Associative property of Rational Numbers

Rational numbers follow the associative property for addition and multiplication.

Suppose a/b, c/d and e/f are rational, then the associativity of addition can be written as: 

(a/b) + [(c/d) + (e/f)] = [(a/b) + (c/d)] + (e/f)

Similarly, the associativity of multiplication can be written as:

(a/b) × [(c/d) × (e/f)] = [(a/b) × (c/d)] × (e/f)

Example: Show that (½) + [(¾) + (⅚)] = [(½) + (¾)] + (⅚) and (½) × [(¾) × (⅚)] = [(½) × (¾)] × (⅚).

Solution: (1/2) + [(3/4) + (5/6)] = (1/2) + [(9 + 10)/12]

= (1/2) + (19/12)

= (6 + 19)/12

= 25/12

[(1/2) + (3/4)] + (5/6) = [(2 + 3)/4] + (5/6)

= (5/4) + (5/6)

= (15 + 10)/12

= 25/12

Therefore, (½) + [(¾) + (⅚)] = [(½) + (¾)] + (⅚)

Now, (1/2) × [(3/4) × (5/6)] = (1/2) × (15/24) = 15/48 = 5/16

[(1/2) × (3/4)] × (5/6) = (3/8) × (5/6) = 15/48 = 5/16

Therefore, (½) × [(¾) × (⅚)] = [(½) × (¾)] × (⅚)