Question 1

An express train travelled at an average speed of 100 km/hr, stopping for 3 minutes after every 75 km. How long did it take to reach its destination 600 km from the starting point? (M.A.T., 2003)

Accepted Answer

Concept & Logic: Total travel time includes the active driving time plus the accumulated stationary time from all intermediate stops. The final destination does not count as a stop that adds to the journey time. Total Time = Driving Time + (Number of Stops × Stop Duration) Step-by-Step Solution: 1. Calculate Continuous Driving Time: • Total Distance = 600 km • Speed = 100 km/hr • Driving Time = (600/100) = 6 hours. 2. Calculate Number of Intermediate Stops: • The train stops every 75 km. • Total intervals = (600/75) = 8. • Since the 8th stop is the final destination, the number of intermediate stops is 8 - 1 = 7. 3. Calculate Total Stopping Time: • Each stop lasts 3 minutes. • Total stopping time = 7 × 3 = 21 minutes. 4. Calculate Final Total Time: • Total Time = 6 hours + 21 minutes = 6 hrs 21 min. Exam Strategy & Shortcut: To quickly find the number of stops, divide the total distance by the stop interval distance. If the result is a whole number N, the number of stops is N - 1. If it has a remainder, the number of stops is the quotient. Here, (600/75) = 8 perfectly, so 8 - 1 = 7 stops. Common Pitfall: Forgetting to subtract 1 from the total number of stops when the distance is perfectly divisible by the interval. Including the destination as a "stop" would erroneously yield 24 minutes of stopping time. Final Answer: Therefore, the correct answer is 6 hrs 21 min.

Question 2

A certain distance is covered by a cyclist at a certain speed. If a jogger covers half the distance in double the time, the ratio of the speed of the jogger to that of the cyclist is :

Accepted Answer

Concept & Formula: Speed is directly proportional to distance and inversely proportional to time. To find a ratio of speeds, substitute the relative distances and times into the speed formula. Speed = (Distance/Time) Step-by-Step Solution: 1. Define Cyclist Variables: • Let the distance covered by the cyclist be D. • Let the time taken by the cyclist be T. • Speed of cyclist, S_c = (D/T). 2. Define Jogger Variables: • Distance covered by jogger = (D/2) (half the distance). • Time taken by jogger = 2T (double the time). • Speed of jogger, S_j = (\frac{D/2)}{2T} = (D/4T). 3. Calculate the Ratio: • Ratio = S_j : S_c • Ratio = (D/4T) : (D/T) • Cancel out the common (D/T) term. • Ratio = (1/4) : 1 • Multiply by 4 to clear the fraction: 1 : 4. Exam Strategy & Shortcut: Use fractional multipliers directly on the speed formula S = (D/T). For the jogger, Distance is × (1/2) and Time is × 2. So, new Speed multiplier = (\frac{1/2)}{2} = (1/4). The jogger is (1/4) the speed of the cyclist, meaning the ratio is 1:4. Common Pitfall: Mixing up the order of the ratio. The question asks for "jogger to cyclist", which is 1:4. Reversing it to 4:1 is a very common error. Final Answer: Therefore, the correct answer is 1 : 4.

Question 3

A motor car starts with the speed of 70 km/hr with its speed increasing every two hours by 10 kmph. In how many hours will it cover 345 kms?

Accepted Answer

Concept & Logic: When speed changes in discrete time intervals, calculate the distance covered in each interval cumulatively until the target distance is reached. Distance = \sum (Speed_i × Time_i) Step-by-Step Solution: 1. Calculate Distance for First Interval (Hours 1 & 2): • Speed = 70 km/hr. Time = 2 hours. • Distance covered = 70 × 2 = 140 km. • Remaining distance = 345 - 140 = 205 km. 2. Calculate Distance for Second Interval (Hours 3 & 4): • Speed increases by 10 km/hr. New speed = 80 km/hr. Time = 2 hours. • Distance covered = 80 × 2 = 160 km. • Remaining distance = 205 - 160 = 45 km. 3. Calculate Time for Remaining Distance (Hour 5+): • Speed increases by 10 km/hr again. New speed = 90 km/hr. • Distance to cover = 45 km. • Time required = (Distance/Speed) = (45/90) = (1/2) hour. 4. Calculate Total Time: • Total Time = 2 hours (first interval) + 2 hours (second interval) + (1/2) hour. • Total Time = 4(1/2) hours. Exam Strategy & Shortcut: Maintain a running tally of distance and time: 2 hrs -> 140 km 4 hrs -> 140 + 160 = 300 km Need 45 km more. Next speed is 90 km/h. 45 is exactly half of 90, so it takes 0.5 hours. 4 + 0.5 = 4.5 hours. Common Pitfall: Assuming the speed increases continuously (like continuous acceleration) rather than in stepped blocks every two hours. Using arithmetic progression formulas for continuous acceleration will yield an incorrect time. Final Answer: Therefore, the correct answer is 4(1/2) hrs.

Question 4

A bus moving at a speed of 24 m/s begins to slow at a rate of 3 m/s each second. How far does it go before stopping? (N.D.A., 2007)

Accepted Answer

Concept & Formula: This is a problem of uniform deceleration. We can use the kinematic equation of motion that relates initial velocity, final velocity, acceleration, and distance. v^2 - u^2 = 2as Where v is final velocity, u is initial velocity, a is acceleration, and s is distance. Step-by-Step Solution: 1. Identify Given Values: • Initial velocity, u = 24 m/s. • Final velocity, v = 0 m/s (since it comes to a stop). • Acceleration, a = -3 m/s^2 (negative because it is slowing down). • Distance, s is what we need to find. 2. Apply the Kinematic Equation: • 0^2 - (24)^2 = 2 × (-3) × s • 0 - 576 = -6s • -576 = -6s 3. Solve for Distance (s): • s = (576/6) • s = 96 metres. Exam Strategy & Shortcut: Alternatively, use the concept of average speed. Time to stop = \frac{Initial Speed}{Deceleration Rate} = (24/3) = 8 seconds. Since deceleration is uniform, Average Speed = (Initial + Final/2) = (24 + 0/2) = 12 m/s. Distance = Average Speed × Time = 12 × 8 = 96 metres. This avoids calculating 24^2. Common Pitfall: Forgetting the negative sign on the deceleration rate when using the v^2 - u^2 = 2as formula, which can cause confusion or lead to a negative distance result. Final Answer: Therefore, the correct answer is 96 m.

Question 5

A car starts running with the initial speed of 40 kmph, with its speed increasing every hour by 5 kmph. How many hours will it take to cover a distance of 385 km?

Accepted Answer

Concept & Arithmetic Progression: When an object travels with an initial distance in the first hour and the distance increases by a constant amount each subsequent hour, the total distance covered over a number of hours forms an Arithmetic Progression (A.P.). S_n = (n/2)[2a + (n-1)d] Step-by-Step Solution: • Identify A.P. Parameters: Initial distance (first term a) = 40 km. Hourly increase (common difference d) = 5 km. Total distance (S_n) = 385 km. • Apply Formula: 385 = (n/2)[2(40) + (n-1)5] • Simplify: 770 = n[80 + 5n - 5] \Rightarrow 770 = n(75 + 5n) • Form Quadratic Equation: 5n^2 + 75n - 770 = 0 \Rightarrow n^2 + 15n - 154 = 0 • Factorize: n^2 + 22n - 7n - 154 = 0 \Rightarrow n(n + 22) - 7(n + 22) = 0 • Solve for n: (n - 7)(n + 22) = 0. Since time cannot be negative, n = 7. Exam Strategy & Shortcut: Instead of solving the quadratic equation, plug the options into the sum formula or just add manually if the numbers are small: 40 + 45 + 50 + 55 + 60 + 65 + 70. The sum of the unit digits is 0 + 5 + 0 + 5 + 0 + 5 + 0 = 15, ending in 5, matching 385. Seven terms is exactly 7 hours. Common Pitfall: Misidentifying the hourly increase as acceleration and trying to use physics kinematic equations (s = ut + (1/2)at^2), which applies to continuous smooth acceleration, not discrete hourly jumps. Final Answer: Therefore, the correct answer is 7 hours.

Question 6

The speeds of A and B are in the ratio 3 : 4. A takes 20 minutes more than B to reach a destination. In what time does A reach the destination?

Accepted Answer

Concept & Ratios in Kinematics: When distance is constant, the ratio of times taken by two bodies is the inverse of the ratio of their speeds. Step-by-Step Solution: • Set up Ratios: The ratio of speeds of A and B is 3 : 4. • Therefore, the ratio of time taken by A and B is 4 : 3. • Assign Variables: Let the time taken by A be 4x and by B be 3x. • Use Given Condition: A takes 20 minutes more than B. So, 4x - 3x = 20, which means x = 20. • Calculate A's Time: Time taken by A = 4x = 4 × 20 = 80 minutes. • Format Answer: 80 minutes is equal to 1 hour and 20 minutes, or 1(20/60) = 1(1/3) hours. Exam Strategy & Shortcut: Notice the gap in the time ratio parts: A takes 4 parts of time, B takes 3 parts. The difference is 1 part. We are given this 1 part difference is exactly 20 minutes. Therefore, A's time (4 parts) is simply 4 × 20 = 80 minutes without formal algebra. Common Pitfall: Solving for x correctly but mistakenly calculating the time for B (60 minutes) instead of A, or botching the conversion from 80 minutes into a mixed fractional hour. Final Answer: Therefore, the correct answer is 1(1/3) hours.

Question 7

An aeroplane covers a certain distance at a speed of 240 kmph in 5 hours. To cover the same distance in $1\frac{2}{3}$ hours, it must travel at a speed of

Accepted Answer

Concept & Distance Formula: For a fixed distance, the product of speed and time remains constant. Calculate the total distance first, then use it to find the new speed. D = S_1 × T_1 = S_2 × T_2 Step-by-Step Solution: • Find total distance: Speed = 240 kmph, Time = 5 hours. • Distance = 240 × 5 = 1200 km. • New time required = 1(2/3) hours = (5/3) hours. • Let the required speed be S. • S × (5/3) = 1200 • S = 1200 × (3/5) • S = 240 × 3 = 720 kmph. Exam Strategy & Shortcut: Using S_1 × T_1 = S_2 × T_2: 240 × 5 = S_2 × (5/3). The 5s cancel out immediately across the equation. S_2 = 240 × 3 = 720. This calculation can easily be done in under 10 seconds mentally. Common Pitfall: Incorrectly converting the mixed fraction 1(2/3) into a decimal and struggling with repeating decimals (1.666...) during division. Always use improper fractions for these calculations. Final Answer: Therefore, the correct answer is 720 kmph.

Question 8

A salesman travels a distance of 50 km in 2 hours and 30 minutes. How much faster, in kilometres per hour, on an average, must he travel to make such a trip in $\frac{5}{6}$ hour less time?

Accepted Answer

Concept & Change in Speed: Calculate the initial speed and the required new speed based on the altered time, then find the absolute difference between the two speeds. Step-by-Step Solution: • Initial Scenario: Distance = 50 km. Time = 2 hours 30 mins = 2.5 hours = (5/2) hours. • Initial Speed = (50/\frac{5){2}} = 50 × (2/5) = 20 km/hr. • New Scenario: The new time should be (5/6) hour less. • New Time = (5/2) - (5/6) = (15 - 5/6) = (10/6) = (5/3) hours. • New Speed = \frac{Distance}{New Time} = (50/\frac{5){3}} = 50 × (3/5) = 30 km/hr. • Difference: Increase in speed = 30 - 20 = 10 km/hr. Exam Strategy & Shortcut: Initial time is 2.5 hrs (150 mins). (5/6) of an hour is 50 mins. New time = 100 mins ((5/3) hrs). Initial speed = (50/2.5) = 20. New speed = (50/5/3) = 30. The difference is 10. Thinking purely in minutes for the time steps can sometimes prevent fraction addition errors. Common Pitfall: Selecting the new speed (30) as the final answer instead of the *increase* in speed (10). Always re-read the final question prompt carefully to see exactly what is being asked. Final Answer: Therefore, the correct answer is 10.

Question 9

A person has to cover a distance of 6 km in 45 minutes. If he covers one-half of the distance in two-thirds of the total time; to cover the remaining distance in the remaining time, his speed (in km/hr) must be

Accepted Answer

Concept & Formula: The core concept is calculating the required speed for the second leg of a journey given the total constraints and the first leg's performance. Speed = \frac{Distance}{Time} Step-by-Step Solution: • Given: • Total distance = 6 km • Total time = 45 minutes • Calculation for First Half: • Distance covered = (1/2) × 6 = 3 km • Time taken = (2/3) × 45 = 30 minutes • Calculation for Remaining Journey: • Remaining distance = 6 - 3 = 3 km • Remaining time = 45 - 30 = 15 minutes • Final Speed Calculation: • Convert time to hours: 15 minutes = (15/60) hours = (1/4) hour • Required Speed = \frac{Remaining Distance}{Remaining Time} = (3/1/4) = 3 × 4 = 12 km/hr Exam Strategy & Shortcut: Break the problem down into two distinct phases. Always convert minutes to hours before the final speed calculation if the options are in km/hr. Common Pitfall: Forgetting to convert the remaining time from minutes to hours, which would lead to an incorrect speed of 3 / 15 = 0.2 km/min, and possibly misinterpreting the options. Final Answer: Therefore, the correct answer is 12.

Question 10

A man performs $\frac{3}{5}$ of the total journey by rail, $\frac{7}{20}$ by bus and the remaining $6.5	ext{ km}$ on foot. His total journey is

Accepted Answer

Concept & Logic: This is a fraction-based distance problem. The entire journey represents a whole (1). The remaining fractional part corresponds to the remaining absolute distance. Step-by-Step Solution: • Given: • Fraction by rail = (3/5) • Fraction by bus = (7/20) • Distance on foot = 6.5 km • Calculation: • Let the total journey be x km. • Fraction of journey covered = (3/5) + (7/20) • To add these, find a common denominator (20): (12/20) + (7/20) = (19/20) • Remaining fraction of the journey = 1 - (19/20) = (1/20) • This remaining fraction equals the distance travelled on foot: (1/20) of x = 6.5 x = 6.5 × 20 = 130 km Exam Strategy & Shortcut: Work purely with fractions. Since the parts sum to (19/20), the remaining 1 part out of 20 is 6.5 km. Thus, 20 parts = 6.5 × 20 = 130 km. This mental math avoids algebra entirely. Common Pitfall: Making arithmetic errors when finding a common denominator for the fractions. Final Answer: Therefore, the correct answer is 130 km.

Question 11

A person wishes to reach his destination 90 km away in 3 hours but for the first half of the journey his speed was 20 km/hr. His average speed for the rest of the journey should be

Accepted Answer

Concept & Formula: This problem tests your ability to track time and distance accurately, and critically, checks your attention to units in the final options. Speed = \frac{Distance}{Time} Step-by-Step Solution: • Given: • Total distance = 90 km • Total time available = 3 hours • First Half of Journey: • Distance = (90/2) = 45 km • Speed = 20 km/hr • Time taken = (45/20) = 2.25 hours • Remaining Journey: • Remaining distance = 90 - 45 = 45 km • Remaining time = 3 - 2.25 = 0.75 hours • Required Speed: • Speed = (45/0.75) = (4500/75) = 60 km/hr • Unit Conversion: • Notice that 60 km/hr is not in the options (A is 40). Look at options B and C which are in km/min. • Convert 60 km/hr to km/min: \frac{60 km}{60 mins} = 1 km/min. Exam Strategy & Shortcut: Once you get 60 km/hr, quickly scan the options. A common trap is selecting "None of these" when a unit conversion of the correct answer is present. Always check units! Common Pitfall: Calculating 60 km/hr, not seeing it in options (a), and immediately choosing "None of these" without checking the alternative units in options (b) and (c). Final Answer: Therefore, the correct answer is 1 km/min.

Question 12

A train is scheduled to cover the distance between two stations 46 km apart in one hour. If it travels 25 km at a speed of 40 km/hr, find the speed for the remaining journey to complete it in the scheduled time.

Accepted Answer

Concept & Formula: Find the time remaining by subtracting the time already spent from the total scheduled time, then calculate the required speed for the remaining distance. Time = \frac{Distance}{Speed} Step-by-Step Solution: • Given: • Total distance = 46 km • Scheduled total time = 1 hour • First Part of Journey: • Distance = 25 km • Speed = 40 km/hr • Time taken = (25/40) = (5/8) hours • Remaining Journey: • Remaining distance = 46 - 25 = 21 km • Remaining time = 1 - (5/8) = (3/8) hours • Required Speed: • Speed = \frac{Distance}{Time} = (21/3/8) = 21 × (8/3) • Speed = 7 × 8 = 56 km/hr Exam Strategy & Shortcut: Using fractions ((5/8) and (3/8)) instead of decimals (0.625 and 0.375) makes the final multiplication much faster and less prone to arithmetic mistakes. Common Pitfall: Using decimals for (25/40) can make the subsequent division (21/0.375) overly complicated to calculate manually under time pressure. Final Answer: Therefore, the correct answer is 56 km/hr.

Question 13

A can complete a journey in 10 hours. He travels first half of the journey at the rate of 21 km/hr and second half at the rate of 24 km/hr. Find the total journey in km.

Accepted Answer

Concept & Formula: When an object travels two equal distances at speeds v_1 and v_2, its average speed for the entire journey is the harmonic mean of the two speeds. Average Speed = (2 \cdot v_1 \cdot v_2/v_1 + v_2) Total Distance = Average Speed × Total Time. Step-by-Step Solution: • Given: • Total time = 10 hours • Speed for first half (v_1) = 21 km/hr • Speed for second half (v_2) = 24 km/hr • Average Speed Calculation: • Average Speed = (2 × 21 × 24/21 + 24) • = (1008/45) • Simplify by dividing numerator and denominator by 9: (112/5) = 22.4 km/hr. • Total Distance Calculation: • Total Distance = Average Speed × Total Time • Total Distance = 22.4 × 10 = 224 km. Exam Strategy & Shortcut: Alternatively, let total distance be 2x. The time equation is (x/21) + (x/24) = 10. x \cdot ((8+7/168)) = 10 \Rightarrow x \cdot ((15/168)) = 10 \Rightarrow x = (1680/15) = 112. Total distance is 2x = 224. This algebra approach often avoids large multiplications if simplified early. Common Pitfall: Taking the simple arithmetic average of the speeds ((21 + 24/2) = 22.5) instead of the harmonic mean, which would give an incorrect total distance of 225 km. Final Answer: Therefore, the correct answer is 224 km.

Question 14

A person travels equal distances with speeds of 3 km/hr, 4 km/hr and 5 km/hr and takes a total time of 47 minutes. The total distance (in km) is

Accepted Answer

Concept & Equation: Set up a total time equation based on equal distance segments. Always ensure units of time match before solving. Time = \frac{Distance}{Speed} Step-by-Step Solution: • Given: • Let each equal distance segment be x km. • Total distance = 3x km. • Speeds are 3, 4, and 5 km/hr. • Total time = 47 minutes = (47/60) hours. • Equation Setup: • (x/3) + (x/4) + (x/5) = (47/60) • Solve for x: • LCM of 3, 4, 5 is 60. • (20x + 15x + 12x/60) = (47/60) • (47x/60) = (47/60) • x = 1 km • Calculate Total Distance: • Total distance = 3x = 3 × 1 = 3 km. Exam Strategy & Shortcut: Assume the equal distance is the LCM of the speeds (60 km). The times would be 20, 15, and 12 hours, totaling 47 hours. Since the actual time is 47 minutes ((47/60) hours), the actual distance must be (1/60) of our assumed distance. Distance per segment = 60 × (1/60) = 1 km. Total = 3 × 1 = 3 km. Common Pitfall: Finding x = 1 and forgetting that the total distance is 3x. Luckily, 1 is not an option here, but this is a frequent trap. Also, forgetting to convert 47 minutes to hours. Final Answer: Therefore, the correct answer is 3.

Question 15

A person travels 285 km in 6 hours in two stages. In the first part of the journey, he travels by bus at the speed of 40 km per hour. In the second part of the journey, he travels by train at the speed of 55 km per hour. How much distance did he tra…

Accepted Answer

Concept & Logic: Use a system of linear equations or a single variable equation splitting the total time across two different speeds. Distance = Speed × Time Step-by-Step Solution: • Given: • Total distance = 285 km • Total time = 6 hours • Speed of bus = 40 km/hr • Speed of train = 55 km/hr • Equation Setup: • Let the time travelled by train be t hours. • Then, time travelled by bus = (6 - t) hours. • Distance by train + Distance by bus = Total Distance • 55t + 40(6 - t) = 285 • Solve for t: • 55t + 240 - 40t = 285 • 15t = 45 • t = 3 hours • Calculate Train Distance: • Distance by train = Speed × Time = 55 × 3 = 165 km. Exam Strategy & Shortcut: Use the Alligation method. Average speed for the whole trip = (285/6) = 47.5 km/hr. Bus speed (40) ............. Train speed (55) ................. Avg (47.5) ................. (55 - 47.5) = 7.5 ........ (47.5 - 40) = 7.5 Ratio of times = 7.5 : 7.5 = 1 : 1. This means time is divided equally: 3 hours bus, 3 hours train. Distance by train = 55 × 3 = 165 km. Common Pitfall: Calculating the distance traveled by bus (120 km) and accidentally selecting it if it were an option, instead of the required train distance. Final Answer: Therefore, the correct answer is 165 km.

Question 16

A car covers a distance from Town I to Town II at the speed of 56 km/hr and from Town II to Town I at the speed of 53 km/hr. What is the average speed of the car?

Accepted Answer

Concept & Average Speed: When a certain distance is traveled at speed x and the exact same distance is traveled at speed y, the average speed for the whole journey is given by the harmonic mean of the two speeds. Average\ Speed = (2xy/x + y) Step-by-Step Solution: • Given speed x = 56 km/hr. • Given speed y = 53 km/hr. • Apply the formula: • Average Speed = (2 × 56 × 53/56 + 53) • Average Speed = (5936/109) • Average Speed \approx 54.458 km/hr. • Since 54.458 km/hr does not match any of the provided numerical options exactly, the answer is "None of these". Exam Strategy & Shortcut: Whenever distances are equal, never just average the two speeds (e.g., (56+53)/2 = 54.5). The correct average speed is always slightly less than the simple arithmetic average. Here, 54.5 is the arithmetic average, so the true average is slightly less. Calculating exactly yields \approx 54.458, confirming none of the fixed options are precisely correct. Common Pitfall: Calculating the simple average of the speeds (56 + 53/2) = 54.5 km/hr, or approximating 54.458 to 54 km/hr and choosing option (b). Average speed must be exact unless an approximation is explicitly asked for. Final Answer: Therefore, the correct answer is None of these.

Question 17

A man can walk uphill at the rate of $2 \frac{1}{2}$ km/hr and downhill at the rate of $3 \frac{1}{4}$ km/hr. If the total time required to walk a certain distance up the hill and return to the starting point was 4 hr 36 min, then what was the dista…

Accepted Answer

Concept & Time and Distance: The core relationship is Distance = Speed × Time. Since the distance traveled uphill and downhill is the same, we can equate the total time taken to the sum of the times taken for the individual legs of the journey. Total\ Time = (Distance/Speed_{uphill)} + (Distance/Speed_{downhill)} Step-by-Step Solution: • Convert speeds to improper fractions: • Uphill speed = 2 (1/2) = (5/2) km/hr. • Downhill speed = 3 (1/4) = (13/4) km/hr. • Convert total time to hours: • Total time = 4 hr 36 min = 4 + (36/60) hours = 4 + (3/5) = (23/5) hours. • Let the distance be d. • The time equation is: (d/5/2) + (d/13/4) = (23/5) • Simplify the fractions: (2d/5) + (4d/13) = (23/5) • Find a common denominator for the left side (65): • (26d + 20d/65) = (23/5) • (46d/65) = (23/5) • Solve for d: • d = (23 × 65/46 × 5) • d = (23/46) × (65/5) • d = (1/2) × 13 = (13/2) • d = 6.5 km, or 6 (1/2) km. Exam Strategy & Shortcut: You can also find average speed first: (2 × (5/2) × (13/4)/(5/2) + (13/4)) = (65/4/33/4) = (65/33) km/hr. Then total distance (both ways) = Average speed × total time = (65/33) × (23/5). But the standard equation method is often less error-prone with mixed fractions. Common Pitfall: Forgetting to convert the 36 minutes into a proper fraction of an hour, or multiplying instead of dividing the time components. Final Answer: Therefore, the correct answer is 6 (1/2) km.

Question 18

A man drives 150 km to the seashore in 3 hours 20 min. He returns from the shore to the starting point in 4 hours 10 min. Let $r$ be the average rate for the entire trip. Then the average rate for the trip going exceeds $r$, in kilometres per hour, …

Accepted Answer

Concept & Average Rate: Average rate (speed) is always calculated as the Total Distance divided by the Total Time. Do not average the two speeds. Average\ Speed = (Total\ Distance/Total\ Time) Step-by-Step Solution: • Step 1: Calculate the average rate for the trip going. • Distance = 150 km. • Time going = 3 hours 20 min = 3 + (20/60) = 3 (1/3) = (10/3) hours. • Rate going = (150/10/3) = (150 × 3/10) = 45 km/hr. • Step 2: Calculate the average rate for the entire trip (r). • Total Distance = 150 km (going) + 150 km (returning) = 300 km. • Total Time = 3 hr 20 min + 4 hr 10 min = 7 hr 30 min = 7.5 hours = (15/2) hours. • Average rate r = (300/15/2) = (300 × 2/15) = 40 km/hr. • Step 3: Find the difference. • The question asks how much the rate going (45 km/hr) exceeds the average rate r (40 km/hr). • Difference = 45 - 40 = 5 km/hr. Exam Strategy & Shortcut: Convert minutes to fractions of hours immediately to keep calculations clean. Work out each distinct piece of the puzzle (rate going, then total rate) sequentially rather than trying to build one giant formula. Common Pitfall: A common mistake is finding the return rate (which is 36 km/hr) and averaging the two rates to find r ((45 + 36/2) = 40.5 km/hr), which would lead to an incorrect difference of 4.5 km/hr. Final Answer: Therefore, the correct answer is 5.

Question 19

A car covers the first 39 kms of its journey in 45 minutes and covers the remaining 25 km in 35 minutes. What is the average speed of the car? (Bank P.O., 2007)

Accepted Answer

Concept & Average Speed: Average speed over multiple segments is always the sum of all distances divided by the sum of all time periods. Ensure time units are consistent with the requested speed units (km/hr requires time in hours). Average\ Speed = (Total\ Distance/Total\ Time) Step-by-Step Solution: • Step 1: Calculate Total Distance. • Total Distance = 39 km + 25 km = 64 km. • Step 2: Calculate Total Time in hours. • Total Time in minutes = 45 mins + 35 mins = 80 minutes. • Convert minutes to hours: (80/60) hours = (4/3) hours. • Step 3: Calculate Average Speed. • Average Speed = (64/4/3) • Average Speed = 64 × (3/4) • Average Speed = 16 × 3 = 48 km/hr. Exam Strategy & Shortcut: Instead of converting 45 mins and 35 mins to hours individually (which leads to adding fractions (3/4) + (7/12)), sum the minutes first to get 80 minutes, and then perform a single, much simpler conversion to (4/3) hours. Common Pitfall: Failing to convert the total time from minutes to hours before dividing, which would result in (64/80) = 0.8, a non-sensical speed for a car in this context. Final Answer: Therefore, the correct answer is 48 km/hr.

Question 20

A train travels at a speed of 30 km/hr for 12 minutes and at a speed of 45 km/hr for the next 8 minutes. The average speed of the train for this journey is (S.S.C., 2005)

Accepted Answer

Concept & Average Speed Calculation: To find the average speed when given speeds and times, first use Distance = Speed × Time to find the distance of each segment, then apply the master average speed formula. Time must be in hours. Average\ Speed = (d_1 + d_2/t_1 + t_2) Step-by-Step Solution: • Step 1: Calculate distance for the first segment. • Speed = 30 km/hr, Time = 12 mins = (12/60) hours = (1/5) hours. • Distance 1 = 30 × (1/5) = 6 km. • Step 2: Calculate distance for the second segment. • Speed = 45 km/hr, Time = 8 mins = (8/60) hours = (2/15) hours. • Distance 2 = 45 × (2/15) = 3 × 2 = 6 km. • Step 3: Calculate Total Distance and Total Time. • Total Distance = 6 + 6 = 12 km. • Total Time = 12 mins + 8 mins = 20 mins = (20/60) hours = (1/3) hours. • Step 4: Calculate Average Speed. • Average Speed = (12/1/3) = 12 × 3 = 36 km/hr. Exam Strategy & Shortcut: Notice that the distances for both segments are equal (6 km each). When distances are equal, the average speed is the harmonic mean of the speeds: (2xy/x+y) = (2 × 30 × 45/30 + 45) = (2700/75) = 36 km/hr. However, calculating the total distance and time is often just as fast. Common Pitfall: Multiplying the speed by the time in minutes without converting to hours (e.g., 30 × 12 = 360), which fundamentally breaks the unit relationship. Final Answer: Therefore, the correct answer is 36 km/hr.

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