96. At how many points between 10 O'clock and 11 O'clock are the minute hand and hour hand of a clock at an angle of 30 degrees to each other?

Solution:
Between 10 and 11, the minute hand and hour hand are at an angle of 30o to each at minutes past 10 = minutes past 10. The next time they will be at angle of 30o to each other will be at 11.

97. A clock loses 1% time during the first week and then gains 2% time during the next one week. If the clock was set right at 12 noon on a Sunday, what will be the time that the clock will show exactly 14 days from the time it was set right?

Solution:
The clock loses 1% time during the first week.
In a day there are 24 hours and in a week there are 7 days. Therefore, there are 7 * 24 = 168 hours in a week.
If the clock loses 1% time during the first week, then it will show a time which is 1% of 168 hours less than 12 Noon at the end of the first week = 1.68 hours less.
Subsequently, the clock gains 2% during the next week. The second week has 168 hours and the clock gains 2% time = 2% of 168 hours = 3.36 hours more than the actual time.
As it lost 1.68 hours during the first week and then gained 3.36 hours during the next week, the net result will be a -1.68 + 3.36 = 1.68 hour net gain in time.
So the clock will show a time which is 1.68 hours more than 12 Noon two weeks from the time it was set right.
1.68 hours = 1 hour and 40.8 minutes = 1 hour + 40 minutes + 48 seconds.
i.e. 1 : 40 : 48 P.M.

98. What is the angle between the minute hand and the hour hand when the time is 1540 hours?
Solution:
The total angle made by the minute hand during an hour is 360o. If it takes 360o for an hour, it will take = 240o. The angle between the hour hand the minute hand will therefore, be somewhere between 240 - 90 = 150o, as the hour hand is between 3 and 4.
The angle made by the hour hand when it moves from say 3 to 4 will be 30o. That is the hour hand makes 30o during the course of an hour.
The hour hand will therefore, move = 20o.
Therefore, the net angle between the hour hand and the minute hand will be 150 - 20 = 130o