2015 AMC 10A Problems/Problem 14

Problem

Clock - radius 20 cm Disk - radius 10 cm, externally tangent at 12 o'clock, arrow pointing up

If the disk rolls clockwise, where on the clock will it be when the arrow points up?

Solution

The circumference of the clock is twice that of the disk. So, a quarter way around the clock (3:00), the point halfway around the disk will be tangent. The arrow will point to the left. We can see the disk made a 75% rotation from 12 to 3, and 3 is 75% of 4, so it would make 100% rotation from 12 to 4. The answer is o'clock.