Tick Tock (More Algebra Than Geometry 3)

Over a period of $12$ hours, the hands align exactly $11$ times (just think about how many times the minute hand sweeps past the hour hand each hour). The time between consecutive alignments is always the same; so it must be $\frac{12}{11}$ hours. Expressed in H:M:S, this is $1:5:\frac{300}{11}$ , giving the answer $\boxed{317}$ .

The time has to be a little after 1:00, because the minutes hand needs to make more than a full revolution before it can point in the same direction as the hour hand.

At 1:00, the hour hand points at the 5 minute mark. We can call the amount of minutes that need to pass before the hands point in the same direction $m$ . The minute hand travels $6^\circ$ every minute, and the hour hand travels $\frac{1}{12}$ that speed, because it travels 5 minute marks every 60 minutes. So, the hour hand travels $0.5^\circ$ every minute. Because the hands need to point in the same direction and the hour hand is ahead by 5 minutes, which is $30^\circ$ , we can form the equation $6m=30+0.5m$ . Solving gives $m=\frac{30}{5.5}$ , or $m=\frac{60}{11}$ , which is $m=5\frac{5}{11}$ as a mixed number.

So, $5\frac{5}{11}$ minutes need to pass after 1:00 before the hands point in the same direction. The $\frac{5}{11}$ minutes represents the number of seconds. Since there are 60 seconds in a minute, $\frac{5}{11}$ minutes is equivalent to $\frac{300}{11}$ seconds.

So, the result is $1:05:\frac{300}{11}$ . Now that we know that $Seconds$ can be written as $\frac{a}{b}$ for co-prime natural numbers $a$ and $b$ , the answer is $1+5+300+11=\boxed{317}$