Clocks

Concepts:
When the minute hand passes over the 60 minutes spaces, the hour hand goes over the 5 minute spaces. That is in 60 minutes, the minute hand gains 55 minutes over the hour hand or 55/60 minute spaces in one minute.
In one minute, minute hand moves 6 degree. In one hour, hour hand moves 30 degree. Hence in one minute hour hand moves ½ degree.
In every hour the hands coincide once.
In every hour hands are twice at right angles (90 degree apart) and in these positions hands are 15 minute spaces apart.
In every hour the hands point in opposite directions (180 degree apart) once and in this position, the hands are 30 minute spaces apart.
The hands are in same straight line when they are coincident or opposite to each other. The hands coincide 11 times in every 12 hours. Between 11 and 1 o clock there is only one common position at 12 o clock. Hence hands coincide 22 times a day.
If both the hands start moving together from the same position, both the hands will coincide after 360*2/11 = 65 5/11 minutes.
The hands of a clock are at right angles twice in every hour, but in 12 hours they are at right angles 22 times since there are two common positions in every 12 hours.
Angle between minute hand and hour hand when time is x hours and y minutes is 30x – 11y/2 degree.
Questions:
At what time between 4 and 5 will the hands of a watch coincide?
At 4 o clock the hands are 20 minutes apart.
In order to coincide minute hand must gain 20 minutes.
Minute hand gains 55 minutes in 60 minutes.
Hence it will gain 20 minutes in > 60*20/55 = 240/11 = 21 9/11 minutes
Hence hands will coincide at 21 9/11 minutes past 4.At what time between 4 and 5 will the hands of a watch point in opposite directions?
They will be in opposite direction when there is 30 minute space between them.
This will happen when minute hand gains 50 minutes (at 4 o clock hands are 20 minutes apart).
Minute hand gains 50 minutes in 60 * 50/55 = 600/11 =54 6/11 minutes.
Hence hands point in opposite directions at 54 6/11 past 4.My watch which gains uniformly is 2 minutes slow at noon on Sunday and is 4 minutes 48 seconds fast at 2 p m on the following Sunday. When it was correct?
Sunday noon to next Sunday 2 PM = (24 * 7) + 2 = 170 hours
In 170 hours watch gains 2 + 4 + 48/60 minutes
In 170 hours watch gains 6 4/5 minutes.
Hence in order to gain 2 minute watch will take 170 * 5 * 2/34 = 50 hours
50 hours means 2 days and 2 hours.
That means at 2 PM on Tuesday it was correct.From noon by how many degrees has the minute hand moved to 2:40 ?
One hour means minute hand moves 360 degree
2 hours means minute hand moves 720 degree
40 minute means minute hand moves 240 degree
Hence total = 720+240 = 960 degrees.How many degrees does an hour hand move in 20 minutes?
In one hour hour hand moves 30 degrees.
Hence in 20 minutes hour hand moves 30/3 = 10 degrees.What is the angle between the minute hand and hour hand of a clock when the time is 7:30?
Angle = 30x11y/2
= 30 * 7 – 11 * 30/2
= 210 – 165
= 45 degree.