Question

The Situation

Imagine that you have three mechanical clocks in your home. (These are traditional, analog (nondigital), 12-hour clocks.) Unfortunately, none of your clocks keeps time properly. Each clock consistently gains or loses a fixed number of minutes each hour, but different clocks may gain or lose different amounts.

Your Task
At noon today, all the clocks are correct. Investigate when, if ever, the clocks will be correct again. Start by looking at these three specific clocks.
If a clock gains 10 minutes each hour, when will it be correct again?
If a clock loses 3 minutes each hour, when will it be correct again?
If a clock gains 7 minutes each hour, when will it be correct again?
Then explore this more general situation: Suppose you have three clocks in your house. The grandfather clock loses g minutes every hour, the alarm clock gains c minutes every hour, and the watch is permanently stuck at the same time. Find a general rule for the next time that all three clocks will be correct.

Answer 1

If a clock gains 10 min/hr in 6 hrs it will have gained 1 hr.

But the clock must gain 12 hrs to agree again with an accurate clock.

So in 6 * 12 hrs it would again agree with the correct time.

If a clock loses 3 min/hr in 20 hrs it will have lost 1 hr.

Then is 20 * 12 = 240 hrs it will again agree with an accurate clock.

So if a clock gains 7 min/hr then in 60 / 7 hrs it will have gained 1 hr,

and in 12 * 60 / 7 = 102 and 6/7 hrs it will have gained the necessary 12 hrs.

So if G loses g min /hr then in 60 /g hrs it will lose 1 hr

and in 12 * 60 / g it will lose12 hrs and again be in sync with the correct time.

Using this rule on the clock that loses 3 min/hr we get

12 * 60 / 3 = 240 which is what we said above.