# Whats the time

I am preparing for the KVPY exam which will be held on 27th of October,2013.I was going through the previous years question paper that I found this problem.

A 12 hour digital clock displays the hour and the minute of a day.Due to 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?

Thank You

Note by A Former Brilliant Member
6 years, 3 months ago

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$\frac{1}{4}$

- 6 years, 3 months ago

Sorry, I mean $\frac{1}{2}$

- 6 years, 3 months ago

There are 45 minutes which doesn't show one in an hour, and there are 16 hours which doesn't show one in a day. There are 60 minutes in an hour, and 24 hours in a day. Therefore, the fraction is $\frac{45 \times 16}{60 \times 24}$ which can be simplified into $\frac{1}{2}$

- 6 years, 3 months ago

It is a 12 hour digital clock. So, how can there be 16 hours when clock doesn't show 1?

The hour display shows 1 only when it is 1 o' clock, be it a.m. or p.m. And the minute display shows 1 when it displays 01,11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51.

So, I think the answer should be $\frac{11 \times 14 +60}{60 \times 12}$

- 6 years, 3 months ago

Well, Takeda did it over $24$ hours, which is basically multiplying by $\dfrac{2}{2}$.

Your problem is that the hour display also shows 1 at 10, 11, and 12. The clock shows the correct time if there are no ones. For each hour without a 1 (there are 8), there are 45 minutes without a 1.

Then, the answer is $\dfrac{8}{12}\cdot\dfrac{45}{60}=\dfrac{1}{2}$.

- 6 years, 3 months ago

Thank you very much!

- 6 years, 3 months ago

Yes, you are correct. I had merged these cases with the other cases. It was a mistake. Thank you for pointing it out.

- 6 years, 3 months ago

Takeda is correct Maharnab. There are really 16 hours when the clock does not show 1. In 12 hours the clock does show 1 at 1, 10, 11, and 12 o clock and in 24 hour system it will show these all two times which sums up to 8 hours and 24 - 8 is 16 hours.

- 5 years, 5 months ago

173/180

- 6 years, 3 months ago