Difference between revisions of "2013 AIME II Problems/Problem 1"
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==Solution== | ==Solution== | ||
− | There are <math>24 | + | There are <math>24 \cdot 60=1440</math> normal minutes in a day , and <math>10 \cdot 100=1000</math> metric minutes in a day. The ratio of normal to metric minutes in a day is <math>\frac{1440}{1000}</math>, which simplifies to <math>\frac{36}{25}</math>. This means that every time 36 normal minutes pass, 25 metric minutes pass. From midnight to <math>\text{6:36}</math> AM, <math>6 \cdot 60+36=396</math> normal minutes pass. This can be viewed as <math>\frac{396}{36}=11</math> cycles of 36 normal minutes, so 11 cycles of 25 metric minutes pass. Adding <math>25 \cdot 11=275</math> to <math>\text{0:00}</math> gives <math>\text{2:75}</math>, so the answer is <math>\boxed{275}</math>. |
− | == See also == | + | ==Solution 2== |
+ | First we want to find out what fraction of a day has passed at 6:36 AM. One hour is <math>\frac{1}{24}</math> of a day, and 36 minutes is <math>\frac{36}{60}=\frac{3}{5}</math> of an hour, so at 6:36 AM, <math>6 \cdot \frac{1}{24} + \frac{1}{24} \cdot \frac{3}{5}=\frac{1}{4}+\frac{1}{40}=\frac{11}{40}</math> of a day has passed. Now the metric timing equivalent of <math>\frac{11}{40}</math> of a day is <math>\frac{11}{40}\cdot 1000=275</math> metric minutes, which is equivalent to 2 metric hours and 75 metric minutes, so our answer is <math>\boxed{275}</math> - mathleticguyyy | ||
+ | == See also == | ||
{{AIME box|year=2013|n=II|before=First Problem|num-a=2}} | {{AIME box|year=2013|n=II|before=First Problem|num-a=2}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 19:03, 3 September 2023
Contents
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
.
Solution 2
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
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.