Question

Which of the following expressions shows that rational numbers are associative under multiplication?

• A. $\frac{2}{3}\times \left(\frac{6}{7}\times \frac{3}{5}\right)=\left(\frac{2}{3}\times \frac{6}{7}\right)\times \frac{3}{5}$
• B. $\frac{2}{3}\times \left(\frac{6}{7}\times \frac{3}{5}\right)=\frac{2}{3}\times \left(\frac{3}{5}\times \frac{6}{7}\right)$
• C. $\frac{2}{3}\times \left(\frac{6}{7}\times \frac{3}{5}\right)=\left(\frac{2}{3}\times \frac{3}{5}\right)\times \frac{6}{7}$
• D. $\left(\frac{2}{3}\times \frac{6}{7}\right)\times \frac{3}{5}=\left(\frac{6}{7}\times \frac{2}{3}\right)\times \frac{3}{5}$

Answer 1

The correct option is A $\frac{2}{3}\times \left(\frac{6}{7}\times \frac{3}{5}\right)=\left(\frac{2}{3}\times \frac{6}{7}\right)\times \frac{3}{5}$

In associativity, the order of the numbers doesn’t change whereas in commutativity, the order of the numbers changes.

E.g.,

a × (b × c) = (a × b) × c ⇒ Associativity
a × b × c = b × c × a ⇒ Commutativity
Hence, $\frac{2}{3}\times \left(\frac{6}{7}\times \frac{3}{5}\right)=\left(\frac{2}{3}\times \frac{6}{7}\right)\times \frac{3}{5}$ is associative because the order of the numbers is not changing.