An easier way to do this is to subtract the mixed numbers without conversion to an improper fraction. For this to happen, the fraction part of the mixed number in the first number (\(\frac{1}{6}\) in our case) needs to be larger than the \(\frac{3}{5}\).

Let's compare.

\(\frac{1}{6} = \frac{5}{30}\)

\(\frac{3}{5} = \frac{18}{30}\)

\(\frac{18}{30}>\frac{5}{30}\).

Since the second fraction is larger, we need to modify the first fraction to make it larger.

So from \(42\frac{1}{6}\), we are going to take \(1\) (Which is equal to \(\frac{6}{6}\)) out of the \(42\), and add it to the fractions part giving us \(41\frac{7}{6}\)

\(\frac{7}{6} = \frac{35}{30}\).

\(\frac{35}{30}>\frac{18}{30}\)

Now we can subtract easily.

\(41\frac{35}{30} - 27\frac{18}{30} = 14\frac{17}{30}\)

So our answer is yet again \(\boxed{14\frac{17}{30}}\)