Find the times in a day when the hour′s, minute′s and second′s hand of a clock occupy the same angular position. [old question reposted]


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Question Number 67208 by mr W last updated on 24/Aug/19

$F i n d t h e t i m e s i n a d a y w h e n$ $t h e {h o u r}^{'} s, {m i n u t e}^{'} s a n d {s e c o n d}^{'} s$ $h a n d o f a c l o c k o c c u p y t h e s a m e$ $a n g u l a r p o s i t i o n .$ $[o l d q u e s t i o n r e p o s t e d]$
Commented by Kunal12588 last updated on 24/Aug/19
at 12 am and 12 pm?
Commented by mr W last updated on 24/Aug/19

$12 a m a n d 12 p m a r e c e r t a i n l y$ $c o r r e c t . b u t t h e r e a r e o t h e r t i m e s .$ $o b s e r v e h o w a (a n a l o g, m e c h a n i c a l)$ $c l o c k w o r k s .$
Commented by Tinkutara@ last updated on 24/Aug/19

$22 t i m e s l i k e 1 : 05, 2 : 10, 3 : 15, 4 : 20, . . . .$ $?$
Commented by Kunal12588 last updated on 24/Aug/19

$a t t h o s e t i m e s {h o u r}^{'} s h a n d a n d {m i n u t e}^{'} s h a n d s$ $h a v e s a m e a n g u l a r p o s i t i o n b u t {s e c o n d}^{'} s h a n d$ $h a s d i f f e r n t a n g u l a r p o s i t i o n . {i s n}^{'} t i t .$
Commented by mr W last updated on 24/Aug/19

$b u t t h e q u e s t i o n m e a n s t h a t a l l t h r e e$ $h a n d s m u s t b e a t t h e s a m e p o s i t i o n .$
Commented by behi83417@gmail.com last updated on 24/Aug/19

$θ =∣ \frac{11}{2} m - 30 h ∣$ $θ = angle between clock and minute hands .$ $now : θ should be zero . so :$ $\frac{11}{2} m - 30 h = 0 \Rightarrow \frac{m}{h} = \frac{60}{11} .$
	Answered by mr W last updated on 25/Aug/19

	Commented by mr W last updated on 25/Aug/19

	$t h i s p i c t u r e s h o w s a n a n a l o g c l o c k .$ $i t s h a n d s {d o n}^{'} t m o v e c o n t i n u o u s l y,$ $b u t c a n o n l y o c c u p y 60 p o s s i b l e$ $a n g u l a r p o s i t i o n s :$ $0, 1, 2, . . ., 58, 59$ $t h e h a n d s m o v e i n f o l l o w i n g w a y :$ $t h e {s e c o n d}^{'} s h a n d m o v e s a s t e p$ $e v e r y s e c o n d . t h e {m i n u t e}^{'} s h a n d$ ${d o e s n}^{'} t m o v e w i t h e a c h m o v e o f$ $t h e {s e c o n d}^{'} s h a n d . o n l y a f t e r 60$ $s t e p s o f t h e {s e c o n d}^{'} s h a n d (i . e .$ $a f t e r 60 s e c o n d s) t h e$ ${m i n u t e}^{'} s h a n d m o v e s a s t e p, a n d$ $s o o n . s i m i l a r l y t h e {h o u r}^{'} s h a n d$ ${d o e s n}^{'} t m o v e w i t h e a c h m o v e o f$ $t h e {m i n u t e}^{'} s h a n d . o n l y a f t e r$ $12 s t e p s o f t h e {m i n u t e}^{'} s h a n d (i . e .$ $a f t e r 12 m i n u t e s o r 720 s e c o n d s)$ $t h e {h o u r}^{'} s h a n d m o v e s a s t e p, a n d$ $s o o n .$ $w i t h t h i s k n o w l e d g e a b o u t h o w$ $t h e h a n d s o f a c l o c k w o r k, w e$ $c a n d e t e r m i n e t h e m o m e n t s w h e n$ $t h e t h r e e h a n d s o f a c l o c k a r e a t$ $t h e s a m e a n g u l a r p o s i t i o n .$ ${l e t}^{'} s s a y t h e t i m e i s h o^{'} c l o c k e x a c t l y .$ $(0 ⩽ h ⩽ 11) . a t t h i s m o m e n t t h e$ ${h o u r}^{'} s h a n d i s a t t h e p o s i t i o n 5 h,$ $a n d t h e {m i n u t e}^{'} s a n d {s e c o n d}^{'} s$ $h a n d s a r e a t p o s i t i o n 0 . a f t e r x$ $s e c o n d s, w i t h 0 < x < 3600,$ $t h e p o s i t i o n o f t h e$ ${s e c o n d}^{'} s h a n d i s m o d (x, 60) . t h e$ $p o s i t i o n o f t h e {m i n u t e}^{'} s h a n d i s$ $⌊ \frac{x}{60} ⌋, t h e p o s i t i o n o f t h e {h o u r}^{'} s$ $i s 5 h + ⌊ \frac{x}{720} ⌋ . s u c h t h a t t h e t h r e e$ $h a n d s a r e a t t h e s a m e p o s i t i o n :$ $5 h + ⌊ \frac{x}{720} ⌋ = ⌊ \frac{x}{60} ⌋ = m o d (x, 60)$ $w e g e t f o l l o w i n g r e s u l t s :$ $f o r h = 0 : x = 0 \Rightarrow t i m e 0 h 0 m 0 s$ $f o r h = 1 : x = 305 \Rightarrow t i m e 1 h 5 m 5 s$ $f o r h = 2 : x = 610 \Rightarrow t i m e 2 h 10 m 10 s$ $f o r h = 3 : x = 976 \Rightarrow t i m e 3 h 16 m 16 s$ $f o r h = 4 : x = 1281 \Rightarrow t i m e 4 h 21 m 21 s$ $f o r h = 5 : x = 1647 \Rightarrow t i m e 5 h 27 m 27 s$ $f o r h = 6 : x = 1952 \Rightarrow t i m e 6 h 32 m 32 s$ $f o r h = 7 : x = 2318 \Rightarrow t i m e 7 h 38 m 38 s$ $f o r h = 8 : x = 2623 \Rightarrow t i m e 8 h 43 m 43 s$ $f o r h = 9 : x = 2989 \Rightarrow t i m e 9 h 49 m 49 s$ $f o r h = 10 : x = 3294 \Rightarrow t i m e 10 h 54 m 54 s$ $f o r h = 11 : x = 3599 \Rightarrow t i m e 11 h 59 m 59 s$ $t h a t m e a n s i n a d a y t h e r e a r e 24$ $t i m e s w h e n t h e t h r e e h a n d s o f$ $a c l o c k a r e a t t h e s a m e a n g u l a r$ $p o s i t i o n .$ $t h e p i c t u r e a b o v e s h o w s e . g . t h e$ $m o m e n t 4 h 21 m 21 s (a m o r p m)$ $w h e n t h e h a n d s a r e a t t h e s a m e$ $p o s i t i o n .$
	Commented by JDamian last updated on 25/Aug/19
	$But many other analog clocks are not so discrete − second′s hand moves quickly than one-second step (fractions of one second)$
	$B u t m a n y o t h e r a n a l o g c l o c k s a r e n o t s o$ $d i s c r e t e - {s e c o n d}^{'} s h a n d m o v e s q u i c k l y t h a n$ $o n e - s e c o n d s t e p (f r a c t i o n s o f o n e s e c o n d)$
	Commented by mr W last updated on 25/Aug/19

	$t h e s o l u t i o n d e p e n t s v e r y m u c h o n$ $h o w t h e h a n d s m o v e .$ $i f i t i s a s s u m e d t h a t t h e h a n d s$ $m o v e c o n t i n u o u s l y, t h e n t h e r e i s$ $o n l y o n e m o m e n t w h e n t h e t h r e e$ $h a n d s a r e a t t h e s a m e p o s i t i o n,$ $i . e . a t 12 : 00 : 00 .$
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