Clocks

The face of a clock or a watch is a circle which is divided into 60 minutes spaces.
The minute hand passes over 60 minute spaces whilst the hour hand goes over 5 minute spaces.
In 60 minutes the minute hand gains 55 minutes on the hour hand.
Hands are straight 22 times in a day.
Hands coincide 22 times in a day.
Hands are at right angle twice per hour.

Example: At what time between 4 and 5 will the hands of a watch.
(a) Point in opposite directions? (b) Coincide (c) Be at right angle
Solution (a) They will be opposite each other (i.e., angle between them is 180 degrees) when there is a space of 30 minutes between them. This will happen when minute hand gains (30 20) minutes, i.e.,

minutes past 4. (b) At 4 O'clock the hands are 20 minutes apart. Clearly the minute hand must gain 20 minutes before the hands can be coincident (I,e angle between them ZERO). But the minute hand gains 55 minutes in 60 minutes. Hence to find out in what time minute hand will gain 20 minutes, we have the following proportion:

55 : 20 60 : x

?x minutes

?Hands will be coincident at 21 minutes past 4. (c) At 4 O'clock the hands are 20 minutes apart. They will be at right angles when there is a space of 15 minutes between them. This will happen twice (i) when the minute hand has gained (20 - 15) or 5 minutes; (ii) when the minute hand has gained (20 + 15) or 35 minutes.

Minute hand gains 5 minutes in minutes.

i.e., 5 minutes past 4.

Or minute hand gains 35 minutes in minutes

i.e., 38 minutes past 4.

Example: Two clocks begin to strike 12 together. One strikes its stroke in 33 seconds and the other in 22 seconds. What is the interval between the 6th stroke of the first and the 8th stroke of the second?

Since the first clock strikes (12 - 1) or 11 strokes in 33 seconds the interval between its two successive strokes is or 3 seconds.
Similarly, the interval between two successive strokes of the other clock is or 2 seconds.
Now the 6th stroke of the first clock will come after 5 × 3 or 15 seconds and the 8th stroke of the second clock will come after 7 × 2 or 14 seconds.

Example: My watch which gains uniformly is 2 min. slow at noon on Sunday, and is 4 minutes 48 seconds fast at 2 p.m. on the following Sunday. When was it correct?

From Sunday noon to the following Sunday at 2 p.m. 7 days 2 hrs 170 hrs.
The watch gains 2 + 4 + = 6 + minutes of 6 minutes in 170 hrs.
?The watch gains 2 minutes in × 170 or 50 hrs.
Now 50 hrs = 2 days 2 hrs.
2 days 2 hrs from Sunday noon = 2 p.m. on Tuesday.