Question

Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $9{:}99$ just before midnight, $0{:}00$ at midnight, $1{:}25$ at the former $3{:}00$ AM, and $7{:}50$ at the former $6{:}00$ PM. After the conversion, a person who wanted to wake up at the equivalent of the former $6{:}36$ AM would set his new digital alarm clock for $A{:}BC,$ where $A,$ $B,$ and $C$ are digits. Find $100A + 10B + C.$

Answer

Solution:

An ordinary day has $60 \cdot 24 = 1440$ minutes, and $6{:}36$ AM comes $6 \cdot 60 + 36 = 396$ minutes after midnight. A metric day has $10 \cdot 100 = 1000$ metric minutes, so the equivalent metric time is $\frac{396}{1440} \cdot 1000 = 275$ metric minutes after midnight, which the new clock displays as $2{:}75.$

Therefore $100A + 10B + C = 275.$

2013 AIME II Problem 1

Solution:

Problem 1 in Other Years