Question 54925
This is actually a neat problem.  Before we begin, are we allowed to use unreduced fractions?  For example, one solution is 2/3, but also you could use 6/9, which has the same value.  Do these both count?  I think they do.

So, begin with the values of P and Q, which are as follows:
P=1, P=2, P=3,P=4, P=5, P=6, P=7, P=8, P=9
Q=1, Q=2, Q=3,Q=4, Q=5, Q=6, Q=7, Q=8, Q=9


The ratio of these numbers {{{P/Q}}} must be between {{{1/2 }}} and {{{1}}}, not including ratios that are equal to {{{1/2}}} or {{{1}}}.


Start with P=1, and try 1/1, 1/2, 1/3, etc. down to 1/9.  None of these work.


Now, try P=2:  2/1 and 2/2 do not work.  NOTE THAT 2/3 works, but 2/4, 2/5, etc do not work.


Now, try P=3:  3/1, 3/2, and 3/3 do NOT work.  NOTE THAT 3/4, 3/5 work, but none of the rest will work.


Now, try P=4:  4/1, 4/2, 4/3, and 4/4 do NOT work.  NOTE THAT 4/5, 4/6, 4/7 work, but the rest do not work.


Now, try P=5:  Are you getting the pattern?  The ones that work are 5/6, 5/7, 5/8, and 5/9.


For P=6:  6/7, 6/8, and 6/9


For P=7:  7/8, and 7/9


For P=8:  8/9


For P=9:  None work.


Summary:  {{{2/3}}}, {{{3/4}}}, {{{3/5}}}, {{{4/5}}}, {{{4/6}}}, {{{4/7}}},{{{5/6}}},{{{5/7}}}, {{{5/8}}},{{{5/9}}}, {{{6/7}}}, {{{6/8}}}, {{{6/9}}}, {{{7/8}}}, {{{7/9}}}, {{{8/9}}}


I counted 16 such numbers, including both fractions that reduce and those that do not reduce.


R^2 at SCC