how to convert any fraction to decimal
You should implement rational numbers yourself. Store number as two integers. Use the greatest common divisor to factor out superflous representation. Literally write school algorithms for rational numbers - find common denominator for addition and then normalize number. At the end convert it to decimal representation with fixed precision. Namely use symbolic calculation as long as you can.

From your example:
$\frac{1}{3} + \frac{1}{5} = \frac{5}{15} + \frac{3}{15} = \frac{8}{15} \approx 0.5333$

If your task really requires to convert it to string at the first step, then you have to assume some precision, e.g. four numbers in decimal expansion, and then write your own code to operate on numbers. The exact decimal expansion is not always possible, as you have already realized, so $\frac{1}{3}$ would expand to "0.3333" (arbitrary length from my example. It may seem weird at first, but the floats are also inexact, with precision given by fixed length representation (e.g. 64, 80, 128 bits).

$\frac{1}{3} \approx 0.333; \frac{1}{5} \approx 0.2000$
So
$0.3333 + 0.2000 = 0.5333$
It gives the same approximate result.