Question

Find the value of $\frac{3}{5}+\frac{-7}{6}+\frac{2}{5}+\frac{-5}{6}$.

Answer 1

Rearranging and grouping the numbers in pairs in such a way that each group contains a pair of rational numbers with same denominators, we have
$\frac{3}{5}+\frac{-7}{6}+\frac{2}{5}+\frac{-5}{6}$
$=\frac{3}{5}+(\frac{-7}{6}+\frac{2}{5})+\frac{-5}{6}$
$=\frac{3}{5}+(\frac{2}{5}+\frac{-7}{6})+\frac{-5}{6}$
[Since addition of rational numbers is commutative]
$=\frac{3}{5}+\frac{2}{5}+\frac{-7}{6}+\frac{-5}{6}$
$=(\frac{3}{5}+\frac{2}{5})+(\frac{-7}{6}+\frac{-5}{6})$
[Since addition of rational numbers is associative]
$=\frac{3+2}{5}+\frac{(-7)+(-5)}{6}=\frac{5}{5}+\frac{(-12)}{6}=1+(-2)=-1$