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Possible Percent Problems Part 2 (Posted on 2010-01-11) Difficulty: 3 of 5
Given whole numbers a and b we can find the ratio a/b and round it to the nearest whole percent.

A value of b is said to generate any percentage that is a possible value of a/b.

For example b=15 generates seven percentages between 0 and 50 including 27% = 4/15 and 47% = 7/15.
There is no smaller b that generates 47% but there is a smaller b (11) that generates 27% (3/11)

What is the smallest value of b such that for every percentage between 0 and 50 that it generates there is a smaller b that also generates it?

No Solution Yet Submitted by Jer    
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Solution re(2): solution Comment 3 of 3 |
(In reply to re: solution by Charlie)

another way of showing this:

The new percents introduced at the b's under 20:

     new             unrounded
 b    %  a b           value
 1
--->  0  0/1        0.000000000  *
 2
---> 50  1/2        0.500000000  *
 3
---> 33  1/3        0.333333333
 4
---> 25  1/4        0.250000000  *
 5
---> 20  1/5        0.200000000  *
---> 40  2/5        0.400000000  *
 6
---> 17  1/6        0.166666667
 7
---> 14  1/7        0.142857143
---> 29  2/7        0.285714286
---> 43  3/7        0.428571429
 8
---> 13  1/8        0.125000000
---> 38  3/8        0.375000000
 9
---> 11  1/9        0.111111111
---> 22  2/9        0.222222222
---> 44  4/9        0.444444444
 10
---> 10  1/10       0.100000000  *
---> 30  3/10       0.300000000  *
 11
--->  9  1/11       0.090909091
---> 18  2/11       0.181818182
---> 27  3/11       0.272727273
---> 36  4/11       0.363636364
---> 45  5/11       0.454545455  *
 12
--->  8  1/12       0.083333333
---> 42  5/12       0.416666667
 13
---> 15  2/13       0.153846154  *
---> 23  3/13       0.230769231
---> 31  4/13       0.307692308
---> 46  6/13       0.461538462
 14
--->  7  1/14       0.071428571
---> 21  3/14       0.214285714
 15
---> 47  7/15       0.466666667
 16
--->  6  1/16       0.062500000
---> 19  3/16       0.187500000
 17
---> 12  2/17       0.117647059
---> 24  4/17       0.235294118
---> 35  6/17       0.352941176  *
---> 41  7/17       0.411764706
 18
---> 28  5/18       0.277777778
---> 39  7/18       0.388888889
 19
--->  5  1/19       0.052631579  *
---> 16  3/19       0.157894737
---> 26  5/19       0.263157895
---> 32  6/19       0.315789474
---> 37  7/19       0.368421053

All 11 of the percents achievable with b=20 (marked with an *) are present in the above, and each b above has added at least one new percent not present above it.

 

DEFDBL A-Z
DIM did(50)
FOR b = 1 TO 20
 PRINT b
 good = 1
 FOR a = 0 TO b / 2
  pct = INT(100 * a / b + .5)
  IF did(pct) = 0 THEN
   good = 0
   PRINT USING "& ## ##"; "--->"; pct; a;
   PRINT "/"; LTRIM$(STR$(b)); TAB(20);
   PRINT USING "##.#########"; a / b
  END IF
  did(pct) = 1
 NEXT a
NEXT b
PRINT

b = 20
FOR a = 0 TO 10
 PRINT a, INT(100 * a / b + .5)
NEXT


  Posted by Charlie on 2010-01-11 15:01:46
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