2013 AIME II Problems/Problem 1
Suppose that the measurement of time during the day is converted to the metric system so that each day has metric hours, and each metric hour has metric minutes. Digital clocks would then be produced that would read just before midnight, at midnight, at the former AM, and at the former PM. After the conversion, a person who wanted to wake up at the equivalent of the former AM would set his new digital alarm clock for , where , , and are digits. Find .
There are normal minutes in a day , and metric minutes in a day. The ratio of normal to metric minutes in a day is , which simplifies to . This means that every time 36 normal minutes pass, 25 metric minutes pass. From midnight to AM, normal minutes pass. This can be viewed as cycles of 36 normal minutes, so 11 cycles of 25 metric minutes pass. Adding to gives , so the answer is .
First we want to find out what fraction of a day has passed at 6:36 AM. One hour is of a day, and 36 minutes is of an hour, so at 6:36 AM, of a day has passed. Now the metric timing equivalent of of a day is metric minutes, which is equivalent to 2 metric hours and 75 metric minutes, so our answer is - mathleticguyyy
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