Verify the distributive law for rational numbers.

The distributive law for rational numbers states that, for any three rational numbers p, q and r :

⇒ p × (q + r) = p × q + p × r.

Verification with example :

Let $p = \dfrac{1}{2}$, $q = \dfrac{2}{3}$ and $r = \dfrac{1}{4}$.

Using $p = \dfrac{1}{2}$, $q = \dfrac{2}{3}$ and $r = \dfrac{1}{4}$, we verify that $p \times (q + r) = p \times q + p \times r$.

Solving L.H.S. :

$\Rightarrow \dfrac{1}{2} \times \left(\dfrac{2}{3} + \dfrac{1}{4}\right) \\[1em] \Rightarrow \dfrac{1}{2} \times \left(\dfrac{8}{12} + \dfrac{3}{12}\right) \\[1em] \Rightarrow \dfrac{1}{2} \times \dfrac{11}{12} \\[1em] \Rightarrow \dfrac{11}{24}.$

Solving R.H.S. :

$\Rightarrow \dfrac{1}{2} \times \dfrac{2}{3} + \dfrac{1}{2} \times \dfrac{1}{4} \\[1em] \Rightarrow \dfrac{2}{6} + \dfrac{1}{8} \\[1em] \Rightarrow \dfrac{1}{3} + \dfrac{1}{8} \\[1em] \Rightarrow \dfrac{8}{24} + \dfrac{3}{24} \\[1em] \Rightarrow \dfrac{11}{24}.$

Since L.H.S. = R.H.S.

Hence, the distributive law p × (q + r) = p × q + p × r is verified for rational numbers.