Problems on Clock Questions in English

(B) The wall clock gains $2$ minutes in $12$ hours,which is equivalent to gaining $6$ minutes in $36$ hours.
The table clock loses $2$ minutes in $36$ hours.
Since one clock gains and the other loses,the relative difference in time between the two clocks increases by $6 + 2 = 8$ minutes every $36$ hours.
$36$ hours is equal to $1.5$ days.
Therefore,a difference of $8$ minutes is created in $1.5$ days.
For the clocks to show the same time again,the total difference must be $12$ hours,which is $720$ minutes.
To find the number of days for a $720$ minute difference:
$\text{Days} = \frac{1.5 \text{ days}}{8 \text{ minutes}} \times 720 \text{ minutes} = 1.5 \times 90 = 135$ days.
Thus,both clocks will show the same time after $135$ days at $12$ noon.

(A) To find the time when the hands of a clock are together between $6$ and $7$,we use the formula: $T = \frac{60}{11} \times H$,where $H$ is the starting hour.
Here,$H = 6$.
Required time $= \frac{60}{11} \times 6$ minutes past $6$.
$= \frac{360}{11}$ minutes past $6$.
$= 32 \frac{8}{11}$ minutes past $6$.

(A) Suresh's birth date is October $4, 1999$.
Shashikanth was born $6$ days before Suresh,so his birth date is September $28, 1999$.
Independence Day is August $15, 1999$,which was a Sunday.
We need to calculate the number of days from August $15$ to September $28$.
Days remaining in August $= 31 - 15 = 16$ days.
Days in September $= 28$ days.
Total days $= 16 + 28 = 44$ days.
To find the day of the week,divide the total days by $7$ to find the remainder (odd days).
$44 \div 7 = 6$ weeks and $2$ remainder.
Since August $15$ was a Sunday,we add $2$ days to Sunday.
Sunday $+ 2$ days $=$ Tuesday.
Therefore,Shashikanth was born on a Tuesday.

(C) The formula for the angle between the hands of a clock is $\theta = |30H - \frac{11}{2}M|$,where $H$ is the hour and $M$ is the minutes.
For the hands to point in opposite directions,the angle $\theta$ must be $180^{\circ}$.
Given $H = 9$,we have $180 = |30(9) - \frac{11}{2}M|$.
$180 = |270 - \frac{11}{2}M|$.
Since the minute hand must be past the $9$ position to create a $180^{\circ}$ angle,we solve $270 - \frac{11}{2}M = 180$.
$\frac{11}{2}M = 270 - 180 = 90$.
$M = \frac{90 \times 2}{11} = \frac{180}{11} = 16 \frac{4}{11}$ minutes.
Therefore,the time is $16 \frac{4}{11}$ minutes past $9$.

(B) To find the next year when January $5$th falls on a Tuesday,we calculate the number of odd days for each year.
$1$ ordinary year has $1$ odd day,and $1$ leap year has $2$ odd days.
January $5, 1965$ is a Tuesday.
January $5, 1966$: $1965$ is an ordinary year,so $+1$ day $\Rightarrow$ Wednesday.
January $5, 1967$: $1966$ is an ordinary year,so $+1$ day $\Rightarrow$ Thursday.
January $5, 1968$: $1967$ is an ordinary year,so $+1$ day $\Rightarrow$ Friday.
January $5, 1969$: $1968$ is a leap year,so $+2$ days $\Rightarrow$ Sunday.
January $5, 1970$: $1969$ is an ordinary year,so $+1$ day $\Rightarrow$ Monday.
January $5, 1971$: $1970$ is an ordinary year,so $+1$ day $\Rightarrow$ Tuesday.
Thus,the next victory day on the same day of the week will be January $5, 1971$.

(A) Republic Day is $26$ January. Independence Day is $15$ August.
From $26$ January $1996$ to $26$ January $1997$: $1996$ is a leap year,so it has $366$ days. Number of odd days = $366 \pmod 7 = 2$ odd days.
From $26$ January $1997$ to $26$ January $1998$: $1$ odd day.
From $26$ January $1998$ to $26$ January $1999$: $1$ odd day.
From $26$ January $1999$ to $26$ January $2000$: $1$ odd day.
Total odd days from $26$ Jan $1996$ to $26$ Jan $2000 = 2+1+1+1 = 5$ odd days.
So,$26$ January $2000$ was Friday $+ 5$ days = Wednesday.
Now,calculate days from $26$ January $2000$ to $15$ August $2000$:
January: $31 - 26 = 5$ days
February: $29$ days (leap year)
March: $31$ days
April: $30$ days
May: $31$ days
June: $30$ days
July: $31$ days
August: $15$ days
Total days = $5 + 29 + 31 + 30 + 31 + 30 + 31 + 15 = 202$ days.
$202 \div 7 = 28$ weeks and $6$ odd days.
Adding $6$ odd days to Wednesday: Wednesday $+ 6$ days = Tuesday.

(B) If the day after tomorrow is Sunday,then today is Friday.
Tomorrow is Saturday.
The day before yesterday of tomorrow means the day before yesterday of Saturday.
Saturday $- 2$ days $=$ Thursday.