SOLUTION: a digital timer counts down from 3 min to zero, one second at a time. for how many seconds does atleast one of these digits show a 5?

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Question 1208133: a digital timer counts down from 3 min to zero, one second at a time. for how many seconds does atleast one of these digits show a 5?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52877)   (Show Source): You can put this solution on YOUR website!
.
a digital timer counts down from 3 min to zero, one second at a time.
for how many seconds does at least one of these digits show a 5?
~~~~~~~~~~~~~~~~~~~~~~~ 

(a)  First (leftmost) digit is never 5.



(b)  Consider the set of numbers XY5 from 1 to 180 inclusive with the last 
     (rightmost) digit of 5 and the middle digit Y =/=5. 

     These numbers are

           005, 015, 025, 035, 045, , 065, 075, 0.85, 0.95,
           
           105, 115, 125, 135, 145, , 165, 175.


     The crossed numbers are those with 5 in the middle position.

     The amount of such  "survived" numbers is 9 + 7 = 16.

     The counter counts/adds 1 as each such a number appears.



(c)  To it, we should add appearances X5Z.

     There are exactly  10 + 10 = 20  such appearances

         050, 051, 052, 053, 054, 055, 056, 057, 058, 059,

         150, 151, 152, 153, 154, 155, 156, 157, 158, 159.


     The counter adds 1 as each such a number appears.



(d)  Thus the total (the last) number on the counter at the end is the sum 16 + 20 = 36.    ANSWER

Solved.

-----------------------

What I did to solve the problem is THIS:

        I separated the set of numbers, containing at least one digit "5",
        in two disjoint subsets  (b)  and  (c)  in a way that the counter adds  1
        at every appearance of a number from each of the two subsets.



Answer by greenestamps(13209)   (Show Source): You can put this solution on YOUR website!


It appears that tutor @ikleyn solved the problem for a timer which counts down from 180 seconds to 0 seconds.

My guess is that the timer counts down from 3:00 to 0:00. In that case....

The digit for the minutes is never 5.

As the timer counts down, the digits representing the numbers of seconds repeat the same sequence in each minute, so we can count the number of times that a 5 appears as either of those digits in one minute and multiply our answer by 3 (for minutes digit 2, 1, or 0).

In each minute, the digits representing the seconds contain at least one 5 for the following numbers of seconds:
59 through 50 (10 occurrences)
45, 35, 25, 15, and 05 (5 occurrences)

Total number of times in each minute that at least one of the digits is 5: 10+5 = 15.

Number of times in 3 minutes that at least one of the digits is 5: 15*3 = 45.

ANSWER: 45
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