When will the hour hand and the minute hand be in a straight line on a clock?

The clock is split into two halves. Do you notice that every day? Tell you the formula.

I instructed readers how to calculate the time when the hour hand and the minute hand overlap on a clock some time ago. You could check it out. Meanwhile, I am going to guide you how to work out the time when the two hands are in a straight line.

Each time we figure out the time they are in a straight line, we have got to calculate the time they overlap first. When it is twelve o’ clock, the hands overlap for sure. Then we can try to find out the time they are straight. When the clock is cut into two halves, it must be divided into two equal sectors each of which occupy 30 minutes.

Moreover, we have to know that their velocities of the hands are different. The net speed is 60/60 minus 5/60 and we can get the fraction as 11/12. The question is that when the hands will form an adjacent angle (180 degrees) with their different velocities. Thus we have this formula as follows: 11/12 = 30/? wherein “ ? ” is the time needed to split the clock equally.

Accordingly, “ ? ” is (12 x 30) divided by 11 and the result is 32 and 8/11 minutes.

The hands are in a straight line when they strike at 12: 32: 8/11 am/pm after 12 o’ clock and before 1 o’ clock. (8/11 x 60 seconds = 43.6363 seconds)

Overlapping time after 1 o’ clock is 01: 05: 32.72.

Straight-line time after 1 o’ clock and before 2 o’ clock is as follows:

01: 05: 32.72 + 00: 32: 43.63 = 01: 37: 76.35 = 01: 38: 16.35

Help me fill out the table above to let me know you have understood, please!

Note: There is no straight-line time after 6 o’ clock although there is an overlapping time at 6/11 x 60 plus 06: 00: 00. Moreover, the straight-line time after 6 o’ clock uses one o’ clock before. For example, we try to find out the straight-line time after 9 o’ clock.

The overlapping time after 8 o’ clock is 8/11 x 60 and it means 08: 43: 37.80 am/pm.

The straight-line time is 08: 43: 37.80 + 00: 32: 43.63 = 08: 75: 81.43 = 09: 16: 21.43 in the morning or afternoon. Is it fun? Give me some comments please. I am waiting for you.