Multiplying and dividing by powers of 10

Next, give your students these challenges:

• Red Express Train by NRICH

5 on the Clock by NRICH: Note the solution provided by NRICH is wrong. Here's the solution: The hour will read 5 for 60 mins. Likewise, the hour will read 15 for 60 mins. We've got 22 hours left to account for. There will be a 5 in the 10 minute column for 10 minutes per hour. Over the 22 hours we've left to account for, that's 10 * 22 = 220 times. Now we've got 22 hours left to account for, and 50 of those minutes to account for per hour. A 5 occurs in the 1 minute column once per 10 minutes. Thus, over 50 minutes, a 5 will occur in the 1 minute column 5 times. Over 22 hours, that's 5 * 22 = 110 times. We've considered when a 5 appears in the 1 hour column, the 10 minute column, and the 1 minute column, so we must be done. Adding everything up, we have 60 + 60 + 220 + 110 = 450 times. So over 24 hours, at least one 5 will be present 450 times. The answer is the same for a 12 hour clock. On a 12 hour clock you will have 5 AM and 5 PM instead of 5 and 15, which doesn't affect our count. Here's a small Python program you can use to verify my answer is correct:

h, m = 0, 0
n = 0
while h < 24:
	if '5' in str(h) + ':' + str(m):
		n += 1
	if m < 59:
		m += 1
	else:
		m = 0
		h += 1
print(n)