At what time between 3 o'clock and 4 o'clock will the hour hand and minute hand of a clock coincide? Express the time in minutes past 3 as $\frac{p}{q}$ in simplest fractional form. The value of $(p + q)$ is ________ (answer in integer).
Answer Key:
191
Step-by-Step Solution & Explanation
1. At 3:00, the minute hand is at 0 minutes while the hour hand is at the 15-minute mark (separation $= 15\text{ minute spaces} = 90^\circ$).
2. The minute hand moves at $1\text{ minute space/min}$, while the hour hand moves at $\frac{1}{12}\text{ minute space/min}$.
3. Relative speed $= 1 - \frac{1}{12} = \frac{11}{12}\text{ minute spaces/min}$.
4. Time required to overtake $15\text{ minute spaces}$:
$t = \frac{15}{11/12} = \frac{180}{11}\text{ minutes past 3}$
Since 11 is prime and does not divide 180, $\frac{180}{11}$ is in simplest form ($p = 180, q = 11$).
5. Value of $p + q = 180 + 11 = 191$.