Reasoning topic
Clock questions ask where the hands are. Calendar questions ask which day a date falls on. Each has a short rule that turns the question into a line of arithmetic.
A typical question asks for the angle between the two hands at a given time. Other questions ask for a clock’s mirror image, or the weekday of a date.
What is the angle between the hour hand and the minute hand of a clock at 9:22?
149°
30 × 9 − 5.5 × 22 = 270 − 121 = 149°, which is less than 180°.
Hands: at H hours and M minutes, the angle is 30 × H − 5.5 × M, without its sign. Read H on the 12-hour dial (15:40 is H = 3). If the result is more than 180°, take it from 360°.
Mirror: the real time and the time in the mirror add up to 11:60, so mirror time = 11:60 − real time. When the hour is 12, read it as 0. An answer of :60 is the next hour (8:60 is 9:00), and an hour of 0 is 12 (0:15 is 12:15).
Weekdays: count the days after the first date, up to and including the second, and divide by 7. The remainder, the odd days, is how far the weekday moves forward.
If 26 February of a certain non-leap year was a Sunday, what day of the week was 18 September in the same year?
Answer: Monday
1. What is the angle between the hour hand and the minute hand of a clock at 9:55?
Use 30 × H − 5.5 × M, and drop the sign.
32.5°
30 × 9 − 5.5 × 55 = 270 − 302.5 = −32.5. Without the sign, 32.5°. That is less than 180°, so it is the angle.
2. What is the mirror image of a clock showing the time 9:39?
Take the time from 11:60.
2:21
11:60 − 9:39: the minutes are 60 − 39 = 21 and the hours are 11 − 9 = 2. The mirror shows 2:21.
3. If today is Monday, what day of the week will it be after 70 days?
How many whole weeks are in 70 days, and what is left over?
Monday
70 ÷ 7 = 10 weeks, remainder 0. No odd days, so the weekday does not move: it is Monday.
Practise Clocks & Calendars in the app. The questions follow your level, and what you practise comes back just before you'd forget it.
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