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Properties of Rational Numbers - Associative Property of Rational Numbers

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Associative Property of Rational Numbers:

1. Associativity of Addition of Rational Numbers:

We have,

`(- 2)/3  + [3/5 + ((-5)/6)] = (-2)/3 + ((-7)/30) = (-27)/30 = (-9)/10`.

`[(-2)/3 + 3/5] + ((-5)/6) = (-1)/15 + ((-5)/6) = (-27)/30 = (-9)/10`.

`(-2)/3 + [3/5 + ((-5)/6)] = [(-2)/3 + 3/5] + ((-5)/6)`

`(-1)/2 + [3/7 + ((-4)/3)] and [(-1)/2 + 3/7] + ((-4)/3)`

If the two sums are equal. We find that addition is associative for rational numbers. That is, for any three rational numbers a, b and c, a + (b + c) = (a + b) + c.

2. Associativity of Subtraction of Rational Numbers:

You already know that subtraction is not associative for integers, then what about rational numbers.

`(- 2)/3 - [(- 4)/5 - 1/2] = (- 2)/3 - [(- 8 – 5)/10] = (-2)/3 + 13/10 = 19/30`

`[2/3 - ((-4)/5)] – 1/2 = [(10 + 12)/15] – 1/2 = 22/15 – 1/2 = 29/30`

Subtraction is not associative for rational numbers i.e., for any three rational numbers a, b and c, a - (b - c) ≠ (a - b) - c. 

3. Associativity of Multiplication of Rational Numbers:

`(-7)/5 xx (5/4 xx 2/9) = (- 7)/3 xx 10/36 = (- 70)/108 = (- 35)/54`.

`((-7)/3 xx 5/4) xx 2/9 = (- 35)/12 xx 2/9 = (- 70)/108`

`(-7)/5 xx (5/4 xx 2/9) = ((-7)/3 xx 5/4) xx 2/9` 

Multiplication is associative for rational numbers. That is for any three rational numbers a, b and c, a × (b × c) = (a × b) × c.

 4. Associativity of Division of Rational Numbers:

Let us see if `1/2 ÷ [(-1)/3 ÷ 2/5] = [1/2 ÷ ((-1)/3)] ÷ 2/5 `

We have,  LHS  = `1/2 ÷ [(-1)/3 ÷ 2/5]  = (- 30)/10`

RHS =` [1/2 ÷ ((-1)/3)] ÷ 2/5  = (-15)/4`

We say that division is not associative for rational numbers. That is, for any three rational numbers a, b and c, a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c.

Example

Find: `3/7 + ((-6)/11) + ((-8)/21) + (5/22)`.

`3/7 + ((-6)/11) + ((-8)/21) + (5/22)`.

`= 198/462 + ((-252)/462) + ((-176)/462) + (105/462)`  .....(462 is the LCM 7, 11, 21, and 22)

`= (198 - 252 - 176 + 105)/(462)`

`= (-125)/462`.

`3/7 + ((-6)/11) + ((-8)/21) + (5/22)`.

`= [3/7 + ((-8)/21)] + [(-6)/11 + 5/22]`          .....(by using commutativity and associativity)

`= [(9 + (-8))/21] + [(-12 + 5)/22]`                .....(LCM of 7 and 21 is 21; LCM of 11 and 22 is 22)

`= 1/21 + ((-7)/22)`

`= (22 - 147)/462`

`= (-125)/462`.

Example

Find: `(-4)/5 xx 3/7 xx 15/16 xx ((-14)/9)`.

`(-4)/5 xx 3/7 xx 15/16 xx ((-14)/9)`

`= (- (4 xx 3)/(5 xx 7)) xx ((15 xx (-14))/(16 xx 9))`

`= (-12)/35 xx ((-35)/24)`

`= (-12 xx (-35))/(35 xx 24)`

`= 1/2`.

`(-4)/5 xx 3/7 xx 15/16 xx ((-14)/9)`

`= ((-4)/5 xx 15/16) xx [3/7  xx ((-14)/9)]`        .....(Using commutativity and associativity)

`= (-3)/4 xx ((-2)/3)`

`= 1/2`.

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