He ran 4 miles per hour for one hour, which means he ran 4 miles.
He cycled 6 miles at 12 miles per hour, which means he cycled for one half an hour.
His total miles was 4 + 6 = 10 total miles.
His total time was 1 hour + one half an hour = 1.5 hours.
His average speed is total distance divided by total time:
10 / 1.5 = 6 2/3 miles per hour.