Difference between revisions of "2013 AIME II Problems/Problem 1"
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+ | ==Problem 1== | ||
Suppose that the measurement of time during the day is converted to the metric system so that each day has <math>10</math> metric hours, and each metric hour has <math>100</math> metric minutes. Digital clocks would then be produced that would read <math>\text{9:99}</math> just before midnight, <math>\text{0:00}</math> at midnight, <math>\text{1:25}</math> at the former <math>\text{3:00}</math> AM, and <math>\text{7:50}</math> at the former <math>\text{6:00}</math> PM. After the conversion, a person who wanted to wake up at the equivalent of the former <math>\text{6:36}</math> AM would set his new digital alarm clock for <math>\text{A:BC}</math>, where <math>\text{A}</math>, <math>\text{B}</math>, and <math>\text{C}</math> are digits. Find <math>100\text{A}+10\text{B}+\text{C}</math>. | Suppose that the measurement of time during the day is converted to the metric system so that each day has <math>10</math> metric hours, and each metric hour has <math>100</math> metric minutes. Digital clocks would then be produced that would read <math>\text{9:99}</math> just before midnight, <math>\text{0:00}</math> at midnight, <math>\text{1:25}</math> at the former <math>\text{3:00}</math> AM, and <math>\text{7:50}</math> at the former <math>\text{6:00}</math> PM. After the conversion, a person who wanted to wake up at the equivalent of the former <math>\text{6:36}</math> AM would set his new digital alarm clock for <math>\text{A:BC}</math>, where <math>\text{A}</math>, <math>\text{B}</math>, and <math>\text{C}</math> are digits. Find <math>100\text{A}+10\text{B}+\text{C}</math>. | ||
Revision as of 16:50, 6 April 2013
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
.
Solution
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
.
See also
2013 AIME II (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |