Here's the question.

"It is now between 10:00 and 11:00. Six minutes from now, the minute hand of a watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now?"

It's from Art of Problem Solving Volume 1, Chapter 4 Proportions. I haven't solved clock problems in the past and I'm not sure how to do this. The solution didn't help and was extremely complicated, so I was hoping someone could provide a solution that I might be able to understand. Any tips for future problems that might be similar to this would also be greatly appreciated!

  • 1
    $\begingroup$ Try to replace the clock's hands by their respective angles, then write down the equation the problem gives you. $\endgroup$ – Fabien Jul 30 '20 at 15:53
  • $\begingroup$ What are the angular velocities of the hour and minute hands, respectively? $\endgroup$ – Andrew Chin Jul 30 '20 at 15:57
  • $\begingroup$ Is this actually from a math contest? $\endgroup$ – halrankard Jul 30 '20 at 17:14

I'd set $t=0$ to correspond to 10:00. Let's write formulas for the position of each hand $t$ minutes after that, where position is measured as the angle (in degrees) clockwise from the "12".

Minute hand: $m(t)=\frac{360\text{ deg}}{60\text{ min}}t=6t$.

Hour hand: $h(t)=\frac{360\text{ deg}}{12\cdot60\text{ min}}t+300=0.5t+300$.

Now let $T$ be the time now. Your information says: $$ \begin{align} m(T+6)&=h(T-3)-180&&\text{(Note $h(T)$ is between 300 and 330.)}\\ 6(T+6)&=0.5(T-3)+300-180 \end{align} $$

You can solve this linear equation to get $T=15$. So it's 10:15.


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