How many times in a day are the hour hand and the minute hand of a wall clock straight (i.e., the angle between them is 180°)?
20
21
22
24
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC CISF-AC-EXE – 2018
The hands are 180° apart once in every hour interval, except for the interval between 6 o’clock and 7 o’clock, where they are 180° apart exactly at 6 o’clock. The 6 o’clock position is counted in both the 5-6 interval and the 6-7 interval if we consider specific time points, but considering distinct occurrences in a 12-hour cycle, it happens 11 times.
For example, between 12 pm and 12 am, the hands are opposite at approximately 12:33, 1:38, 2:44, 3:49, 4:55, 6:00, 7:05, 8:11, 9:16, 10:22, 11:27. (These are approximate times, the exact times are fractions). This is 11 distinct times in a 12-hour period.
A full day is 24 hours, which consists of two 12-hour periods.
Therefore, in 24 hours, the hands will be straight (180° apart) 11 times + 11 times = 22 times.
For the hands to be straight, $\theta = 180^\circ$.
$t = \frac{2}{11} (30H \pm 180)$.
Let’s check for H from 1 to 12.
For H=6, $t = \frac{2}{11} (180 \pm 180)$. $t = \frac{2}{11} (360) \approx 65.45$ min (past 6) or $t = \frac{2}{11} (0) = 0$ min (past 6). This confirms 6:00 is one time.
For H=5, $t = \frac{2}{11} (150 \pm 180)$. $t = \frac{2}{11} (330) = 60$ min (past 5, which is 6:00) or $t = \frac{2}{11} (-30)$ (not valid in this hour).
The hands are exactly opposite at 6:00. This instance is the boundary point that is counted only once in a 12-hour period when counting the intervals between hours.
The number of times the hands are 180° apart is 11 in 12 hours, and 22 in 24 hours.