Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$.
$$\left|\frac pq-\frac{22}7\right|=\left|\frac{7p-22q}{7q}\right|$$
The point is to solve the Diophantine equation $$7p-22q=\pm1$$ and get the greatest possible value for $q$.
It turns out that $7\cdot 311-22\cdot99=-1$, so your fraction is $\frac{311}{99}$