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(A) $12$-hour clock cycles through $12$ hours and $60$ minutes. $A$ full day consists of $24$ hours,or $1440$ minutes. First,identify the minutes in a

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$\frac{1}{2}$

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(A) $12$-hour clock cycles through $12$ hours and $60$ minutes. $A$ full day consists of $24$ hours,or $1440$ minutes. First,identify the minutes in an hour where the digit $1$ appears: $01, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51$. There are $15$ such minutes. In each hour,the clock shows the incorrect time for these $15$ minutes. Additionally,in hours $01, 10, 11, 12$,the clock shows the incorrect time for all $60$ minutes. Total incorrect minutes in a $12$-hour cycle: For the $8$ hours where $1$ does not appear in the hour digit $(02, 03, 04, 05, 06, 07, 08, 09)$,the clock is incorrect for $15$ minutes per hour: $8 	imes 15 = 120$ minutes. For the $4$ hours where $1$ does appear $(01, 10, 11, 12)$,the clock is incorrect for all $60$ minutes: $4 	imes 60 = 240$ minutes. Total incorrect minutes in $12$ hours = $120 + 240 = 360$ minutes. Since a day has two $12$-hour cycles,total incorrect minutes in a day = $360 	imes 2 = 720$ minutes. Total minutes in a day = $24 	imes 60 = 1440$ minutes. Correct minutes = $1440 - 720 = 720$ minutes. Fraction of the day showing correct time = $\frac{720}{1440} = \frac{1}{2}$.

$A$ certain $12$-hour digital clock displays the hour and the minute of a day. Due to a defect in the clock,whenever the digit $1$ is supposed to be displayed,it displays $7$. What fraction of the day will the clock show the correct time?

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