For each set of rational numbers, given below, verify the associative property of addition of rational numbers:

$\dfrac{-2}{5} , \dfrac{4}{15}$ and $\dfrac{-7}{10}$

To prove:

$\Big(\dfrac{-2}{5} + \dfrac{4}{15}\Big) + \dfrac{-7}{10} = \dfrac{-2}{5} + \Big(\dfrac{4}{15} + \dfrac{-7}{10}\Big)$

Taking LHS:

$\Big(\dfrac{-2}{5} + \dfrac{4}{15}\Big) + \dfrac{-7}{10} \\[1em]$ LCM of 5 and 15 is 3 x 5 = 15

$= \Big(\dfrac{-2 \times 3}{5 \times 3} + \dfrac{4 \times 1}{15 \times 1}\Big) + \dfrac{-7}{10} \\[1em] = \Big(\dfrac{-6}{15} + \dfrac{4}{15}\Big) + \dfrac{-7}{10} \\[1em] = \Big(\dfrac{-6 + 4}{15}\Big) + \dfrac{-7}{10} \\[1em] = \dfrac{-2}{15}+ \dfrac{-7}{10} \\[1em]$ LCM of 15 and 10 is 2 x 3 x 5 = 30

$= \dfrac{-2 \times 2}{15 \times 2} + \dfrac{-7 \times 3}{10 \times 3} \\[1em] = \dfrac{-4}{30} + \dfrac{-21}{30} \\[1em] = \dfrac{-4 + (-21)}{30} \\[1em] = \dfrac{-25}{30} \\[1em] = \dfrac{-5}{6} \\[1em]$

Taking RHS: $\dfrac{-2}{5} + \Big(\dfrac{4}{15} + \dfrac{-7}{10}\Big)$

LCM of 15 and 10 is 2 x 3 x 5 = 30 $\dfrac{-2}{5} + \Big(\dfrac{4 \times 2}{15 \times 2} + \dfrac{-7 \times 3}{10 \times 3}\Big) \\[1em] = \dfrac{-2}{5} + \Big(\dfrac{8}{30} + \dfrac{-21}{30}\Big) \\[1em] = \dfrac{-2}{5} + \Big( \dfrac{8 +(-21)}{30}\Big) \\[1em] = \dfrac{-2}{5} + \dfrac{-13}{30} \\[1em]$

LCM of 5 and 30 is 2 x 3 x 5 = 30

$= \dfrac{-2 \times 6}{5 \times 6} + \dfrac{-13 \times 1}{30 \times 1} \\[1em] = \dfrac{-12}{30} + \dfrac{-13}{30} \\[1em] = \dfrac{-12 + (-13)}{30} \\[1em] = \dfrac{-25}{30} \\[1em] = \dfrac{-5}{6} \\[1em]$

∴ LHS = RHS

$\Big(\dfrac{-2}{5} + \dfrac{4}{15}\Big) + \dfrac{-7}{10} = \dfrac{-2}{5} + \Big(\dfrac{4}{15} + \dfrac{-7}{10}\Big)$

So, the associative property for the addition of the rational number is verified.