You're exactly right, the answer is $11$ times every $12$ hours, or $1$ time every $\frac{12}{11}$ of an hour, which works out to about once every $1$ hour, $5$ minutes, and $27.27$ seconds. This isn't an average either; the amount of time that passes between each meeting of the two hands is constant.
Just think about the time 11:59, and count through $12$ hours. You'll notice that there is a meeting for each hour $12,1,2,3,\ldots,10$, but no meeting for the $11$th hour because you stop at 11:59. This makes $11$ meetings in $12$ hours.
It's not too hard to see why the time passing between two meetings is constant. Just think about the rotational symmetry.
Another (and more difficult way) to look at the problem is to calculate the angular speed of each hand. Using convenient units, we have the minute hand traveling at a rate of $60$ units per hour, and the hour hand travels at a rate of $5$ units per hour. Thus, we would like to solve the following equation:
$$60t=5t\operatorname{mod}60$$
$0$ clearly solves the equation, so we look for the smallest positive value of $t$ solving it. Thus we want to solve:
$$55t=60$$
which gives us $t=\frac{12}{11}$ hours as before.
We will measure angles in degrees from $12:00$ in a clockwise direction.
Let $h(t)$ be the position of the hour hand, in degrees, at $t$ minutes after $9$:$00$ and let $m(t)$ be the position of the minute hand, in degrees, at $t$ minutes after $9$:$00$.
Then $$h(t) = 270 + \frac 12t$$
and $$m(t) = 6t$$
We need to solve
\begin{align}
h(t+4) &= m(t-3) + 180 \\
270 + \frac 12(t+4) &= 6(t-3) + 180 \\
t &= 20\; \text{minutes}
\end{align}
Best Answer
Oh, let's do this the annoying way.
The minute hand moves at $360\frac {\text{degrees}}{\text{revolutions}}\frac {1 \text{revolution}}{60 \text{minutes}}=6\frac{\text{degrees}}{\text{minutes}}$
The hour hand moves at $\frac{360 degrees}{1revolution}\frac{1revolution}{12hours}\frac{1hour}{60 minute}= \frac 12 \frac{degree}{minute}$.
If a minute hand is $x$ degrees away from the hour hand how long will it take for the minute hand to become $x + 180$ degrees away from the hour hand?
Well that's a matter of solving $x + 6t = x + \frac 12t + 180$ or $t= \frac {180}{\frac {11}{2}}=\frac {360}{11}$. (Roughly $32$ minutes.)
At $3:00$ the hour hand and the minute hand are perpendicular. The next time that will happen will be when the minute hand moves for $90$ degrees before the hour had to $90$ degrees after the hour. That will happen in about $32$ minutes. (At $3:32$). Every $\frac {360}{11}$ theminute hand will move 180 degrees further than the hour hand and the hands will be perpendicular.
In a $12$ hour (or a $60*12 = 720$ minute period) this will happen $\frac {720}{\frac {360}{11}} = 22$ times.
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Here's a better way, although it was easy to make annoying sign and logic errors if you aren't careful.
The minute hand travels $6$ degrees a minute. ($\frac {360}{60} = 6$). The hour hand travels $.5$ degrees a minute. ($\frac {360}{12*60} = .5$).
In a 12-hour period there are $12*60=720$ minutes.
Let's say $\theta_t = 6t$ is the angle of the minute hand after $t; 0 \le t < 720$ minutes. Note: it's very possible that $\theta(t) > 360$.
Let's say $\phi_t = .5t$ is the angle of the hour hand after $t$ minutes..
Obviously $\theta_t \ge \phi_t$.
$\theta_t$ and $\phi_t$ are perpendicular if $\theta_t = \phi_t + 90 + k*180$ for some non-negative integer $k$.
So we need the find out how many solutions there are to:
$6t = .5t + 90 + k*180; 0 \le t < 720$ there are.
So $t = \frac {90 + k*180}{5.5}$ will have one solution for each integer $k$.
So how many $k$ are there so that $0 \le t = \frac {90 + k*180}{5.5} < 720$ are there?
Multiply everything by $11$: $0 \le 2(90 + k*180)< 11*720$
Divide everything by $180$: $0 \le 1 + 2k < 11*4$
$-1 \le 2k < 11*4 -1$
$\frac 12 \le k < 2*11 - \frac 12$
And as $k$ is a non-negative integer $0 \le k < 22$. There are $22$ possible $k$s which give a solution.
This also gives us the times this occurs:
$t = \frac {90 + k*180}{5.5} = \frac {180}11 + k *\frac {360}{11} \approx 16.36 + 32.72k$ (or a little more than every half-hour):
So at 12:16, 12:49, 1:22, 1:54, 2:27, 3:00, etc.