Question 1

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 2

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 3

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 4

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.

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