A clock is set right at 8:00 AM in the morning. The minute hand and the hour hand of the clock come together after every 65 minutes. What is the actual time when the clock shows 8:00 PM, approximated to the nearest minutes?

Correct — C, 7:55 PM. On a clock that keeps correct time the hands coincide not every hour but every 720/11 minutes, which is 65 and 5/11 minutes — the minute hand gains 55 minutes of dial on the hour hand in every hour, so it needs a little over 65 minutes to make up a full lap. This clock's hands come together every 65 minutes of true time, meaning it performs in 65 real minutes what a correct clock takes 65 and 5/11 real minutes to do. Its dial therefore advances faster than reality in the ratio (720/11) : 65 = 144 : 143. From 8:00 AM to 8:00 PM the dial shows 720 minutes, so the true elapsed time is 720 x 143/144 = 715 minutes, that is 11 hours 55 minutes. Adding that to 8:00 AM gives 7:55 PM.

• (a)8:05 PM — This is the answer for a clock that loses time. A slow clock shows less than has passed, so the real time is later than the dial; this clock gains, so the real time is earlier.
• (b)8:00 PM — Would mean the clock is accurate. It is not: 65 minutes between coincidences is 5/11 of a minute short of the correct interval, and over twelve hours that accumulates to five minutes.
• (d)8:02 PM — Right direction of error only if the clock were slow, and the wrong size in any case. The gain works out at 720/143, about 5.03 minutes over the twelve dial hours.

The two hands coincide eleven times in twelve hours, not twelve, because the hour hand is itself moving. The minute hand travels 360 degrees an hour and the hour hand 30, so the minute hand gains 330 degrees an hour and needs 360/330 of an hour — 65 and 5/11 minutes — to lap it. That number is the yardstick for every faulty-clock question: a clock whose hands meet more often than that is fast, and one whose hands meet less often is slow.

The direction of the error is what candidates get wrong, and the option list is built around exactly that. Set it out as a ratio of dial time to true time. This clock packs one coincidence into 65 true minutes where a good clock needs 65 and 5/11, so its dial gets through 65 and 5/11 minutes' worth of work in 65 real minutes — dial runs ahead. A dial that runs ahead reaches 8:00 PM before the world does, so the real time is earlier, and 715 minutes after 8:00 AM is 7:55 PM.

• On a correct clock the hands coincide every 720/11 minutes, which is 65 and 5/11 minutes.
• The hands coincide 11 times in 12 hours and 22 times in a day.
• If the coincidence interval is shorter than 65 and 5/11 true minutes, the clock is running fast.
• Here the dial-to-true ratio is (720/11) : 65 = 144 : 143.
• 720 dial minutes correspond to 720 x 143/144 = 715 true minutes, a gain of about 5 minutes.

• Assuming the hands coincide every 65 minutes on a correct clock.
• Getting the direction backwards and adding the gain to the dial reading.
• Using 65 as an exact figure at the end instead of the ratio 144 : 143.

A faulty-clock item where the arithmetic is small and the whole difficulty is deciding whether the clock is fast or slow.

• practice — not a real PYQ

  How many times do the hour hand and the minute hand of a correct clock coincide in 24 hours?

  • (a)20
  • (b)22
  • (c)24
  • (d)44

  Answer(b) 22 — eleven coincidences in every twelve hours, because the hour hand is also moving.

• practice — not a real PYQ

  A clock's hands coincide every 64 minutes of correct time. Over a full day this clock

  • (a)loses time
  • (b)gains time
  • (c)keeps correct time
  • (d)gains in the morning and loses in the evening

  Answer(b) gains time — the interval is shorter than the correct 65 and 5/11 minutes, so the dial runs ahead.