Logical Reasoning · Micro-test

Clocks and Calendars

Clock questions run on two speeds and calendar questions run on remainders when dividing by seven. Once those are fixed in mind, each question is a short calculation.

10 questions · 5 minutes · instant scoring

What this topic actually tests

The minute hand turns 360 degrees in 60 minutes, which is 6 degrees a minute. The hour hand turns 360 degrees in 12 hours, which is 30 degrees an hour or half a degree a minute. At H hours and M minutes the hour hand stands at 30H + M/2 degrees and the minute hand at 6M degrees, so the angle between them is the difference between 30H and 5.5M; if that comes to more than 180, subtract it from 360. In twelve hours the hands coincide 11 times, not 12, because the coincidence that would fall between 11 and 1 happens exactly at 12. A clock seen in a mirror shows the time subtracted from 12 hours: 3:15 in the mirror is 8:45. Calendar questions use odd days, the days left over after complete weeks. An ordinary year has 365 days, which is 52 weeks and 1 odd day, so the same date falls one weekday later the next year. A leap year has 366 days and 2 odd days. A year is a leap year if it is divisible by 4, except that a century year must be divisible by 400: 2000 was a leap year, 1900 was not and 2100 will not be. To find the weekday a number of days ahead, divide by 7 and count forward by the remainder. Worked example: 100 days after a Wednesday. 100 divided by 7 leaves 2, so the day is two after Wednesday, which is Friday.

The common trap on this topic

In angle questions the usual mistake is to leave the hour hand on the hour mark. At 3:30 the hour hand is halfway between 3 and 4, which is why the angle is 75 degrees and not 90. In calendar questions the trap is the century year: 1900 and 2100 are divisible by 4 and are still not leap years. A second calendar slip is to count a leap year's extra day when the period in question does not include 29 February. Finally, when counting days between two dates, decide once whether the first day is included and stay with it; counting both ends adds a day and shifts the answer by one weekday.

Clocks and Calendars: answers and explanations

What is the angle between the hour hand and the minute hand of a clock at 3:30?
Answer: A. 75 degrees
The hour hand is at 30 × 3 + 30 ÷ 2 = 105 degrees and the minute hand is at 6 × 30 = 180 degrees. The difference is 75 degrees. Ninety would be right only if the hour hand were still exactly on the 3.
What is the angle between the hour hand and the minute hand of a clock at 4:20?
Answer: D. 10 degrees
The hour hand is at 30 × 4 + 20 ÷ 2 = 130 degrees and the minute hand is at 6 × 20 = 120 degrees. The difference is 10 degrees. The hands are not together, because the hour hand has moved on from the 4.
How many times do the hour hand and the minute hand of a clock coincide in 12 hours?
Answer: B. 11
The hands coincide once in each hour except that the coincidences for the hours beginning at 11 and at 12 are the same one, at exactly 12 o'clock. That gives 11 in twelve hours.
A clock seen in a mirror appears to show 3:15. What is the actual time?
Answer: C. 8:45
The mirror image of a clock shows the time subtracted from 12 hours. 12:00 − 3:15 = 8:45.
1 January 2023 was a Sunday. What day of the week was 1 January 2024?
Answer: A. Monday
2023 was an ordinary year of 365 days, which is 52 weeks and 1 odd day. The same date a year later falls one weekday later: Monday.
Which of the following years is a leap year?
Answer: A. 2000
A century year is a leap year only if it is divisible by 400. 2000 is; 1900 and 2100 are not. 2022 is not divisible by 4.
If today is Wednesday, what day of the week will it be 100 days from today?
Answer: C. Friday
100 = 14 × 7 + 2, so there are 2 odd days. Two days after Wednesday is Friday.
26 January 2025 was a Sunday. What day of the week was 15 August 2025?
Answer: D. Friday
Days after 26 January: 5 left in January, then 28 + 31 + 30 + 31 + 30 + 31 for February to July, then 15 in August, a total of 201. 201 = 28 × 7 + 5, so the day is five after Sunday: Friday.
A clock gains 5 minutes every hour. It is set right at 8:00 am. What time will it show when the correct time is 8:00 pm on the same day?
Answer: A. 9:00 pm
From 8:00 am to 8:00 pm is 12 hours. The clock gains 12 × 5 = 60 minutes, so it shows 9:00 pm.
How many odd days are there in a period of 100 years that contains 76 ordinary years and 24 leap years?
Answer: D. 5
Each ordinary year has 1 odd day and each leap year has 2. That is 76 + 48 = 124 odd days, and 124 = 17 × 7 + 5, leaving 5.

FAQ

What is the formula for the angle between the hands of a clock?

At H hours and M minutes, the angle is the difference between 30H and 5.5M degrees. If the result is more than 180, subtract it from 360 to get the smaller angle.

How many times do the hands of a clock coincide in a day?

22 times in 24 hours: 11 times in every 12 hours. They are at right angles 44 times in 24 hours.

How do I know whether a year is a leap year?

It is a leap year if it is divisible by 4, unless it is a century year, in which case it must be divisible by 400. So 2024 and 2000 are leap years; 1900 and 2100 are not.

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