A quarter-hour is a unit of time equal to 15 minutes. 8:15 is a quarter-hour later than 8:00. Below is a representation of a quarter-hour on a clock.
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Notice the "quart" in "quarter." There are four quarts in a gallon, four quarters in a dollar, four quarter-hours in an hour, and four quarters in a football game. Quart comes from a Latin word meaning four or fourth.