Question:Prove that the calendar for 1995 will for 2009. 

In order that the calendar for 1995 and 2006 be the same, 

1st January of both the years must be on the same days week.

For this, the total number of odd days between 3 1st Dec, 1994 

and 31 st Dec, 2005  must be zero.

We know that an ordinary year has 1 odd day and a leap year has 2 odd days. 

During this period there are 3 leap years, namely  1996 , 2000 and 2004 and 8 

ordinary years. 

:. Total number of odd days during this period (6 + 8) + 14 = 0

i.e. 0 odd day.

Hence, the calendar for 1995 will serve for 2006.

Question:Prove that any date in March is the same day of the week as the corresponding date in November of that year. 

In order to prove the required result, we have to show that the number of odd days between last day of February and last day of October is Zero.

Number of days between these dates are.

March    April     May     June   July     Aug     Sep     Oct

31      +   30   +   31  +  30  +  31  + 31  +   30   +   31

`= 241 days = 35 weeks 

:. Number of odd days during this period = 0.

Hence, the result follows.

Question:At what time between 2 and 3 o'clock will hands of a clock together ? 

At 2 o'clock, the hour hand is it 2 and the minute hand is at 12, i,e they are 10 min spaces apart.

To be together the minute hand must gain 10 minutes over the hour hand.

Now, 35 minutes are gained by it in 60 min.

:. 10 minutes will be gained in` (60/55 xx 10)`

:. The hands will coincide at `10 10/11` min past 2.

Question:At what time between 4 and 5 o'clock will the hands of a clock be at right angle ? 

At 4 o'clock the minute hand will be 20 min.spaces behind the hour hand.

Now, when the two hands are at right angles, they are 15 min. spaces apart.

So they are at right angles in following two cases.

Question:Find at what time between 8 and 9 o'clock will the hands of a clock be in the same straight line but not together. 

At 8 o'clock the hour hand is at 8 and the minute hand is at 12 i,e. the two hands are 20 min. spaces apart.

To be in the same straight line but not together they will be 30 minute spaces apart.
So, the minute hand will have to gain (30 - 20) = 10 minute spaces over the hour hand.

55 minute spaces are gained in 60 min.

10 minute spaces will be gained in` (60/55 xx 10)` min` = 10 10/11`

:. The hands will be in the same straight line but not together at

`10 10/11` min. part 8.

Question:At what time between 5 and 6 o'clock are the hands of a 3 minute apart ? 

At 5 o'clock, the minute hand is 25 min. space behind the hour hand.

Case I: Minute hand is 3 min.spaces behind the hour hand.

In this case, the minute hand has to gain (25 - 3) = 22 minute space 

55 min are gained in 60 min.

22 min. are gained in` (60/55 xx 22)` min. = 24 min

:. The hands will be 3 min. apart at` 31 5/11` min. past 5.

Case II : Minute hand is 3 min.spaces ahead of the hour hand.

In this case, the minute hand has to gain (25 + 3) = 28 minute spaces.

55 min.are gained in 60 min.

28 min. are gained in` (60/55 xx 28)` `= 31 5/11` min.

:. The hands will be 3 min. apart at `31 5/11` min. past 5.

Question:A watch which gains uniformly's is 5 min. slow at 8 o'clock in the morning on Sunday and it is 5 min. 48 sec. fast at 8 p.m. on following Sunday. When was it correct ? 

Time from 8 a.m. on Sunday to 8 p.m. on following Sunday

       = 7 days 12 hours = 180 hours.

:. The watch gains `(5 + 5 4/5)` min or `54/5` min. in 180 hrs.

Now` 54/5` min. are gained in 180 hrs.

:. 5 min. are gained in `(180 xx 5/54 xx 5)` hrs.

= 83 hrs 20 min. = 3 days 11 hrs 20 min.

:. watch is correct 3 days 11 hrs 20 min. past 7 p.m.on Wednesday.

:. It will be correct at 20 min.past 7 p.m. on Wednesday.

Question:A clock is set right at 5 a.m. The clock loses 16 minutes in 24 hours. What will be the true time when the clock indicates 10 p.m. on 4 th day ? 

Time from 5 a.m. on 4th day = 89 hours.

Now 23 hrs 44 min. of this clock = 24 hours of correct clock.

`:. 356/15` hrs of this clock = 24 hours of correct clock.

89 hrs of this clock` = (24 xx 15/356 xx 89)` hrs hours of correct clock.

   = 90 hrs of correct clock.

So, the correct time is 11 p.m.

Question:A clock is set right at 8 a.m. The clock gains 10 minutes in 24 hours. What will be the true time when the clock indicates 1 p.m. on the following day ? 

Time from 8 a.m. on a day to 1 p.m. on the following day = 29 hrs.

24 hours 10 min.of this clock = 24 hours of the correct clock.

`145/6` hrs of this clock = 24 hrs of the correct clock

29 hrs of this clock` = (24 xx 6/145 xx 29)` hrs of the correct clock

   = 28 hrs 48 min of correct clock

:. The correct time is 28 hrs 48 min. after 8 a.m.

This is 48 min. past 12.

Question:Find the cash required to purchase `Rs.3200,7 1/2%` stock at 107 `(brokerage 1/2%)` 

Cash required to purchase Rs. 100 stock

` = Rs.(107 + 1/2) = Rs.215/2.`

 Cash required to purchase Rs. 3200 stock

 ` = Rs. (215/2 xx 1/100 xx 3200)`

  `= Rs. 3440.`