Reasoning · Grade 3-1 Length and Time

Problem

Broken Clock: Constant Error Rate

One clock gains 6 minutes an hour, the other loses 4. Both drift at a steady rate. Two faces show the same time when their gap is a whole number of 12-hour laps. Set both to 12 now and find how many days until they first agree again.
Your answer
How to solve
Strategy Look for a Pattern — Every hour the fast clock pulls ahead of the slow clock by a fixed amount, so the gap between them grows in a steady pattern. I find how fast the gap grows per hour, then ask how big the gap must become before the two faces line up again, which happens after a whole 12-hour lap of difference.
1STEP 1

Find how fast the gap grows each hour

One runs ahead and one behind, so the gap grows 10 minutes an hour.

6 + 4 = 10 minutes of gap per hour
2STEP 2

Decide how big the gap must be to match again

The faces agree again when the gap is one lap: 720 minutes.

12 × 60 = 720 minutes in one full lap
3STEP 3

Find how many hours give a 720-minute gap

At 10 a hour that takes 720 ÷ 10 = 72 hours.

720 ÷ 10 = 72 hours
4STEP 4

Change hours into days

72 hours is 3 days.

72 ÷ 24 = 3 days
Answer
3 days
720 ÷ 10 = 72 hours = 3 days
After 3 days = 72 hours, the fast clock has gained 72 * 6 = 432 minutes and the slow clock has lost 72 * 4 = 288 minutes, so the gap is 432 + 288 = 720 minutes = exactly 12 hours, which is one whole lap of the clock face. A 12-hour difference makes both faces read the same, so 3 days checks out, and no smaller whole-lap gap happens first.
Takeaway

When one clock speeds up and another slows down, just add their errors to see how fast they drift apart!

  • Find how fast the gap grows each hour
  • Decide how big the gap must be to match again
  • Find how many hours give a 720-minute gap
  • Change hours into days