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 Possible Percent Problems Part 2 (Posted on 2010-01-11)
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 No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 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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