Question 1

Person A starts walking from a point towards a place 7 miles away at a speed of 1 mile in 8 minutes. Person B starts from the same point 4 minutes after A. When A reaches the destination, turns back and walks 1 mile, he meets B. If A's speed is 1 mi…

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Introduction / Context: This question combines time, distance and relative motion for walkers starting at different times. Instead of asking for speed directly in km/h or miles per hour, it asks for how many minutes B takes per mile, which is a slightly different but equivalent measure of speed. Given Data / Assumptions: Distance from starting point to destination = 7 miles. Person A speed = 1 mile in 8 minutes. Person B starts 4 minutes after A from the same point. A reaches the destination, turns back, walks 1 mile back and meets B at that point. We assume straight path and constant speeds for both persons. Concept / Approach: Compute the total time A has been walking before meeting B. Use that to find how long B has been walking. Since we know B covers 6 miles (from start to meeting point 1 mile short of destination), we can determine how many minutes B takes per mile by dividing his walking time by his distance. Step-by-Step Solution: A walks from start to destination: 7 miles. Time for A to reach destination = 7 * 8 = 56 minutes. A then walks 1 mile back: extra 8 minutes. Total walking time for A before meeting B = 56 + 8 = 64 minutes. B starts 4 minutes after A, so B has been walking for 64 - 4 = 60 minutes by the meeting time. Meeting point is 1 mile from destination, so it is 7 - 1 = 6 miles from start. Therefore B walks 6 miles in 60 minutes. Minutes per mile for B = 60 / 6 = 10 minutes per mile. Verification / Alternative check: B's speed in miles per hour = 60 minutes / 10 minutes per mile = 6 miles per hour. In 1 hour B travels 6 miles; in 60 minutes in the scenario he also travels 6 miles, which is consistent. Why Other Options Are Wrong: 9 minutes per mile would mean 6.67 miles in 60 minutes, so B would be closer to or past the destination, which contradicts the meeting point. 12 minutes per mile gives only 5 miles in 60 minutes, so B would not have reached the meeting point that is 6 miles away. 8 minutes per mile implies B walks as fast as A, in which case their meeting pattern and start time difference would not match the given situation. Common Pitfalls: Some learners forget to include the time A spends walking back 1 mile, thinking they meet exactly at the destination. Another mistake is to assume 7 miles is the distance B walks and then divide the time incorrectly. Always clearly mark distances on a mental or rough diagram and carefully account for each phase of the motion. Final Answer: B's speed is such that he takes 10 minutes per mile.

Question 2

Car P and Q started from point A and point B respectively running towards each other. Car P running at $4Z$ km/h and meets Q after 10 hours. After reaching point A, driver of car Q realized that car R (running towards point B) is $\left(\frac{3Y}{5}…

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Concept & Relative Speed: When two bodies move towards each other, relative speed is the sum of their speeds. When one chases another, it's the difference. Distance = Relative Speed × Time Step-by-Step Solution: • Given: P and Q meet in 10 hours. Speed of P = 4Z. Let Q's speed be V_q. Total distance AB = Y. • 10 × (4Z + V_q) = Y \implies 40Z + 10V_q = Y \implies 10V_q = Y - 40Z. • Q reaches A, meaning Q travels distance Y from B to A. • When Q is at A, Q starts chasing R. R is ((3Y/5) - 360) km ahead. • Speed of R = 2Z. Relative speed of Q chasing R = V_q - 2Z. • Q catches R in 6 hours: 6 × (V_q - 2Z) = (3Y/5) - 360. • Multiply by 5: 30(V_q - 2Z) = 3Y - 1800. • Substitute 3Y = 3(40Z + 10V_q) = 120Z + 30V_q. • 30V_q - 60Z = 120Z + 30V_q - 1800. • The term 30V_q cancels out perfectly: -60Z = 120Z - 1800 \implies 180Z = 1800 \implies Z = 10. • Speed of car P = 4Z = 4 × 10 = 40 km/hr. Exam Strategy & Shortcut: Recognize that the speed of Q (V_q) acts as an intermediate variable. By aligning the coefficients of Y in both equations, V_q completely cancels out, allowing for a direct solution for Z without needing Q's actual speed. Common Pitfall: Assuming Q meets P and *then immediately* chases R from the meeting point. The problem explicitly states Q chases R "After reaching point A", meaning Q completed the full distance Y. Final Answer: Therefore, the correct answer is 40 km/hr.

Question 3

Directions: Read the table carefully and answer the following question.  The table shows the data about distance (km) covered from one point to another point by the vehicle used between each point and speed of each vehicle (km/h). Note: (i) If one p…

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Concept & Table Interpretation: The problem requires extracting the direct travel distances from the given data table for specific origin-destination pairs. Difference = Distance(P 	o S) - Distance(Q 	o P) Step-by-Step Solution: 1. Identify Vasu's travel route: Vasu travels from point P to point S. 2. According to the table, reading the row for origin P and column for destination S, the direct distance is 780 km. 3. Identify Sanjeet's travel route: Sanjeet travels from point Q to point P. 4. According to the table, reading the row for origin Q and column for destination P, the direct distance is 540 km. 5. Calculate the difference between these two primary direct distances: 780 - 540 = 240. Exam Strategy & Shortcut: When questions use confusing terminology like "maximum distance" but provide options that strictly match the difference between direct table lookups, prioritize the direct values. Always double-check row vs. column mapping (Origin is row, Destination is column). Common Pitfall: A common mistake is assuming the distance from P to Q is the same as Q to P. In this specific table, P 	o Q = 360 km, but Q 	o P = 540 km. Always read the exact directional value. Final Answer: Therefore, the correct answer is 240.

Question 4

Directions: Read the table carefully and answer the following question.  The table shows the data about distance (km) covered from one point to another point by the vehicle used between each point and speed of each vehicle (km/h). Note: (i) If one p…

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Concept & Minimum Time Path: To find the minimum time required, we must calculate the time taken for all possible routes from P to R and select the smallest value. Time = \frac{Distance}{Speed} Step-by-Step Solution: 1. Direct Route (P 	o R): The table shows "", meaning no direct connection exists. 2. Route via Q (P 	o Q 	o R): - P 	o Q: Distance = 360 km. Vehicle at P = Train (60 km/h). Time = 360 / 60 = 6 hours. - Q 	o R: The table shows "" (no direct connection). This route is invalid. 3. Route via S (P 	o S 	o R): - P 	o S: Distance = 780 km. Vehicle at P = Train (60 km/h). Time = 780 / 60 = 13 hours. - S 	o R: Distance = 900 km. Vehicle at S = Bike (45 km/h). Time = 900 / 45 = 20 hours. - Total time for this route = 13 + 20 = 33 hours. 4. Since the only valid logical path with provided direct links to R from reachable nodes is via S, 33 hours is the minimum time. Exam Strategy & Shortcut: Quickly scan the destination column (R) for available incoming routes. Only S and P can potentially reach R, but P has no direct connection. Thus, any viable path must go through S. Calculating P 	o S 	o R directly saves time mapping out impossible routes. Common Pitfall: Applying the origin vehicle's speed to the entire journey. Note (i) specifies using the vehicle available at the starting point of the leg. For S 	o R, you must use the Bike (45 km/h), not the Train. Final Answer: Therefore, the correct answer is 33 hours.

Question 5

Directions: Read the table carefully and answer the following question.  The table shows the data about distance (km) covered from one point to another point by the vehicle used between each point and speed of each vehicle (km/h). Note: (i) If one p…

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Concept & Linear Equations: This problem combines distance interpretation with proportional algebra. We can set up a linear equation representing the total cost of the journey by assigning a variable to the base fare. Total Fare = (Distance_1 × Fare_1) + (Distance_2 × Fare_2) Step-by-Step Solution: 1. Define the variables: Let the fare per km from P to Q (which uses the Train) be x. 2. The fare per km from Q to S is 25% more than from P to Q, which makes it 1.25x. 3. Extract the distances from the table: - Distance P 	o Q = 360 km. - Distance Q 	o S = 600 km. 4. Formulate the equation for total fare: - Cost for P 	o Q = 360 × x = 360x - Cost for Q 	o S = 600 × 1.25x = 750x - Total Cost = 360x + 750x = 1110x 5. Solve for x: - 1110x = 19980 - x = (19980/1110) = (1998/111) = 18 6. The train fare per km (from P to Q) is ₹18. Exam Strategy & Shortcut: To calculate 600 × 1.25 quickly mentally, break it down: 1.25 = 1 + (1/4). So, 600 + (600 / 4) = 600 + 150 = 750. This avoids messy decimal multiplication. Common Pitfall: A common error is applying the 25% increase to the total distance or total cost rather than strictly to the per-km rate for the specific second leg of the trip. Final Answer: Therefore, the correct answer is ₹18.

Question 6

Directions: Read the table carefully and answer the following question.  The table shows the data about distance (km) covered from one point to another point by the vehicle used between each point and speed of each vehicle (km/h). Note: (i) If one p…

Accepted Answer

Concept & Shortest Path Time: We need to independently calculate the minimum time to travel from P to S and from R to S, then sum these two minimum values. Time = \frac{Distance}{Speed} Step-by-Step Solution: 1. Find minimum time for P 	o S: - Direct route: P 	o S = 780 km at 60 km/h (Train) = 780 / 60 = 13 hours. - Route via Q: P 	o Q 	o S. - P 	o Q: 360 km at 60 km/h = 6 hours. - Q 	o S: 600 km at 50 km/h = 12 hours. - Total = 6 + 12 = 18 hours. - Minimum time for P 	o S = 13 hours. 2. Find minimum time for R 	o S: - Direct route: R 	o S = "" (No direct connection). - Route via P: R 	o P 	o S. - R 	o P: 420 km at 40 km/h (Bus) = 420 / 40 = 10.5 hours. - P 	o S: 780 km at 60 km/h (Train) = 780 / 60 = 13 hours. - Total = 10.5 + 13 = 23.5 hours. - Route via Q: R 	o Q 	o S. - R 	o Q: 720 km at 40 km/h = 720 / 40 = 18 hours. - Q 	o S: 600 km at 50 km/h = 600 / 50 = 12 hours. - Total = 18 + 12 = 30 hours. - Minimum time for R 	o S = 23.5 hours. 3. Calculate the sum: - Sum = 13 hours + 23.5 hours = 36.5 hours. Exam Strategy & Shortcut: Whenever checking paths, start with the direct path. If indirect paths have distances that are vastly larger, you can often visually skip the full calculation. For instance, R 	o Q 	o S covers 720 + 600 = 1320 km at relatively slow speeds compared to 420 + 780 = 1200 km. Common Pitfall: A frequent error is misreading the direction of travel (e.g., using S 	o P distances instead of P 	o S). Always strictly read rows as origins and columns as destinations. Final Answer: Therefore, the correct answer is 36.5.

Question 7

The speeds of three cars are in the ratio 5 : 4 : 6. The ratio between the time taken by them to travel the same distance is

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Concept & Logic: Time and speed are inversely proportional when the distance is constant. Time \propto (1/Speed) Step-by-Step Solution: 1. The given ratio of speeds of the three cars is 5 : 4 : 6. 2. Because the distance is the same, the ratio of the time taken will be the inverse of their speeds: Time Ratio = (1/5) : (1/4) : (1/6) 3. To simplify this ratio into whole numbers, find the Least Common Multiple (LCM) of the denominators (5, 4, 6). LCM(5, 4, 6) = 60. 4. Multiply each term of the inverse ratio by 60: ((1/5) × 60) : ((1/4) × 60) : ((1/6) × 60) 5. Simplify to get the final integer ratio: 12 : 15 : 10 Exam Strategy & Shortcut: To find the inverse ratio of a : b : c, simply hide one number and multiply the other two. For the first term (hide 5): 4 × 6 = 24 For the second term (hide 4): 5 × 6 = 30 For the third term (hide 6): 5 × 4 = 20 Ratio: 24 : 30 : 20. Divide by 2 to get 12 : 15 : 10. Common Pitfall: A frequent error is simply reversing the order of the speeds to get the time ratio (e.g., 6 : 4 : 5). This only works for two variables, not three. Final Answer: Therefore, the correct answer is 12 : 15 : 10.

Question 8

A motorist travelled between two towns, which are 65 km apart, in 2 hours and 10 minutes. Find the speed in metres per minute.

Accepted Answer

Concept & Specific Unit Conversion: The problem asks for speed in a non-standard unit: metres per minute. Therefore, the most efficient approach is to convert the total distance into metres and the total time into minutes before dividing. Speed = \frac{Distance in metres}{Time in minutes} Step-by-Step Solution: • Given: Distance = 65 km, Time = 2 hours and 10 minutes. • Convert Distance: Since 1 km = 1000 m, Distance = 65 × 1000 = 65,000 metres. • Convert Time: Since 1 hour = 60 minutes, 2 hours = 120 minutes. Total Time = 120 + 10 = 130 minutes. • Calculate Speed: • Speed = (65000/130) • Cancel out the trailing zeros: (6500/13) • We know 13 × 5 = 65, so 13 × 500 = 6500. • Speed = 500 metres per minute. Exam Strategy & Shortcut: Convert exactly into the units requested by the prompt directly, rather than converting to standard m/sec first and then scaling up to m/min. This prevents multi-step conversion errors and keeps the arithmetic cleaner. Common Pitfall: A common error is converting the time to a pure hour decimal incorrectly (e.g., writing 2 hours 10 minutes as 2.1 hours instead of 2 (10/60) hours) or converting the final answer into meters per second out of habit. Final Answer: Therefore, the correct answer is 500.

Question 9

Which of the following trains is the fastest?

Accepted Answer

Concept & Unit Equivalency : To compare different speeds to see which is "fastest," you must convert all the given options into a single, common unit (e.g., meters per second) and compare their numerical values. km/hr to m/sec = × (5/18) m/min to m/sec = \div 60 Step-by-Step Solution: • Analyze Option A: The speed is already in m/sec. • Speed = 25 m/sec. • Analyze Option B: Convert 1500 m/min to m/sec. • Speed = \frac{1500 m}{60 sec} = 25 m/sec. • Analyze Option C: Convert 90 km/hr to m/sec. • Speed = 90 × (5/18) = 5 × 5 = 25 m/sec. • Conclusion: Options A, B, and C all represent the exact same speed (25 m/sec). Because they are equal, no single train is faster than the others. Exam Strategy & Shortcut: Convert everything to m/sec because it is the standard SI unit and usually yields clean integers. As soon as you see two options calculate out to the exact same value in a "which is greatest" question with a "None of these" option, you know the answer must be "None of these". Common Pitfall: Students might intuitively assume that the option with the largest raw number (1500 or 90) is the fastest without standardizing the units first. Final Answer: Therefore, the correct answer is None of these.

Question 10

A car covers a distance of 432 km at the speed of 48 km/hr. In how many hours will the car cover this distance?

Accepted Answer

Concept & Time Calculation: The fundamental formula relating distance, speed, and time is that time is equal to the total distance divided by the speed. Time = \frac{Distance}{Speed} Step-by-Step Solution: • Given: Distance = 432 km, Speed = 48 km/hr. • Calculation: Substitute the given values into the formula. • Time = (432/48) • Simplify the fraction by dividing the numerator and denominator by 12: 432 / 12 = 36 and 48 / 12 = 4. • Time = (36/4) = 9 hours. Exam Strategy & Shortcut: Use the unit digit strategy to quickly find the answer without full division. You need a number that, when multiplied by 48, ends in 2 (since the distance is 432). Looking at the options: 6 × 8 = 48 (ends in 8), 7 × 8 = 56 (ends in 6), 9 × 8 = 72 (ends in 2). Only 9 yields the correct unit digit. Common Pitfall: A common mistake is trying to perform long division mentally and making a calculation error. Relying on fraction simplification or the unit digit method prevents these arithmetic mistakes. Final Answer: Therefore, the correct answer is 9 hours.

Question 11

The average speed of a bus is one-third of the speed of a train. The train covers 1125 km in 15 hours. How much distance will the bus cover in 36 minutes?

Accepted Answer

Concept & Unit Conversion in Distance: When speeds and times are given in mixed units (hours vs minutes), they must be converted to a uniform metric before calculating final distances. Speed = \frac{Distance}{Time} Minutes to Hours = \frac{Minutes}{60} Step-by-Step Solution: • Find the speed of the train: • Train Speed = \frac{1125 km}{15 hours} = 75 km/hr. • Find the speed of the bus: The bus is one-third as fast as the train. • Bus Speed = (75/3) = 25 km/hr. • Convert the bus's travel time to hours: • 36 minutes = (36/60) hours = (3/5) hours. • Calculate the distance covered by the bus: • Distance = Speed × Time • Distance = 25 × (3/5) = 5 × 3 = 15 km. • Since 15 km is not present among options (a), (b), (c), or (d), the correct choice is "None of these". Exam Strategy & Shortcut: To quickly divide 1125 by 15, break it down: 15 × 100 = 1500. 1125 is 375 less than 1500. Since 15 × 25 = 375, the multiplier is 100 - 25 = 75. Mental math techniques like this save time compared to scratchpad long division. Common Pitfall: A major pitfall is failing to convert 36 minutes into hours. A student might mistakenly calculate Distance = 25 × 36 = 900 km, completely ignoring unit standardizations. Final Answer: Therefore, the correct answer is None of these.

Question 12

A star is $8.1 	imes 10^{13}$ km away from the earth. Suppose light travels at the speed of $3.0 	imes 10^5$ km per second. How long will it take the light from the star to reach the earth?

Accepted Answer

Concept & Time-Distance Formula: Time is calculated by dividing distance by speed. Pay attention to the units and convert seconds to hours if necessary. Time = \frac{Distance}{Speed} Step-by-Step Solution: • Given Distance = 8.1 × 10^{13} km. • Given Speed = 3.0 × 10^5 km/s. • Calculate time in seconds: Time = \frac{8.1 × 10^{13}}{3.0 × 10^5} = 2.7 × 10^8 seconds. • Convert seconds to hours (since 2.7 × 10^8 is not an option): Divide by 3600 (60 minutes × 60 seconds). • Time in hours = (2.7 × 10^8/3600) = (2700 × 10^5/36 × 10^2) = 75 × 10^3 = 7.5 × 10^4 hours. Exam Strategy & Shortcut: Work with powers of 10 separately. 8.1 / 3 = 2.7 and 10^{13} / 10^5 = 10^8. You immediately get 2.7 × 10^8 seconds. Recognizing 3600 seconds in an hour, 270/36 = 7.5, giving you the mantissa 7.5 for the hour conversion. Common Pitfall: Forgetting to convert the calculated time from seconds to hours, or miscalculating the exponents when dividing terms with powers of 10. Final Answer: Therefore, the correct answer is 7.5 × 10^4 hours.

Question 13

Deepa rides her bike at an average speed of 30 km/hr and reaches her destination in 6 hours. Hema covers the same distance in 4 hours. If Deepa increases her average speed by 10 km/hr and Hema increases her average speed by 5 km/hr, what would be th…

Accepted Answer

Concept & Time and Speed Variation: First establish the constant total distance using the initial conditions. Then, calculate the new speeds and the corresponding new times to find the time difference. Distance = Speed × Time Step-by-Step Solution: • Calculate Distance: Deepa travels at 30 km/hr for 6 hours. Distance = 30 × 6 = 180 km. • Calculate Hema's Initial Speed: Hema covers 180 km in 4 hours. Speed = (180/4) = 45 km/hr. • Calculate New Speeds: Deepa's new speed = 30 + 10 = 40 km/hr. Hema's new speed = 45 + 5 = 50 km/hr. • Calculate New Times: Deepa's new time = (180/40) = 4.5 hours. Hema's new time = (180/50) = 3.6 hours. • Find Time Difference: Difference = 4.5 - 3.6 = 0.9 hours. Convert to minutes: 0.9 × 60 = 54 minutes. Exam Strategy & Shortcut: Once the distance is found as 180, calculate the new times as fractions: 180/40 = 9/2 and 180/50 = 18/5. The difference is (9/2) - (18/5) = (45 - 36/10) = (9/10) hours. Multiply by 60 to directly get 54 minutes. Common Pitfall: Converting 0.9 hours to 90 minutes instead of correctly multiplying by 60 to get 54 minutes. Final Answer: Therefore, the correct answer is 54 minutes.

Question 14

A train travels at an average of 50 miles per hour for $2\frac{1}{2}$ hours and then travels at a speed of 70 miles per hour for $1\frac{1}{2}$ hours. How far did the train travel in the entire 4 hours?

Accepted Answer

Concept & Formula: Total distance traveled in a journey with multiple legs is simply the sum of the distances traveled in each leg. Distance = Speed × Time Total Distance = D_1 + D_2 Step-by-Step Solution: 1. Calculate Distance for the First Leg (D_1): • Speed = 50 miles per hour. • Time = 2(1/2) hours = 2.5 hours. • D_1 = 50 × 2.5 = 125 miles. 2. Calculate Distance for the Second Leg (D_2): • Speed = 70 miles per hour. • Time = 1(1/2) hours = 1.5 hours. • D_2 = 70 × 1.5 = 105 miles. 3. Calculate Total Distance: • Total Distance = D_1 + D_2 • Total Distance = 125 + 105 = 230 miles. Exam Strategy & Shortcut: You can mentally calculate the parts: First leg: 50 × 2 = 100, plus half of 50 (25) = 125. Second leg: 70 × 1 = 70, plus half of 70 (35) = 105. 125 + 105 = 230. This skips writing down decimal conversions. Common Pitfall: A common mistake is trying to average the speeds first (50 and 70) and then multiplying by total time. Since the times driven at these speeds are not equal, a simple average of the speeds will yield an incorrect total distance. Final Answer: Therefore, the correct answer is 230 miles.

Question 15

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 16

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 17

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 18

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 19

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 20

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.

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