/s = 20 m/s. The distance is 20 × 5 = 100 m. Common Pitfall: A major mistake is forgetting to convert units and simply multiplying 72 × 5, leading to incorrect units and answers. Always ensure unit consistency. Final Answer: Therefore, the correct answer is 100 m.

Question 12

A man travels for 5 hours 15 minutes. If he covers the first half of the journey at 60km/h and rest at 45km/h. Find the total distance travelled by him. [SSC—CHSL (10 + 2) Exam, 2015]

Accepted Answer

Concept & Formula: When a journey is split into two equal halves (distances are the same) covered at different speeds S_1 and S_2, the relationship between total distance D and total time T can be evaluated by summing the time taken for each half. Total Time = \frac{Distance_1}{Speed_1} + \frac{Distance_2}{Speed_2} Step-by-Step Solution: 1. Convert Time: The total time is 5 hours and 15 minutes. 5 hours 15 minutes = 5 + (15/60) = 5 + (1/4) = (21/4) hours 2. Define Distance: Let the total distance be 2d. Thus, the first half is d and the second half is d. 3. Set Up Equation: The sum of times for both halves equals the total time. (d/60) + (d/45) = (21/4) 4. Solve for d: LCM of 60 and 45 is 180. (3d + 4d/180) = (21/4) (7d/180) = (21/4) d = (21 × 180/4 × 7) d = 3 × 45 = 135 km 5. Calculate Total Distance: Total Distance = 2d = 2 × 135 = 270 km Exam Strategy & Shortcut: Use the average speed formula for equal distances: Average Speed = (2 × S_1 × S_2/S_1 + S_2). Average Speed = (2 × 60 × 45/60 + 45) = (5400/105) = (360/7) km/h. Total Distance = Average Speed × Total Time = (360/7) × (21/4) = 90 × 3 = 270 km. Common Pitfall: A common mistake is forgetting to multiply d by 2 at the end if you assumed the total distance was 2d. If you assume the total distance is D, the equation is (D/2/60) + (D/2/45) = (21/4), which helps avoid this. Final Answer: Therefore, the correct answer is 270 km.

Question 13

In order to reach his office on time, Mr. Roy goes through the middle passage of a round fort which he takes 14 minutes to pass through. However, on a certain day, due to repairs, the straight road being blocked, he had to take the roundabout way as…

Accepted Answer

Concept & Circular Motion: The middle passage of a round fort represents its diameter. The roundabout way represents its semicircular arc. The time taken is directly proportional to the distance traveled if the speed remains constant. Distance_{diameter} = 2r Distance_{semicircle} = \pi r Step-by-Step Solution: • Let the radius of the round fort be r. • Distance through the middle passage (diameter) = 2r. • Time taken for distance 2r = 14 minutes. • Distance through the roundabout way (semicircle) = \pi r. • Let the time taken for the roundabout way be t. • Since speed is constant, the ratio of times equals the ratio of distances: (t/14) = (\pi r/2r) • t = 14 × (\pi/2) = 7\pi • Using \pi \approx (22/7): t = 7 × (22/7) = 22 minutes. • The time he was late = Time taken for roundabout way - Usual time • Late by = 22 - 14 = 8 minutes. Exam Strategy & Shortcut: The ratio of the length of a semicircle to its diameter is (\pi r/2r) = (\pi/2) \approx (22/7/2) = (11/7). Multiply the usual time by this ratio to get the new time: 14 × (11/7) = 22 mins. Difference = 22 - 14 = 8 mins. Common Pitfall: A common error is confusing the "roundabout way" with the full circumference. Since he crosses the fort to reach his office on the other side, he only travels half the circumference (a semicircle). Final Answer: Therefore, the correct answer is 8 min.

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