what is the approximation value of pi/3+7-2root3 as a fraction

Answer 1

pi/3+7-2root3
= π/3 + 7 - 2√3
= 4.583095936... on my calculator

There is a field of study in "numerical methods", where one has to find the best fraction that could serve as an approximation for a given number.
This deals with a topic called "continued fractions", which is an interesting topic on its own

the best fraction with neither numerator or denominator more than 3 digits is 55/12

the best fraction with neither numerator or denominator more than 4 digits is 4826/1053

Using the number given as a comparison
55/12 = 4.5833..., off by .000237, not bad
but...
4826/1053 = 4.583095916 , off by .000000019, now I would say that is close

4826/1053 is the winner, without going into more digits in my fraction