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}…

Accepted Answer

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…

Accepted Answer

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

$A$ is travelling at 72 km per hour on a highway while $B$ is travelling at a speed of 25 metres per second. What is the difference in their speeds in metres per second?

Accepted Answer

Concept & Unit Standardization: Before comparing or finding the difference between two speeds, they must be expressed in the exact same units. We will convert km/hr to m/sec using the standard multiplier. Speed in m/sec = Speed in km/hr × (5/18) Step-by-Step Solution: • Given: Speed of A = 72 km/hr, Speed of B = 25 m/sec. • Convert A's speed: • Speed of A in m/sec = 72 × (5/18) • = 4 × 5 = 20 m/sec. • Calculate the difference: • Difference = Speed of B - Speed of A • Difference = 25 - 20 = 5 m/sec. Exam Strategy & Shortcut: Recognize common km/hr to m/sec conversions instantly. 18 km/hr = 5 m/sec, 36 km/hr = 10 m/sec, 54 km/hr = 15 m/sec, 72 km/hr = 20 m/sec. Memorizing these multiples of 18 allows you to bypass the calculation step entirely and immediately see the difference is 25 - 20 = 5. Common Pitfall: Subtracting the raw numbers (72 - 25) without noticing the mixed units (km/hr vs m/sec). Always standardize units before performing arithmetic operations. Final Answer: Therefore, the correct answer is 5 m/sec.

Question 9

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 10

In track meets both 100 yards and 100 metres are used as distances. By how many metres is 100 metres longer than 100 yards?

Accepted Answer

Concept & Imperial to Metric Conversion: To solve this, you need to know the standard conversion rate between yards and meters. One yard is defined internationally as exactly 0.9144 meters. 1 yard = 0.9144 metres Step-by-Step Solution: • Find the metric equivalent of 100 yards: • 100 yards = 100 × 0.9144 metres • = 91.44 metres • Calculate the difference: • Difference = 100 metres - 91.44 metres • = 8.56 metres Exam Strategy & Shortcut: If you don't remember the exact 0.9144 conversion, remember the rough approximation that a yard is about 90% of a meter (or 1 meter is about 1.1 yards). 100 meters minus roughly 91 meters leaves around 9 meters. Option (d) 8.56 m is the only answer in that logical ballpark. Common Pitfall: Guessing that a yard and a meter are practically identical and choosing 1 m, or misplacing the decimal point during subtraction (choosing 0.856 m). Final Answer: Therefore, the correct answer is 8.56 m.

Question 11

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 12

A person crosses a 600 m long street in 5 minutes. What is his speed in km per hour?

Accepted Answer

Concept & Multi-Step Conversion: This problem requires calculating speed and then standardizing the units. First, calculate the speed in meters per second (m/sec), then convert that result into kilometers per hour (km/hr). Speed = \frac{Distance}{Time} Speed in km/hr = Speed in m/sec × (18/5) Step-by-Step Solution: • Given: Distance = 600 m, Time = 5 minutes. • Convert Time to seconds: • 5 minutes = 5 × 60 = 300 seconds. • Calculate Speed in m/sec: • Speed = \frac{600 m}{300 s} = 2 m/sec. • Convert Speed to km/hr: • Speed in km/hr = 2 × (18/5) • = (36/5) • = 7.2 km/hr. Exam Strategy & Shortcut: Alternatively, convert the original inputs directly to km and hours. Distance = 0.6 km. Time = 5/60 hours = 1/12 hours. Speed = 0.6 / (1/12) = 0.6 × 12 = 7.2 km/hr. This avoids the 18/5 multiplier entirely and is often faster if you are comfortable with fractions. Common Pitfall: Using the 5 minutes directly as seconds (calculating 600/5 = 120 m/s) resulting in an absurdly high and incorrect final km/hr speed. Always ensure your time base is in seconds before using the 18/5 conversion factor. Final Answer: Therefore, the correct answer is 7.2.

Question 13

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 14

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 15

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 16

Akash leaves home for school which is 12 km from his house. After the school, he goes to his club which is 7 km from his school. If his house, school and club all fall in a line, then what is the minimum distance he has to travel to get back home?

Accepted Answer

Concept & 1D Coordinate Geometry: When points lie on a single line, the distance between them is the absolute difference of their coordinates. To find the minimum return distance, arrange the points to minimize the span from the origin (house). Step-by-Step Solution: • Let the house be at position 0 on a number line. • The school is at position 12 (since it is 12 km from the house). • The club is 7 km from the school, meaning its position can be either 12 + 7 = 19 or 12 - 7 = 5. • A position of 19 implies the club is further down the line, making the return trip back to 0 equal to 19 km. • A position of 5 implies the club lies directly between the house and the school, making the return trip equal to 5 km. • The minimum distance to travel back home from the club is therefore 5 km. Exam Strategy & Shortcut: Simply subtract the two given distances: 12 - 7 = 5. Since "minimum" is requested, you must assume the club lies directly on the return path between the school and the house. Common Pitfall: Adding the distances to get 19 km, incorrectly assuming the school is always between the house and the club without considering other linear arrangements. Final Answer: Therefore, the correct answer is 5 km.

Question 17

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 18

A boy is running at a speed of $p$ kmph to cover a distance of 1 km. But, due to the slippery ground, his speed is reduced by $q$ kmph ($p > q$). If he takes $r$ hours to cover the distance, then

Accepted Answer

Concept & Speed-Time Relation: The fundamental algebraic relationship between distance, time, and speed states that speed equals distance divided by time. Set up the equation using the effective speed. Speed = \frac{Distance}{Time} Step-by-Step Solution: • Determine Effective Speed: The original speed is p, but it is reduced by q. Therefore, the new effective speed is (p - q) kmph. • Set Up Equation: The distance to cover is given as 1 km, and the time taken is r hours. • Substitute these into the formula: r = (1/p - q). • Rearrange: Multiply both sides by (p - q) and divide by r (or take the reciprocal of both sides) to match the given options. (1/r) = p - q Exam Strategy & Shortcut: Effective Speed = Distance / Time. The effective speed is directly (p - q) and Distance/Time is 1/r. Equating them gives (p - q) = 1/r instantly. Common Pitfall: Selecting option (b) r = p - q, carelessly confusing the position of the time variable with the reciprocal of time in the formula. Final Answer: Therefore, the correct answer is (1/r) = p - q.

Question 19

A truck covers a distance of 550 metres in 1 minute whereas a bus covers a distance of 33 kms in 45 minutes. The ratio of their speeds is

Accepted Answer

Concept & Logic: To find the ratio of speeds, both speeds must be expressed in the exact same units (e.g., meters per minute or kilometers per hour). Speed = (Distance/Time) Step-by-Step Solution: 1. Calculate Truck's Speed: • Distance = 550 metres • Time = 1 minute • Speed of truck = 550 m/min. 2. Calculate Bus's Speed: • Distance = 33 kms = 33,000 metres • Time = 45 minutes • Speed of bus = (33000/45) m/min. • Simplifying: (33000 \div 15/45 \div 15) = (2200/3) = 733.33 m/min. 3. Find the Ratio: • Ratio = Speed of Truck : Speed of Bus • Ratio = 550 : (2200/3) • Multiply both sides by 3 to remove the fraction: • Ratio = 1650 : 2200 • Divide both sides by 500 (or simplify step-by-step): • Divide by 550: 1650 \div 550 = 3 and 2200 \div 550 = 4. • Ratio = 3 : 4. Exam Strategy & Shortcut: Instead of fully solving the division for the bus, keep it as a fraction. Ratio = 550 : (33000/45). Divide both sides by 550: Ratio = 1 : (60/45) Simplify the fraction (60/45) to (4/3). Ratio = 1 : (4/3) \implies 3 : 4. Common Pitfall: Comparing speeds without converting them to the same unit first. If you compare 550 directly with (33/45), you will get a wildly incorrect ratio. Final Answer: Therefore, the correct answer is 3 : 4.

Question 20

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.

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