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$\frac{a}{5a-1} : \frac{ a^{2} }{25 a^{2}-10a+1 }= \frac{a}{5a-1}* \frac{(5a-1) ^{2} }{ a^{2} }= \frac{5a-1}{a} \\\\ \frac{5* \frac{1}{6}-1 }{ \frac{1}{6} }= \frac{ \frac{5}{6}-1 }{ \frac{1}{6} }= \frac{- \frac{1}{6} }{ \frac{1}{6} }=-1$

$\frac{a}{4a-1} : \frac{ a^{2} }{16 a^{2}-8a+1 } = \frac{a}{4a-1}* \frac{(4a-1) ^{2} }{ a^{2} }= \frac{4a-1}{a}\\\\ \frac{4* \frac{1}{7}-1 }{ \frac{1}{7} }= \frac{ \frac{4}{7}-1 }{ \frac{1}{7} }= \frac{ \frac{3}{7} }{ \frac{1}{7} }= \frac{3*7}{7*1} =3$

$\frac{a}{9a-1} : \frac{ a^{2} }{81 a^{2} -18a+1}= \frac{a}{9a-1}* \frac{(9a-1) ^{2} }{ a ^{2} }= \frac{9a-1}{a}\\\\ \frac{9* \frac{1}{6}-1 }{ \frac{1}{6} }= \frac{ \frac{9}{6} -1}{ \frac{1}{6} }= \frac{- \frac{1}{2} }{ \frac{1}{6} }=- \frac{1*6}{2*1}=-3$