Question 1

How many times in a day, the hands of a clock are straight?

Accepted Answer

Concept & Logic: The hands of a clock are "straight" when they are coincident (angle = 0^\circ) OR when they are pointing in opposite directions (angle = 180^\circ). Step-by-Step Solution: 1. Coincident (0 degrees): The hands coincide 11 times in 12 hours, which means 22 times in 24 hours. (They miss a coincidence between 11 and 1, meeting only at 12). 2. Opposite (180 degrees): The hands are in opposite directions 11 times in 12 hours, which means 22 times in 24 hours. (They miss being opposite between 5 and 7, being opposite only exactly at 6). 3. Total Straight Line Occurrences: Total = (Coinciding) + (Opposite) Total = 22 + 22 = 44 times. Exam Strategy & Shortcut: Memorize the standard daily occurrences for clock hands: - Coincide: 22 times - Opposite: 22 times - Straight line (either of the above): 22 + 22 = 44 times - Right angles: 44 times Common Pitfall: A frequent mistake is interpreting "straight" as ONLY pointing in opposite directions (180^\circ), leading to an answer of 22. "Straight" implies forming a straight line, which includes the overlapping state. Final Answer: Therefore, the correct answer is 44.

Question 2

How many times in a day, are the hands of a clock in straight line but opposite in direction?

Accepted Answer

Concept & Logic: The question asks for the number of times the hands form a 180^\circ angle (pointing in opposite directions). In a 12-hour period, this happens 11 times. The loss of one occurrence happens between 5 o'clock and 7 o'clock, where they are opposite only exactly at 6:00. Step-by-Step Solution: 1. Half-day Analysis: In 12 hours, the hands point in opposite directions 11 times. 2. Full-day Analysis: A full day has 24 hours, which is two 12-hour cycles. Number of times in 24 hours = 11 × 2 = 22 times. Exam Strategy & Shortcut: Direct fact recall: - Straight but opposite (180^\circ) = 22 times/day. - Coincide (0^\circ) = 22 times/day. Common Pitfall: Selecting 24 by assuming it happens exactly once every hour without accounting for the shared occurrence at 6:00. Another pitfall is confusing this with just "straight line" which would be 44. The phrase "but opposite in direction" restricts it to 22. Final Answer: Therefore, the correct answer is 22.

Question 3

A wall-clock takes 9 seconds in tringing at $9O'$ clock. The time, it will take in tringing at $11O'$ clock, is

Accepted Answer

Concept & Logic: When a clock rings a certain number of times, the time taken is measured by the *intervals* between the rings, not the rings themselves. To ring n times, there are (n-1) intervals. Assume each interval takes the same amount of time. Step-by-Step Solution: 1. Analyze 9 O'clock: At 9 O'clock, the clock rings 9 times. Number of intervals = 9 - 1 = 8 intervals. Time taken for 8 intervals = 9 seconds. Time taken for 1 interval = (9/8) seconds. 2. Analyze 11 O'clock: At 11 O'clock, the clock rings 11 times. Number of intervals = 11 - 1 = 10 intervals. 3. Calculate Total Time: Time for 10 intervals = 10 × (Time for 1 interval) Time = 10 × (9/8) Time = (90/8) Time = (45/4) = 11.25 seconds Exam Strategy & Shortcut: Set up a direct proportion using intervals. \frac{Time_1}{Intervals_1} = \frac{Time_2}{Intervals_2} (9/8) = (x/10) x = (90/8) = 11.25 Common Pitfall: The most common mistake is dividing the time by the number of rings (e.g., 9/9 = 1 second per ring) and then multiplying by the new number of rings (1 × 11 = 11 seconds). Always count the gaps (intervals) between the strikes. Final Answer: Therefore, the correct answer is 11.25 seconds.

Question 4

In every 30 minutes the time of a watch increases by 3 minutes. After showing the correct time at 5 a.m; what time will the watch show after 6 hours?

Accepted Answer

Concept & Rate of Gain: The watch is running fast. We need to find the total time gained over the given period and add it to the actual elapsed time. Gain per given interval: 3 minutes gained per 30 minutes. Step-by-Step Solution: 1. Determine the rate of gain: In 30 minutes, the watch gains 3 minutes. Therefore, in 60 minutes (1 hour), the watch gains 3 × 2 = 6 minutes. 2. Calculate total gain for the period: The elapsed time is 6 hours. Total time gained in 6 hours = 6 hours × 6 minutes/hour = 36 minutes. 3. Calculate the displayed time: Actual time after 6 hours from 5:00 a.m. is 5 + 6 = 11:00 a.m. The watch will show the actual time plus the gained time. Time shown = 11:00 a.m. + 36 minutes = 11:36 a.m. Exam Strategy & Shortcut: Find the hourly gain (6 mins/hr) and multiply by total hours (6 hrs) to get 36 mins. Add this directly to the 6 hours elapsed from 5 a.m. (11:00 a.m. + 0:36 = 11:36 a.m.). Common Pitfall: Some students might subtract the gained time instead of adding it, or incorrectly calculate the rate (e.g., assuming it gains 3 mins an hour). Final Answer: Therefore, the correct answer is 11 : 36 a.m.

Question 5

A watch becomes fast by 5 minutes everyday. By what percent does it become fast?

Accepted Answer

Concept & Percentage: To find the percentage by which the watch becomes fast, we must compare the time gained to the total time in a standard day. Percentage = ( \frac{Time Gained}{Total Standard Time} ) × 100 Step-by-Step Solution: 1. Find total minutes in a day: 1 day = 24 hours. 24 hours = 24 × 60 minutes = 1440 minutes. 2. Identify the gained time: The watch gains 5 minutes in this 1440-minute period. 3. Calculate the percentage: Percentage = ( (5/1440) ) × 100 Percentage = (500/1440) Simplify the fraction by dividing numerator and denominator by 10: Percentage = (50/144)% Exam Strategy & Shortcut: Instead of fully reducing the fraction to lowest terms (which would be (25/72)), glance at the options first. The option (d) leaves the fraction partially unreduced at (50/144), saving you an extra calculation step. Common Pitfall: A common mistake is using 24 hours as the denominator without converting the numerator to hours, resulting in (5/24)%. Units must match! Final Answer: Therefore, the correct answer is (50/144)%.

Question 6

A clock goes slow from midnight by 5 minutes at the end of the first hour, by 10 minutes at the end of the second hour, by 15 minutes at the end of the third hour and so on. What will be the time by this clock after 6 hours?

Accepted Answer

Concept & Arithmetic Progression: The clock is losing time in a linear pattern that corresponds directly to the number of elapsed hours. At the end of hour n, the clock is slow by 5 × n minutes. Step-by-Step Solution: 1. Identify the pattern of delay: - End of 1st hour: 5 minutes slow - End of 2nd hour: 10 minutes slow - End of 3rd hour: 15 minutes slow The delay at the end of the n-th hour is 5n minutes. 2. Calculate the delay after 6 hours: Let n = 6. Total delay = 5 × 6 = 30 minutes. 3. Determine the displayed time: The actual elapsed time from midnight is 6 hours, meaning the true time is 6:00 a.m. Since the clock is 30 minutes slow, we subtract this delay from the true time. Displayed time = 6:00 a.m. - 30 minutes = 5:30 a.m. Exam Strategy & Shortcut: Recognize immediately that the delay is simply 5 × hours. For 6 hours, it is 30 minutes slow. 6:00 a.m. actual time minus 30 minutes gives 5:30 a.m. Common Pitfall: Confusing "goes slow" with "goes fast" and adding the time (resulting in 6:30 a.m.) instead of subtracting it. Final Answer: Therefore, the correct answer is 5.30 a.m.

Question 7

A watch goes fast by 15 minutes compared to the right time everyday. If it is corrected and set to the standard time at 12O' clock at noon, which of the following will be the time shown by it at 4 : 00 a.m. in the morning?

Accepted Answer

Concept & Proportionality: The watch gains time uniformly over a 24-hour cycle. We need to calculate the proportional gain for the exact number of hours elapsed between the starting standard time and the target actual time. Step-by-Step Solution: 1. Determine the total elapsed hours: From 12:00 noon to 12:00 midnight is 12 hours. From 12:00 midnight to 4:00 a.m. is 4 hours. Total time elapsed = 12 + 4 = 16 hours. 2. Calculate the proportional time gained: The watch gains 15 minutes in 24 hours. Gain in 1 hour = (15/24) minutes = (5/8) minutes. Gain in 16 hours = 16 × ((5/8)) minutes. 16 × (5/8) = 2 × 5 = 10 minutes. 3. Determine the shown time: The actual time is 4:00 a.m. Because the watch is fast, it will show a time ahead of the actual time. Time shown = 4:00 a.m. + 10 minutes = 4:10 a.m. Exam Strategy & Shortcut: Set up a quick ratio: \frac{Gain}{Actual Hours}. (15/24) = (x/16). x = (15 × 16/24) = 10 minutes. Add to 4:00 to get 4:10. Common Pitfall: A standard trap is applying the full 15-minute gain regardless of the elapsed time, leading to option (c) 4:15 a.m. Always scale the gain to the specific time window. Final Answer: Therefore, the correct answer is 4 : 10 a.m.

Question 8

It is between 3 P.M. and 4 P.M. and the distance between the hour and the minute hand of clock is 18 minute spaces. What time does the clock show?

Accepted Answer

Concept & Relative Speed: In a clock, the minute hand moves faster than the hour hand. The minute hand travels 60 minute spaces in 60 minutes. The hour hand travels 5 minute spaces in 60 minutes. Therefore, the minute hand gains 60 - 5 = 55 minute spaces on the hour hand in 60 actual minutes. Time taken = (60/55) × (Minute spaces to be gained) Step-by-Step Solution: 1. Analyze the initial position (at 3:00 P.M.): At exactly 3:00, the hour hand is at 3 and the minute hand is at 12. The minute hand is 5 × 3 = 15 minute spaces *behind* the hour hand. 2. Determine required relative movement: The problem states the hands are 18 minute spaces apart. Case 1: Minute hand is 18 spaces *behind*. (Not possible, as it starts 15 behind and moves faster, so it only gets closer until they meet). Case 2: Minute hand is 18 spaces *ahead*. To be 18 spaces ahead, it must first cover the 15-space gap to catch up to the hour hand, and then travel an additional 18 spaces ahead of it. Total minute spaces to be gained = 15 + 18 = 33 spaces. 3. Calculate the time taken: Using the relative speed formula: Time = ((60/55)) × 33 Time = ((12/11)) × 33 Time = 12 × 3 = 36 minutes 4. Final Time: 36 minutes past 3:00 P.M. is 3:36 P.M. Exam Strategy & Shortcut: Use the fixed multiplier (12/11) for all "space gaining" problems. Spaces to gain = Initial gap (15) + Desired gap (18) = 33. Time = (12/11) × 33 = 36. Common Pitfall: Forgetting to account for the starting gap at 3:00. Students sometimes just calculate time for 18 spaces, yielding an incorrect answer. Final Answer: Therefore, the correct answer is 3.36 P.M.

Question 9

At what time, in minutes, between 3 o'clock and 4 o'clock, both the needles will coincide each other?

Accepted Answer

Concept & Clock Angles: When the hands of a clock coincide, the angle between them is 0^\circ. The angle 	heta between the hour and minute hand is calculated using the formula: 	heta = |30H - 5.5M| Step-by-Step Solution: • Given: The time is between 3 and 4 o'clock, so H = 3. The angle 	heta = 0. • Substitute into the formula: 0 = |30(3) - 5.5M| • Simplify: 90 - 5.5M = 0 • Solve for M: 5.5M = 90 • Convert decimal to fraction: (11/2)M = 90 • Calculate M: M = (180/11) = 16 (4/11) Exam Strategy & Shortcut: To find the time when hands overlap between H and (H+1) o'clock, simply multiply the starting hour H by (60/11). Shortcut: 3 × (60/11) = (180/11) = 16 (4/11) minutes past 3. Common Pitfall: Misinterpreting the fraction conversion. Sometimes students calculate 90 / 5.5 incorrectly by forgetting to move the decimal point, leading to arithmetic errors. Final Answer: Therefore, the correct answer is 16 (4/11) ' past.

Question 10

At what time between 7 and 8 o'clock will the hands of a clock be in the same straight line but, not together?

Accepted Answer

Concept & Opposite Directions: "Straight line but not together" means the hands are pointing in exactly opposite directions, creating an angle of 180^\circ. 	heta = |30H - 5.5M| Step-by-Step Solution: • Given: H = 7 and 	heta = 180^\circ. • Plug in the values: 180 = |30(7) - 5.5M| • Evaluate the absolute value: 180 = |210 - 5.5M| • Since 210 is greater than 180, we use: 210 - 5.5M = 180 • Rearrange to solve for M: 5.5M = 30 • Convert decimal to fraction: (11/2)M = 30 • Solve for M: M = (60/11) = 5 (5/11) minutes. Exam Strategy & Shortcut: To find when hands are 180^\circ apart, identify the opposite hour on the clock face. The opposite of 7 is 1 (which represents 5 minutes). Multiply this minute equivalent by (12/11). Shortcut: 5 × (12/11) = (60/11) = 5 (5/11) min. Common Pitfall: Mistakenly selecting "5 min. past 7" by just looking at the opposite number on the dial (1) and forgetting that the hour hand advances slightly past 7 during those 5 minutes. Final Answer: Therefore, the correct answer is 5 (5/11) min. past 7.

Question 11

At what time between 4 and 5 o'clock will the hands of a watch point in opposite directions?

Accepted Answer

Concept & Clock Opposites: Hands pointing in opposite directions indicate an angle of 180^\circ. We can find the exact minute using the standard clock angle formula or the relative speed multiplier. 	heta = |30H - 5.5M| Step-by-Step Solution: • Given: H = 4 and 	heta = 180^\circ. • Apply formula: 180 = |30(4) - 5.5M| • Simplify: 180 = |120 - 5.5M| • Since the minute hand must pass the hour hand to be 180^\circ ahead: 5.5M - 120 = 180 • Solve for 5.5M: 5.5M = 300 • Substitute (11/2) for 5.5: (11/2)M = 300 • Solve for M: M = (600/11) = 54 (6/11) minutes. Exam Strategy & Shortcut: Find the hour directly opposite to 4 on the clock face, which is 10. The 10 mark represents 50 minutes. Multiply this by (12/11): 50 × (12/11) = (600/11) = 54 (6/11) minutes past 4. Common Pitfall: Picking a simpler minute mark like 50 minutes (the opposite number 10) without accounting for the relative movement of the hour hand. Final Answer: Therefore, the correct answer is 54 (6/11) min. past 4.

Question 12

At what time between 5.30 and 6 will the hands of a clock be at right angles?

Accepted Answer

Concept & Right Angles: A right angle between the clock hands means 	heta = 90^\circ. This happens twice every hour. Since we are looking for a time between 5:30 and 6:00, the minute hand must be ahead of the hour hand. 	heta = |30H - 5.5M| Step-by-Step Solution: • Given: H = 5 and 	heta = 90^\circ. We need M > 30. • Apply formula: 90 = |30(5) - 5.5M| \Rightarrow 90 = |150 - 5.5M| • Set up the correct equation for when the minute hand is ahead: 5.5M - 150 = 90 • Add 150 to both sides: 5.5M = 240 • Use fraction for 5.5: (11/2)M = 240 • Multiply and divide: M = (480/11) • Convert to mixed fraction: M = 43 (7/11) minutes. Exam Strategy & Shortcut: For right angles, the minute spaces between hands must be 15. At 5 o'clock, the hour hand is at 25 minute spaces. To be at a right angle after 5:30, add 15 spaces: 25 + 15 = 40 minute spaces. Multiply by the constant (12/11): 40 × (12/11) = (480/11) = 43 (7/11). Common Pitfall: Calculating the first right angle (before 5:30) which happens at 25 - 15 = 10 minute spaces (10 × (12/11) = 10 (10/11) min past 5), ignoring the specific condition "between 5.30 and 6". Final Answer: Therefore, the correct answer is 43 (7/11) min. past 5.

Question 13

Imagine that your watch was correct at noon, but then it began to lose 30 minutes each hour. It now shows 4 p.m. but it stopped 5 hours ago. What is the correct time now?

Accepted Answer

Concept & Faulty Clocks: When a clock loses time consistently, it operates at a slower rate compared to real time. We can establish a ratio of "Clock Time" to "Real Time" to determine how much actual time has passed. Real Time Elapsed = \frac{Clock Time Elapsed}{Clock Speed Rate} Step-by-Step Solution: • Given: The watch loses 30 minutes every hour. This means for every 1 hour (60 mins) of real time, the watch advances only 30 minutes. • Clock Speed Rate: The watch runs at half speed (0.5 hours on watch = 1 hour real time). • Clock Time Elapsed: From noon (12:00) to 4 p.m. is 4 hours shown on the watch. • Real Time Elapsed: To show 4 hours, the real time passed must be 4 × 2 = 8 hours. • The watch stopped at 8 p.m. real time. • It stopped 5 hours ago. • Current Real Time: 8 p.m. + 5 hours = 1 a.m. Exam Strategy & Shortcut: Think of it logically: If it loses half an hour every hour, it takes 2 real hours to show 1 hour of progress. 4 hours of progress on the face = 8 hours of real time. 12:00 PM + 8 hours + 5 hours = 1:00 AM. Common Pitfall: Adding the 5 hours to the 4 p.m. shown on the broken clock (getting 9 p.m.) without converting the 4 hours of "broken clock time" into actual elapsed real time first. Final Answer: Therefore, the correct answer is 1 a.m..

Question 14

I have two watches with a 12 hour cycle. One of them gains one minute a day and the other loses $1 \frac{1}{2}$ minutes per day. If I set them both at the correct time, how long will it be before they again tell the correct time together?

Accepted Answer

Concept & Faulty Clocks & Cycles: For a standard 12-hour watch to show the *correct* time again after drifting, it must gain or lose exactly 12 hours (720 minutes). To find when two independent watches show the correct time simultaneously, find the Least Common Multiple (LCM) of their individual correction cycles. Step-by-Step Solution: • Given: 12-hour cycle clocks. Watch 1 gains 1 min/day. Watch 2 loses 1.5 min/day. • To show the correct time again, a watch must accumulate a drift of a full 12 hours (which is 12 × 60 = 720 minutes). • Time for Watch 1 to gain 720 minutes: 720 / 1 min/day = 720 days. • Time for Watch 2 to lose 720 minutes: 720 / 1.5 min/day = 480 days. • To show correct time together, we need the Least Common Multiple (LCM) of their individual cycles (720 days and 480 days). • LCM of 720 and 480: 720 = 240 × 3 480 = 240 × 2 LCM = 240 × 3 × 2 = 1440. Exam Strategy & Shortcut: Determine how many days it takes for each watch to drift by 720 minutes. Watch A takes 720 days. Watch B takes 480 days. Find the LCM of (720, 480) immediately, which is 1440. Common Pitfall: Calculating when they show the *same* time rather than the *correct* time together. If they just needed to show the same time, you would find when their relative drift is 12 hours. Final Answer: Therefore, the correct answer is 1440 days.

Question 15

A watch which gains 5 seconds in 3 minutes was set right at 7 a.m. In the afternoon of the same day, when the watch indicated quarter past 4 o'clock, the true time is

Accepted Answer

Concept & Faulty Clocks Proportions: Establish a ratio between the "incorrect time" and "true time". Use this multiplier on the total elapsed incorrect hours to find the true elapsed hours. True Time Elapsed = Incorrect Elapsed × ( \frac{True Interval}{Incorrect Interval} ) Step-by-Step Solution: • Given: Watch gains 5 seconds every 3 minutes. Set correct at 7 a.m. Current false time = 4:15 p.m. • Calculate the ratio of incorrect time to correct time: 3 minutes of correct time = 3 min 5 sec of incorrect time. • Convert incorrect interval to minutes: 3 + (5/60) = 3 (1/12) = (37/12) minutes. • Total incorrect time elapsed from 7 a.m. to 4:15 p.m. is 9 hours and 15 minutes = 9 (1/4) hours = (37/4) hours. • Set up ratio to find true elapsed hours: True Hours = (37/4) × (3/\frac{37){12}} = (37/4) × (36/37) . • Simplify the calculation: True Hours = (36/4) = 9 hours exactly. • True time = 7 a.m. + 9 hours = 4 p.m. Exam Strategy & Shortcut: Notice the clean fractions. (37/12) mins incorrect = 3 mins true. The total incorrect time is (37/4) hours. The 37s cancel out perfectly when multiplying by the inverse ratio (36/37), immediately yielding 9 true hours. 7 a.m. + 9 = 4 p.m. Common Pitfall: Trying to subtract the accumulated 5-second intervals linearly based on the incorrect time elapsed, rather than setting up an exact proportional ratio. Final Answer: Therefore, the correct answer is 4 p.m..

Question 16

How much does a watch losse per day if its hands coincide every 64 minutes?

Accepted Answer

Concept & Coincidence Interval: In a correct clock, the hands coincide every (720/11) minutes (or 65 (5/11) minutes). If a clock's hands coincide every x minutes, the total time gained or lost in a day (24 hours) is calculated by comparing this interval to the standard interval over the course of a day (1440 minutes). Gain/Loss in a day = ( (720/11) - x ) × ( (24 × 60/x) ) minutes Step-by-Step Solution: • Given: The interval x = 64 minutes. Since 64 < 65 (5/11), the clock is actually running fast (gaining time), despite the question's wording ("losse"). We calculate the magnitude of this shift. • Substitute x = 64 into the formula: Gain/Loss = ( (720/11) - 64 ) × ( (1440/64) ) • Simplify the first bracket: (720 - (64 × 11)/11) = (720 - 704/11) = (16/11) • Simplify the second bracket: (1440/64) = (45/2) • Multiply the two parts: (16/11) × (45/2) = (8 × 45/11) = (360/11) • Convert to a mixed fraction: (360/11) = 32 (8/11) minutes. Exam Strategy & Shortcut: Memorize the true coincidence time: 65 (5/11) minutes. The difference for one coincidence is 1 (5/11) = (16/11) minutes. Since it coincides every 64 minutes, in a full day (1440 minutes), it coincides 1440 / 64 = 45/2 times. Multiply the difference per coincidence by the number of coincidences: (16/11) × (45/2) = (360/11) = 32 (8/11). Common Pitfall: Confusing whether the clock gains or loses. If the interval is less than 65 (5/11) mins, the clock is fast (gaining). If it's more, it's slow (losing). Also, arithmetic errors when subtracting 64 from (720/11). Final Answer: Therefore, the correct answer is 32 (8/11) minutes.

Question 17

At 8:30, the hour hand and the minute hand of clock form an angle of

Accepted Answer

Concept & Clock Hand Angles: The angle 	heta formed between the hands of a clock at H hours and M minutes is calculated using the relative speeds of the minute (6^\circ/min) and hour (0.5^\circ/min) hands. 	heta = |30H - 5.5M| Step-by-Step Solution: • Given: Hours (H) = 8, Minutes (M) = 30. • Substitute the values into the angle formula: 	heta = |30(8) - 5.5(30)| • Evaluate the individual parts: 30 × 8 = 240 5.5 × 30 = 165 • Subtract to find the absolute angle: 	heta = |240 - 165| = 75^\circ. Exam Strategy & Shortcut: Use the visualization method: At 8:30, the minute hand points exactly at 6 (180^\circ from 12). The hour hand is halfway between 8 and 9. The angle from 6 to 8 is two 5-minute spaces (2 × 30^\circ = 60^\circ). The hour hand has moved exactly half an hour (30 × 0.5^\circ = 15^\circ) past the 8. Total angle = 60^\circ + 15^\circ = 75^\circ. Common Pitfall: Assuming the hour hand is exactly on the 8, which would incorrectly give an angle of 60^\circ (the angle between 6 and 8). Final Answer: Therefore, the correct answer is 75^\circ.

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