Clock hands moving to take each others places. This is from 2012 Euclid. I was wondering if someone could point out the flaw in my reasoning here:
Let the minute hand point to $50+x$ minutes. Then the hour hand points to $45 + \dfrac{50 + x}{12}$ minutes because the hour hand moves at a rate 1/12 that of the minute hand. Since the hour hand will move from $45 + \dfrac{50 + x}{12}$ to $50+x$ the number of minutes it travels is
$50+x -45 - \dfrac{50 + x}{12} = \dfrac{11x + 10}{12}$
but since the minute hand travels to $45 + \dfrac{50+x}{12}$ and the hour travels to $50+x$ we know that
x = $\dfrac{45 + \dfrac{50+x}{12}}{12}$ and
x = $\dfrac{590}{143}$
Plugging this into $\dfrac{11x + 10}{12}$ we get 60/13 and we divide by 12 to convert minutes to hours, for a total of 5/13 hours. The correct answer is 12/13. I was hoping someone could point out the error in my solution, rather than providing a new one, as I have seen better ways to solve this using angles, but would like to see the error here.
Your calculation of $x$ is correct, and you are correct to conclude that the hour hand has moved $60/13$ minute marks from start to finish. But the hour hand moves $5$ minute marks per hour, so to obtain hours elapsed you should divide $60/13$ by $5$ (rather than divide by $12$), obtaining $12/13$ hours elapsed.
there are 60 minutes in hour.
So, the minute hand moves 12 times as quickly as the hour hand,so 60-x/x=12 or x=60/13 ---&gt; eq.(1)
In one hour, the hour hand moves through 1/12 x 60 = 5. Since the hour hand is moving for t hours, then
5t = x
Substitute value of x from eq.1
5t = 60/13
t = 12/13