# UKMT Special (Problem $4$)

My $24$-hour digital clock displays hours and minutes only.

How many displayed times in a $24$-hour period contain at least one occurrence of the digit

$5$

?

[UKMT JMO $2019$ Section B, B$5$]

Note by Yajat Shamji
2 weeks, 5 days ago

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You have until this Friday, $3:00$pm!

- 2 weeks, 5 days ago

0:05

0:15

0:25

0:35

0:45

0:50

0:51

0:52

0:53

0:54

0:55

0:56

0:57

0:58

0:59

Well, that's fifteen of 5 occurrences in an hour. We have 24 of these hours, but for the 5th and 15th hours, 5 will be present 60 times, for each minute. Thus,

Number of occurrences of the number $5$ : $22 \times 15 + 2 \times 60$

(22 remaining hours have an occurrence of 5, fifteen times, and the other two hours have it the whole hour or 60 times)

The answer is : $\boxed{450}$

- 2 weeks, 4 days ago

@Yajat Shamji - Why is there a time limit for these probs? Just curious, is it coz' you'll be posting a solution after the given time? Anyway, is my answer correct?

- 2 weeks, 4 days ago

Yes, it's because I'll be posting the official solution...

As for answers, I can check...

- 2 weeks, 4 days ago

Ok :)

- 2 weeks, 4 days ago

But.. the method is different.

- 2 weeks, 4 days ago

I see...okay then, I'll wait for your method...

- 2 weeks, 4 days ago

You have to wait 3 days!

Haha...!

- 2 weeks, 4 days ago