I used to not like analog clocks because they unecessarily made it harder to tell time in a world where digital clocks are a reality. Now, I appreciate them a lot more for all the mathematical fun that they present.
So, here’s a very simple puzzle I thought of while looking at one.
The Puzzle
It is 3:00 right now on an analog clock. How much longer do I have to wait to see the minute and the hour hands cross each other?
The Solution
Lets call the position of the minute hand and the hour hand be $x$,$y$, respectively. If we let the ranges be $x,y \in [0,60)$ then we can conveniently express $y$ as
$$y = \frac{5x}{60} + 15$$Then, the problem becomes as simple as solving the systems of equations of
$$ \begin{cases} y = x \\ y = \frac{x}{12}+15 \end{cases} $$Which is solved with
$$ \begin{align} x &= \frac{x}{12} + 15 \\ 11x &= 180 \\ x &= 180/11 \end{align} $$ $$\implies x = 16.\overline{36}$$Thus, we will see them cross for the first time at 3:16, which makes sense!