$${\frac{\left({\frac{{\mathtt{5}}}{{\mathtt{51}}}}\right)}{\left({\frac{{\mathtt{15}}}{{\mathtt{17}}}}\right)}} = \left({\frac{{\mathtt{5}}}{{\mathtt{51}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{17}}}{{\mathtt{15}}}}\right)$$
$$\left({\frac{{\mathtt{5}}}{{\mathtt{51}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{17}}}{{\mathtt{15}}}}\right) = {\frac{{\mathtt{1}}}{{\mathtt{9}}}} = {\mathtt{0.111\: \!111\: \!111\: \!111\: \!111\: \!1}}$$
$${\frac{\left({\frac{{\mathtt{5}}}{{\mathtt{51}}}}\right)}{\left({\frac{{\mathtt{15}}}{{\mathtt{17}}}}\right)}} = \left({\frac{{\mathtt{5}}}{{\mathtt{51}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{17}}}{{\mathtt{15}}}}\right)$$
$$\left({\frac{{\mathtt{5}}}{{\mathtt{51}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{17}}}{{\mathtt{15}}}}\right) = {\frac{{\mathtt{1}}}{{\mathtt{9}}}} = {\mathtt{0.111\: \!111\: \!111\: \!111\: \!111\: \!1}}$$