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
4 years, 11 months ago

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The answer is 1213\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 00^\circ, the time the person goes to sleep at t1t_1 and wakes up at t2t_2, the angle made by the hour-hand at t1t_1 and minute-hand at t2t_2 be α\alpha and that by the hour-hand at t2t_2 and minute-hand at t1t_1 be β\beta. Note that the hour-hand makes 3030^\circ on the dial in 11 hour while the minute-hand makes 360360^\circ in 11 hour.

Therefore, we have 30t1=α ,360t1=360+β\quad 30t_1 = \alpha \space, \quad 360t_1 = 360 + \beta (as t1>1t_1 > 1 hour) t1=α30=1+β360\quad \Rightarrow t_1 = \dfrac {\alpha}{30} = 1 + \dfrac {\beta}{360}

Similarly, we have t2=β30=2+α360\quad t_2 = \dfrac {\beta}{30} = 2 + \dfrac {\alpha}{360}

The time that the person sleep, t=t2t1=βα30=1βα360t = t_2-t_1 = \dfrac {\beta-\alpha} {30} = 1 - \dfrac {\beta - \alpha}{360}

βα30(1+112)=1βα30=t=1213 \dfrac {\beta-\alpha} {30} \left( 1 + \dfrac {1}{12} \right) = 1\quad \Rightarrow \dfrac {\beta-\alpha} {30} = t = \boxed{\frac {12}{13}} hours.

Chew-Seong Cheong - 4 years, 11 months ago

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thanx a lot!!

Aneesh Kundu - 4 years, 11 months ago

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Why you add 10 to y in the equation 10+y/60

surya Prabha - 2 years, 11 months ago

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he slept at 1:10 and woke up at 2:05

Mehul Arora - 4 years, 11 months ago

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According to the book the answer should be 55513 mins55\dfrac{5}{13}\text{ mins}

Aneesh Kundu - 4 years, 11 months ago

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but the question says that consider minute and hour hand only

Mehul Arora - 4 years, 11 months ago

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@Mehul Arora yes but that doesn't mean that time can't be fractional

Aneesh Kundu - 4 years, 11 months ago

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A similar question once appeared in INMO And I really think this is a gud one So pls reshare and write solutions

Aneesh Kundu - 4 years, 11 months ago

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The book is correct. Let xx and yy be the minutes past 1am1 am and 2am2 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:

10+y60=x5\dfrac { 10+y }{ 60 } =\dfrac { x }{ 5 }

5+x60=y5\dfrac { 5+x }{ 60 } =\dfrac { y }{ 5 }

We get x=125143x=\dfrac { 125 }{ 143 } and y=70143y=\dfrac { 70 }{ 143 } , from which we can work out the time of sleep

(120+5+x)(60+10+y)=72013=55+513(120+5+x)-(60+10+y)=\dfrac { 720 }{ 13 } =55+\dfrac { 5 }{ 13 }

Michael Mendrin - 4 years, 11 months ago

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Thanx a lot!!

Aneesh Kundu - 4 years, 11 months ago

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Why u add 10 to y in the equation (10+y)/60

surya Prabha - 2 years, 11 months ago

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