6. The ratio of the sum of the first $n$ terms of two arithmetic sequences is $\frac{9 n+2}{n+7}$, the ratio of their 5th terms is . $\qquad$

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6. $\frac{83}{16}$.

Let the two arithmetic sequences be $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, and their sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively. Then, $\frac{S_{n}}{T_{n}}=\frac{9 n+2}{n+7}$. Given that $a_{n}=\frac{S_{2 n-1}}{2 n-1}$ and $b_{n}=\frac{T_{2 n-1}}{2 n-1}$, it follows that $\frac{a_{n}}{b_{n}}=\frac{S_{2 n-1}}{T_{2 n-1}}$. Therefore, $\frac{a_{5}}{b_{5}}=\frac{S_{9}}{T_{9}}=\frac{83}{16}$.