Associative Property

Concept Explanation

Associative Property

It states that you can add or multiply numbers regardless of how they are grouped. In both the groups the sum is the same. Addition and multiplication are associative for rational numbers. Subtraction and division are not associative for rational numbers.

Rational numbers follow the associative property for addition and multiplication.

Suppose x, y and z are rational then for addition: x+(y+z)=(x+y)+z

For multiplication: x(yz)=(xy)z.

An important properties that should be remembered are:

0 is an additive identity and 1 is a multiplicative identity for rational numbers.

Associative Property of Addition in Rational Numbers:

The addition of rational numbers is associative i.e. if $frac{a}{b},frac{c}{d};and;frac{e}{f}$ are any three rational numbers, the

$frac{a}{b}+left ( frac{c}{d}+frac{e}{f} right )=left ( frac{a}{b}+frac{c}{d} right )+frac{e}{f}$

Verification: In order to verify this property, let us consider three rational number $frac{-3}{5}, frac{5}{4};and;frac{-4}{9}$ and if the following expression holds true so we can say that addition supports associativity

$frac{-3}{5}+left ( frac{5}{4}+frac{-4}{9} right )=left ( frac{-3}{5}+frac{5}{4} right )+frac{-4}{9}$

The left hand side of the expression can be simplified as

= $frac{-3}{5}+left ( frac{5}{4}+frac{-4}{9} right )$

$=frac{-3}{5}+left ( frac{45}{36}+frac{-16}{36} right )$

$=frac{-3}{5}+frac{45-16}{36}$

$=frac{-3}{5}+frac{29}{36}=frac{-108}{180}+frac{145}{180}=frac{37}{108}$

and, the right hand side of the expression can be simplified as

$R.H.S.=left ( frac{-3}{5}+frac{5}{4} right )+frac{-4}{9}$

$=left ( frac{-12}{20}+frac{25}{20} right )+frac{-4}{9}$

$=frac{-12+25}{20}+frac{-4}{9}$

$=frac{13}{20}+frac{-4}{9}=frac{117}{180}+frac{-80}{180}=frac{117+(-80)}{180}=frac{37}{180}=L.H.S.$

$therefore ;;frac{-3}{4}+left ( frac{5}{6}+frac{-4}{9} right )=left ( frac{-3}{4}+frac{5}{6} right )+frac{-4}{9}$

Similarly, it can be verified for other rational numbers.

Hence, associative property is true under addition.

Associative Property of Subtraction in Rational Numbers:

The subtraction of rational numbers is not associative, i.e. for any three rational numbers $frac{a}{b},frac{c}{d}andfrac{e}{f},$ we have

$left ( frac{a}{b}-frac{c}{d} right )-frac{e}{f}neq frac{a}{b}-left ( frac{c}{d}-frac{e}{f} right )$

Verification: In order to verify this property, let us consider three rational number $frac{2}{9},frac{5}{2};and;frac{3}{4}$ and if the following expression holds true so we can say that subtraction does not support associativity.

$left(frac{2}{9}-frac{5}{2}right)- frac{3}{4}neq frac{2}{9}-left(frac{5}{2}-frac{3}{4}right)$

The left hand side of the expression can be simplified as

$L.H.S.= left(frac{2}{9}-frac{5}{2}right)- frac{3}{4}=left(frac{4}{18}-frac{45}{18}right)-frac{3}{4}$

$=frac{-41}{18}-frac{3}{4}=frac{-82}{36}-frac{27}{36}=frac{-109}{36}$

and, the right hand side of the expression can be simplified as

$R.H.S. =frac{2}{9}-left(frac{5}{2}- frac{3}{4}right)=frac{2}{9}-left(frac{20}{8}- frac{6}{8}right)$

$=frac{2}{9}-frac{14}{8}=frac{16}{72}-frac{126}{72}=frac{116}{72}neq L.H.S.$

Hence LHS $large neq$ RHS

Hence Associative property is not true for subtraction of rational numbers

Associative Property of Multiplication in Rational Numbers:

The multiplication of rational numbers is associative. That is, if $frac{a}{b},frac{c}{d}andfrac{e}{f}$ are three rational numbers, then

$left ( frac{a}{b}times frac{c}{d} right )times frac{e}{f}=frac{a}{b}times left ( frac{c}{d}times frac{e}{f} right )$

Verification: In order to verify this property, let us consider three rational number $frac{3}{2},frac{-1}{5}andfrac{2}{7}$ and if the following expression holds true so we can say that multiplication supports associativity

$left (frac{3}{2}times frac{-1}{5} right )times frac{2}{7}=frac{3}{2}timesleft( frac{-1}{5}times frac{2}{7} right)$

The left hand side of the expression can be simplified as

$L.H.S. =left (frac{3}{2}times frac{-1}{5} right )times frac{2}{7}=frac{-3}{10} times frac{2}{7} =frac{-6}{70}=frac{-3}{35}$

and, the right hand side of the expression can be simplified as

$R.H.S=frac{3}{2}times(frac{-1}{5}times frac{2}{7})=frac{3}{2}timesfrac{-2}{35}=frac{-6}{70}=frac{-3}{35}=L.H.S.$

Hence LHS = RHS

Hence, associative property is true under multiplication.

Associative Property of Division in Rational Numbers:

The division of rational numbers is not associative. That is, if $frac{a}{b},frac{c}{d}andfrac{e}{f}$ are three rational numbers, then

$left ( frac{a}{b}div frac{c}{d} right )div frac{e}{f}neq frac{a}{b}div left ( frac{c}{d}div frac{e}{f} right )$

$left (frac{3}{2}div frac{-1}{5} right )div frac{2}{7}neq frac{3}{2}divleft( frac{-1}{5}div frac{2}{7} right)$

The left hand side of the expression can be simplified as

$L.H.S. =left (frac{3}{2}div frac{-1}{5} right )div frac{2}{7}=left(frac{3}{2}timesfrac{5}{-1} right )divfrac{2}{7}$

$=frac{-15}{2} div frac{2}{7}=frac{-15}{2}times frac{7}{2} =frac{-105}{4}$

and, the right hand side of the expression can be simplified as

$R.H.S=frac{3}{2}divleft(frac{-1}{5}div frac{2}{7}right)=frac{3}{2}divleft(frac{-1}{5} timesfrac{7}{2}right )$

$=frac{3}{2}divfrac{-7}{10}=frac{3}{2}timesfrac{10}{-7}=frac{30}{-14}neq L.H.S.$

Hence LHS $large neq$ RHS

Hence Associative property is not true for division of rational numbers

For every non-zero rational number $frac{a}{b}$ , we have

$(i);frac{a}{b}div frac{a}{b}=1$ $(ii);frac{a}{b}div left ( -frac{a}{b} right )=-1$ $(iii);left ( frac{-a}{b} right )div frac{a}{b}=-1$

The division of rational numbers is neither commutative nor associative.

Sample Questions

(More Questions for each concept available in Login)

Question : 1

Which of the following is an example of the Associative Property of Multiplication?
A. (1/3 + 1/2 ) + 1/4 = 1/3 + (1.4 + 1/2)	B. 5/2 x 4 = 4 x 5/2	C. (1/3 x 1/4) x 1/5 = 1/3 x (1/4 x 1/5)	D. 1/3 x 1/2 = 1/6
Right Option : C
View Explanation

Question : 2

9 x (3 x 2) = (9 x 3) x 2 , is explained by which property?
A. Associative over multiplication	B. Commutative property	C. Distributive property	D. Associative over addition
Right Option : A
View Explanation

Question : 3

Reciprocal of 3 is __________________
A. -3	B. $-frac {1}{3}$	C. $frac {1}{3}$	D. 0
Right Option : C
View Explanation

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Attention !!!!!

Associative Property

Associative Property

Associative Property of Addition in Rational Numbers:

Associative Property of Subtraction in Rational Numbers:

Associative Property of Multiplication in Rational Numbers:

Associative Property of Division in Rational Numbers:

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