Let's evaluate the expression step by step with the given values \( m = 35 \) and \( n = -18 \).
The expression is:
\[ \frac{1}{5} - \frac{m}{3} - \frac{n}{3} \]
Substituting the values of \( m \) and \( n \):
Calculate \(-\frac{m}{3}\): \[ -\frac{35}{3} = -\frac{35}{3} \]
Calculate \(-\frac{n}{3}\): \[ -\frac{-18}{3} = \frac{18}{3} = 6 \]
Now, substitute these values back into the expression:
\[ \frac{1}{5} - \frac{35}{3} + 6 \]
Combine \( 6 \) and \( -\frac{35}{3} \): Convert \( 6 \) to a fraction with a common denominator of 3: \[ 6 = \frac{18}{3} \] Therefore: \[ -\frac{35}{3} + \frac{18}{3} = -\frac{35 - 18}{3} = -\frac{17}{3} \]
Now, combine that with \( \frac{1}{5} \): We need a common denominator for \( \frac{1}{5} \) and \( -\frac{17}{3} \). The least common multiple of 5 and 3 is 15.
Convert each fraction: \[ \frac{1}{5} = \frac{3}{15} \]
\[ -\frac{17}{3} = -\frac{85}{15} \]
Now, add these together: \[ \frac{3}{15} - \frac{85}{15} = \frac{3 - 85}{15} = \frac{-82}{15} \]
So, the final result is: \[ \frac{-82}{15} \]
