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# Help needed!!!!!

A person goes to sleep between 1am and 2 am and he wakes up when his watch shows such a time that the two hands interchange their respective places. He wakes up between 2am and 3am, how long does he sleep?

Consider only hour hand & minute hand.

Note by Aneesh Kundu
2 years, 9 months ago

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The answer is $$\boxed {\frac{12}{13}}$$ hours.

The positions of hour-hand and minute-hand have a certain relationship. They are together at 00:00h or 12:00AM. Let 12:00AM position be $$0^\circ$$, the time the person goes to sleep at $$t_1$$ and wakes up at $$t_2$$, the angle made by the hour-hand at $$t_1$$ and minute-hand at $$t_2$$ be $$\alpha$$ and that by the hour-hand at $$t_2$$ and minute-hand at $$t_1$$ be $$\beta$$. Note that the hour-hand makes $$30^\circ$$ on the dial in $$1$$ hour while the minute-hand makes $$360^\circ$$ in $$1$$ hour.

Therefore, we have $$\quad 30t_1 = \alpha \space, \quad 360t_1 = 360 + \beta$$ (as $$t_1 > 1$$ hour) $$\quad \Rightarrow t_1 = \dfrac {\alpha}{30} = 1 + \dfrac {\beta}{360}$$

Similarly, we have $$\quad t_2 = \dfrac {\beta}{30} = 2 + \dfrac {\alpha}{360}$$

The time that the person sleep, $$t = t_2-t_1 = \dfrac {\beta-\alpha} {30} = 1 - \dfrac {\beta - \alpha}{360}$$

$$\dfrac {\beta-\alpha} {30} \left( 1 + \dfrac {1}{12} \right) = 1\quad \Rightarrow \dfrac {\beta-\alpha} {30} = t = \boxed{\frac {12}{13}}$$ hours. · 2 years, 9 months ago

Why you add 10 to y in the equation 10+y/60 · 8 months, 2 weeks ago

thanx a lot!! · 2 years, 9 months ago

The book is correct. Let $$x$$ and $$y$$ be the minutes past $$1 am$$ and $$2 am$$ where the hands could be. Then we solve this system of equations, for start and end times, the difference being the time of sleeping:

$$\dfrac { 10+y }{ 60 } =\dfrac { x }{ 5 }$$

$$\dfrac { 5+x }{ 60 } =\dfrac { y }{ 5 }$$

We get $$x=\dfrac { 125 }{ 143 }$$ and $$y=\dfrac { 70 }{ 143 }$$, from which we can work out the time of sleep

$$(120+5+x)-(60+10+y)=\dfrac { 720 }{ 13 } =55+\dfrac { 5 }{ 13 }$$ · 2 years, 9 months ago

Why u add 10 to y in the equation (10+y)/60 · 8 months, 2 weeks ago

Thanx a lot!! · 2 years, 9 months ago

A similar question once appeared in INMO And I really think this is a gud one So pls reshare and write solutions · 2 years, 9 months ago

he slept at 1:10 and woke up at 2:05 · 2 years, 9 months ago

According to the book the answer should be $55\dfrac{5}{13}\text{ mins}$ · 2 years, 9 months ago

but the question says that consider minute and hour hand only · 2 years, 8 months ago