Question 632433
The number 0.757575... is {{{75/99}}} and 0.767676... is {{{76/99}}}.
You can figure out 0.757575... as a fraction because
if n=0.757575... , then 100n=75.757575... and subtracting one equation from the other, you get
100n-n=75 --> (100-1)n=75 --> 99n=75 --> {{{n=75/99}}}
You can figure out that 0.767676... = {{{76/99}}} similarly.
You can choose an integer m between {{{75^2=5625}}} and {{{76^2=5776}}}
({{{2625<m<5776}}}),
and then {{{sqrt(m)/99}}} will be an irrational number between 0.757575... and 0.767676... because

0.757575... = {{{75/99=sqrt(5625)/99<sqrt(m)/99<sqrt(5776)/99=76/99}}} = 0.767676...