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I found this problem intriguing: $355 / 113 = 3.14159292035398\ldots$ gives the approximation of $\pi$ in $7$ correct numbers, say $C(355/113)=7$, but it number of digits in numerator + number of digits in denominator is six, say $L(355/113)=6$. How many rationals $a/b$ there are such that $L(a/b)<C(a/b)$?

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Here is a few OEIS sequences which may help answer this question:

A002485 https://oeis.org/A002485 A002486 https://oeis.org/A002486

I hope this helps.

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