Problem:
A particular 12-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a 1, it mistakenly displays a 9. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
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Solution:
The clock will display the incorrect time for the entire hours of , and 12 . So the correct hour is displayed correctly of the time.
The minutes will not display correctly whenever either the tens digit or the ones digit is a 1 , so the minutes that will not display correctly are , 19 , and , and 51 . This is 15 of the 60 possible minutes for a given hour. Hence the fraction of the day that the clock shows the correct time is .
The problems on this page are the property of the MAA's American Mathematics Competitions