# I need help with solving a math problem that involves clocks

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!

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