Twelve Hour Period

In a twelve-hour period of time, for how many minutes is the number of the hour greater than the number of the minute?
Source: NCTM Mathematics Teacher

SOLUTION

From $12\!:\!00$ to $12\!:\!12,12$ minutes
From $11\!:\!00$ to $11\!:\!11,11$ minutes
From $10\!:\!00$ to $10\!:\!10,10$ minutes
etc.
From $3\!:\!00$ to $3\!:\!03,3$ minutes
From $2\!:\!00$ to $2\!:\!02,2$ minutes
From $1\!:\!00$ to $1\!:\!01,1$ minute
Total minutes
$1+2+3+\cdots+11+12=\frac{\left (1+12\right )12}{2}$
$=\frac{\left (13\right )12}{2}$
$=13\times 6$
$=78$

Answer: $78$ minutes