Clock Buzzing: Total Buzzes from 12 Noon to 11 PM

This problem involves calculating the total number of times a clock buzzes over a specific period, given a pattern based on the hour.

Understanding the Buzzing Pattern

The clock buzzes a number of times equal to the hour it indicates:

• At 1 o'clock, it buzzes 1 time.
• At 2 o'clock, it buzzes 2 times.
• ...
• At 12 o'clock, it buzzes 12 times.

Calculating Buzzes from 12 Noon to 11 PM

We need to find the total buzzes from 12 noon up to and including 11 p.m. Let's list the hours and the corresponding buzzes within this time frame:

• 12 Noon: 12 buzzes
• 1 PM: 1 buzz
• 2 PM: 2 buzzes
• 3 PM: 3 buzzes
• 4 PM: 4 buzzes
• 5 PM: 5 buzzes
• 6 PM: 6 buzzes
• 7 PM: 7 buzzes
• 8 PM: 8 buzzes
• 9 PM: 9 buzzes
• 10 PM: 10 buzzes
• 11 PM: 11 buzzes

Summing the Buzzes

To find the total number of buzzes, we need to sum the buzzes for each hour from 12 noon to 11 p.m.

Total Buzzes = Buzzes at 12 noon + Buzzes from 1 PM to 11 PM

Total Buzzes = $12 + (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11)$

Applying the Arithmetic Series Formula

The sum of the first '$n$' natural numbers (1 + 2 + ... + n) can be calculated using the formula:

$ S_n = \frac{n(n+1)}{2} $

In our case, we need the sum of numbers from 1 to 11. So, we set $n = 11$.

Sum (1 to 11) = $S_{11} = \frac{11(11+1)}{2}$

Sum (1 to 11) = $\frac{11 \times 12}{2}$

Sum (1 to 11) = $\frac{132}{2}$

Sum (1 to 11) = $66$

Final Calculation

Now, we add the buzzes from 12 noon to this sum:

Total Buzzes = Buzzes at 12 noon + Sum (1 to 11)

Total Buzzes = $12 + 66$

Total Buzzes = $78$

Conclusion

The clock will buzz a total of 78 times from 12 noon till 11 p.m.