Question

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?

$\dfrac12$

$\dfrac58$

$\dfrac34$

$\dfrac56$

$\dfrac{9}{10}$

Solution:

Among the hours $1$ through $12,$ exactly $1, 10, 11, 12$ contain a $1,$ so the hour is correct $\dfrac{8}{12}=\dfrac23$ of the time.

A minute is displayed wrong when its tens digit is $1$ (minutes $10$ – $19$ ) or its units digit is $1$ ( $01,11,\dots,51$ ), which is $15$ of the $60$ minutes. So the minute is correct $\dfrac{45}{60}=\dfrac34$ of the time.

The clock is correct $\dfrac23\cdot\dfrac34=\dfrac12$ of the day.

Thus, the correct answer is A.

2009 AMC 10B Problem 19

Solution:

Problem 19 in Other Years