A clock shows 3:00 p.m. What does it show when the hour hand has moved 135°?

Solution: 7:30. 135° ÷ 30° per hour = 4.5 hours.

Find the angle between the hands at 9:31.

Solution: 99.5°. |270 - 170.5| = 99.5.

Match: 105°, 45°, 127.5°, 187.5° with 7:15, 9:15, 4:30, 2:30.

Solution: 105° - 2:30, 45° - 4:30, 127.5° - 7:15, 187.5° - 9:15. Each from |30H - 5.5M| (187.5° is the formula value at 9:15; the smaller angle is 172.5°).

What are the common traps in Angle between hands / coincide / opposite questions?

Hour hand drift: at 6:45 the angle is 67.5°, not 90°. Which angle: "reflex" means the larger angle (360° minus the smaller). Mirror: 4:15 in a mirror reads 7:45, not 7:15.

Start here · the first lesson of Angle between hands / coincide / opposite

Angle between hands / coincide / opposite

Clocks and Calendars · for IPMAT Indore, IPMAT Rohtak and JIPMAT

The idea

Clock questions turn on two speeds: the minute hand moves 6° a minute and the hour hand 0.5°, so the minute hand gains 5.5° a minute. From this one fact come the angle at any time, the times when the hands meet, oppose or form a given angle, and how far a hand turns in a given time. JIPMAT 2023 set two (a matching list of angles and times, and the hour hand moving 135°).

Textbook vs exam method

Textbook method

Measure both hands from the 12: minute hand at 6M°, hour hand at 30H + 0.5M°. Their difference is the angle; set it equal to a target angle and solve for M to find when it happens.

Exam method

Angle. $|30H - 5.5M|$ ; take 360° minus it if it is more than 180° (or keep the larger one if the reflex angle is asked).
When at angle θ. solve $|30H - 5.5M| = θ$ : two answers in most hours (minute hand behind, then ahead).
Rotation. hour hand 30° an hour, minute hand 6° a minute.
Counts. together 11 and opposite 11 times in 12 hours; any other angle 22 times.
Clock as a compass. if one hour mark points in a known direction, every other mark is 30° per hour round from it.