The hands of a clock point in opposite directions (in the same straight line) 11 times in every 12 hours. (Because between 5 and 7 they point in opposite directions at 6 o'clcok only).
So, in a day, the hands point in the opposite directions 22 times.
Angle traced by hour hand in 12 hrs = 360º.
Angle traced by hour hand in 5 hrs 10 min. i.e., 31/6 hrs = ${\left(\frac{360}{12}*\frac{31}{6}\right)}^{\xb0}$= 155º
Time from 12 p.m. on Monday to 2 p.m. on the following Monday = 7 days 2 hours = 170 hours.
The Watch gains $\left(2+4\frac{4}{5}\right)$ min. or $\frac{34}{5}$ min.in 170 hrs.
Now, $\frac{34}{5}$min.are gained in 170 hrs.
2 min.are gained in $\left(170*\frac{5}{34}*2\right)$ hrs=50 hrs
Watch is correct 2 days 2 hrs. after 12 p.m. on Monday i.e., it will be correct at 2 p.m. on Wednesday
Angle traced by hour hand in 12 hrs. = 360º.
Angle traced by it in 11/3 hrs= ${\left(\frac{360}{12}*\frac{11}{3}\right)}^{\xb0}$= ${110}^{\xb0}$
Angle traced by minute hand in 60 min. = 360º.
Angle traced by it in 40 min. =(360/60*40) ${\xb0}^{}$=240 ${\xb0}^{}$
Required angle (240 - 110)º = 130º.
At 9 o?clock, the hour hand is at 9 and the minutes hand is at 12, i.e., the two hands are 15 min. spaces apart.
So, the minute hand should gain = (30 - 15) minutes = 15 minutes
55 minutes will be gained in 60 min.
15 minutes spaces will be gained in ((60/55) x 15) min. = 180/11 min.
The hands will be in the same straight line but not together i.e.,in 180 degrees at 180/11 min. past 9.
Since at 4 : 20 the minute hand will be at 4 and the angle between them will be same as the distance covered in degree by the hour hand in 20 minutes.
Required angle = distance of hour hand = speed × time = $\frac{1}{2}$× 20 =10 degrees.
We know that 1 year = 365.25 days
1 day = 24 hours
1 hour = 60 minutes
1 minute = 60 seconds
=> 1 Year = 365.25 x 24 x 60 x 60 Seconds
But 1 billion seconds = ?
1 Year = 365.25 x 24 x 60 x 60 Seconds
? = 1000000000 Seconds
$\mathbf{?}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1000000000}}{\mathbf{365}\mathbf{.}\mathbf{25}\mathbf{}\mathbf{x}\mathbf{}\mathbf{24}\mathbf{}\mathbf{x}\mathbf{}\mathbf{60}\mathbf{}\mathbf{x}\mathbf{}\mathbf{60}}\mathbf{}\mathbf{years}$ = 31. 68 years.
In this problem , it has considered that 65 mins = 1hr
So mins has increased by 5 mins so multiply 5 x 24 = 120 mins extra ,
That is now per day it adds 2hr extra, so divide 1440/26 = 59.384 days =~ 60 days.
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