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# math

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A list of 10 positive integers, each less than 15, has an average of 6. What is the maximum number of ones (the number 1, not the digit) that could be in the list?

Mar 17, 2021

#1
+118015
+1

Here's  my  best  shot

If  the 10 ntegers  have  an average  of  6.....then  the  total sum of the  integers must  be 10 * 6   =  60 because  60 / 10  =  6

Therefore......we  want  to make  the  sum  of  the  "non-one" digits  as small as possible  and  the  number  of ones as large as possible

Possible Number of 1's         Sum  of  the other digits

1                                      59

2                                      58

3                                      57

4                                      56

5                                      55

6                                      54

7                                      53

8                                      52

9                                      51

10                                      50

The last is clealrly not possible  ( all ones)

The next-to-the last  is also  not possible......the  largest  sum  that  could be  created  with 9 ones  is  9 + 14 = 23

The  third-to-the last is also  not possible ...the largest possible sum  =  8 ones + 2 * 14  =  36

The fourth-to-the-last  is also not possible....the largest sum =  7 ones  + 3 * 14  = 49

Six  ones   is possible  because   14 + 14  + 13 + 13  =  54

So....the max number of 1s   =    6

Mar 17, 2021