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?

$\dfrac{1}{2}$

$\dfrac{5}{8}$

$\dfrac{3}{4}$

$\dfrac{5}{6}$

$\dfrac{9}{10}$

Solution:

The hours containing a $1$ are $1, 10, 11, 12,$ so $8$ of the $12$ hours display correctly, a fraction $\dfrac{2}{3}.$

A minute is wrong if either digit is $1$ : the tens digit gives $10\text{-}19$ ( $10$ minutes), and the ones digit adds $01, 21, 31, 41, 51$ ( $5$ more), $15$ in all. So $\dfrac{45}{60} = \dfrac{3}{4}$ of minutes are correct.

The fraction of the day is $\dfrac{2}{3} \cdot \dfrac{3}{4} = \dfrac{1}{2}.$

Thus, the correct answer is A.

2009 AMC 12B Problem 10

Solution:

Problem 10 in Other Years