Question

A digital clock reads hours and minutes. The sum of the digits it displays at 12:00 is 3 (1+2+0+0). At 12:01 it is 4 (1+2+0+1) and so on. What is the sum of all the digits it displays from 12:00 to 12:59?

Answer 1

Correct Answer: 600

To find the sum of all digits displayed on a digital clock from 12:00 to 12:59, we start by analyzing the structure of time representation in this hour range. The hours part is always 12; hence, its digit sum remains constant: 1+2=3. Next, consider the minutes, which range from 00 to 59. We calculate the sum of digits for each minute and accumulate this value:

• Minutes 00 to 09: 0 through 9 leads to sums: 0+1+2+...+9 = 45.
• Minutes 10 to 19: Each sum includes adding 1 to the tens place, resulting in: (1+0)+(1+1)+...+(1+9) = 10+1+2+...+9 = 55.
• Minutes 20 to 29: Similar logic with tens digit fixed as 2: (2+0)+(2+1)+...+(2+9) = 20+1+2+...+9 = 65.
• Minutes 30 to 39: Adding 3 for the tens place: (3+0)+(3+1)+...+(3+9) = 30+1+2+...+9 = 75.
• Minutes 40 to 49: Tens place as 4: (4+0)+(4+1)+...+(4+9) = 40+1+2+...+9 = 85.
• Minutes 50 to 59: Fixed tens digit 5: (5+0)+(5+1)+...+(5+9) = 50+1+2+...+9 = 95.

To find the total sum of all minutes, add up the computed sums: 45+55+65+75+85+95 = 420.
Since the hours part has a constant sum of 3 for all 60 minutes, it contributes 60*3 = 180 to the total.
Total sum of all digits from 12:00 to 12:59 = 180 (hours part) + 420 (minutes part) = 600.
This total falls within the given range of 600 to 600; thus, it is correctly calculated.