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gerald

Let the angle the minute hand covered be $x$ [in degrees]

Let the angle the hour hand covered be $y$ [in degrees]

I believe that $y = 360 - x$ because of how it is shaped.

Hours passed $=\frac{y}{360} = 1 - \frac{x}{360}$

Minutes passed: $=\frac{x}{360}$ This means, $\frac{x}{360 \cdot 60}$ hours.

Equation,

$1 - \frac{30x}{30 \cdot 360} = \frac{x}{30 \cdot 360}$

$1 = \frac{31x}{30 \cdot 360} \implies \frac{x}{360} = \frac{30}{31}$ minutes.

$= \frac{30}{31} \cdot \frac{1}{60} = \frac{1}{62}$ hours.

This is clearly incorrect, but why?

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  • $\begingroup$ Why assume $y=360-x$? That's clearly your error. $\endgroup$ – Noble Mushtak Jan 24 '16 at 13:19
  • $\begingroup$ @NobleMushtak, but if the hour hand ends where the minute hand started and the minute hand ended where the hour hand started, isnt $y = 360 - x$ valid? $\endgroup$ – Amad27 Jan 24 '16 at 13:26
  • $\begingroup$ ...Why? Just because they switch places does not mean they add up to $360$. $\endgroup$ – Noble Mushtak Jan 24 '16 at 13:28
  • $\begingroup$ @NobleMushtak, I see, how can I rectify this? $\endgroup$ – Amad27 Jan 24 '16 at 13:29
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Don't assume $y=360-x$. I'm not sure why you assumed that, but unfortunately, it's wrong.

Instead, for the time that Jimmy starts, let $x$ be the angle measure of the angle from $12$ to the minute hand, going clockwise, and let $y$ be the angle measure of the angle from $12$ to the hour hand, going clockwise. This is basically what you said, but more specific.

Since we know that Jimmy starts between $9:00$ and $10:00$, we can deduce that $y$ is at least $\frac{9}{12}*360^\circ=270^\circ$. Then, since the minute hand goes $360^\circ$ (one full rotation) in one hour and the hour hand goes $30^\circ$ (one twelfth of a rotation, from $9$ to $10$) in one hour, the hour hand must move $(\frac{x}{12})^\circ$ after $9:00$. Thus, we have the following equation: $$y=270+\frac{x}{12}$$

Now, for the time that Jimmy ends, we know that the minute hand and the hour hand switch. Thus, we just need to switch our definitions: $y$ is now the angle measure of the angle from $12$ to the minute hand, going clockwise, and $x$ is now angle measure of the angle from $12$ to the hour hand, going clockwise.

Since we know that Jimmy ends between $10:00$ and $11:00$, we can deduce that $x$ is at least $\frac{10}{12}*360^\circ=300^\circ$. Then, since the minute hand goes $360^\circ$ (one full rotation) in one hour and the hour hand goes $30^\circ$ (one twelfth of a rotation, from $10$ to $11$) in one hour, the hour hand must move $(\frac{y}{12})^\circ$ after $10:00$. Thus, we have the following equation: $$x=300+\frac{y}{12}$$

Substitute the first equation into the second equation: $$x=300+\frac{270+\frac{x}{12}}{12}$$ Multiply everything by $12$: $$12x=3600+270+\frac{x}{12}$$ Multiply everything by $12$ again: $$144x=43200+3240+x=46440+x$$ Subtract both sides by $x$: $$143x=46440$$ Divide both sides by $143$: $$x=\frac{46440}{143}$$ Now, we have found $x$. Using this, we can find the number of minutes before $10:00$ in the starting time and the number of minutes past $10:00$ in the ending time.

Starting Time: The minute hand moves $360^\circ$ in one hour and there are $60$ minutes in one hour. This means that if the minute hand has moved $x^\circ$, $\frac x 6$ minutes have passed. Thus, we just need to divide $x$ by $6$ to find the number of minutes past $9:00$. This turns out to be $7740/143$. However, we want to find the number of minutes before $10:00$. Since the difference between $9:00$ and $10:00$ is just one hour and there are $60$ minutes in an hour, we just need to subtract $7740/143$ from $60$, which yields $840/143$ minutes before $10:00$.

Ending Time: First, subtract $300^\circ$ from $x$ so we can figure out how much the hour hand has moved since $10:00$. This yields $\frac{3540}{143}$. The hour hand moves $30^\circ$ in one hour and there are $60$ minutes in an hour. This means that if the hour hand has moves $x^\circ$ since the hour, $2x$ minutes have passed. Thus, we just need to multiply $\frac{3540}{143}$ by $2$, which yields $\frac{7080}{143}$ minutes past $10:00$.

Final Answer: The difference between the starting time and ending time is the sum of the number of minutes the starting time was before $10:00$ plus the number of minutes the ending time was after $10:00$. This sum is $\frac{840}{143}+\frac{7080}{143}=\frac{7920}{143}$. However, we want our answer in hours, so we need to divide this by $60$, which yields $\frac{132}{143}$ which can be simplified to $\frac{12}{13}$. Thus, Jimmy spent $\frac{12}{13}$ hours painting.

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