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

$-1 , \dfrac{5}{6}$ and $\dfrac{-2}{3}$

To prove: $\Big(-1 + \dfrac{5}{6}\Big) + \dfrac{-2}{3} = -1 + \Big(\dfrac{5}{6} + \dfrac{-2}{3}\Big)$

Taking LHS:

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

$= \Big(\dfrac{-1 \times 6}{1 \times 6} + \dfrac{5 \times 1}{6 \times 1}\Big) + \dfrac{-2}{3} \\[1em] = \Big(\dfrac{-6}{6} + \dfrac{5}{6}\Big) + \dfrac{-2}{3} \\[1em] = \Big(\dfrac{-6 + 5}{6}\Big) + \dfrac{-2}{3} \\[1em] = \dfrac{-1}{6}+ \dfrac{-2}{3} \\[1em]$

LCM of 6 and 3 is 2 x 3 = 6

$= \dfrac{-1 \times 1}{6 \times 1} + \dfrac{-2 \times 2}{3 \times 2} \\[1em] = \dfrac{-1}{6} + \dfrac{-4}{6} \\[1em] = \dfrac{-1 + (-4)}{6} \\[1em] = \dfrac{-5}{6} \\[1em]$

Taking RHS: $-1 + \Big(\dfrac{5}{6} + \dfrac{-2}{3}\Big) \\[1em] = \dfrac{-1}{1} + \Big(\dfrac{5}{6} + \dfrac{-2}{3}\Big) \\[1em]$

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

LCM of 1 and 6 is 2 x 3 = 6

$= \dfrac{-1 \times 6}{1 \times 6} + \dfrac{1 \times 1}{6 \times 1} \\[1em] = \dfrac{-6}{6} + \dfrac{1}{6} \\[1em] = \dfrac{-6 + 1}{6} \\[1em] = \dfrac{-5}{6} \\[1em]$

∴ LHS = RHS

$\Big(-1 + \dfrac{5}{6}\Big) + \dfrac{-2}{3} = -1 + \Big(\dfrac{5}{6} + \dfrac{-2}{3}\Big)$

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