7. Calculate in the most rational way:

\[
3 \frac{1}{117} \cdot 4 \frac{1}{119}-1 \frac{116}{117} \cdot 5 \frac{118}{119}-\frac{5}{119}
\]
7. Solution. Let \(a=\frac{1}{117}\) and \(b=\frac{1}{119}\). Then \(3 \frac{1}{117}=3+a\),

\[
4 \frac{1}{119}=4+b, \quad 1 \frac{116}{117}=2-a, \quad 5 \frac{118}{119}=6-b
\]

therefore, the given expression can be written as:

\[
(3+a)(4+b)-(2-a)(6-b)-5 b
\]

After some simple transformations, we get that this algebraic expression is identically equal to \(10 a=\frac{10}{117}\).