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Properties of Rational Numbers - Distributive Property of Multiplication Over Addition for Rational Numbers

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Distributive Property of Multiplication Over Addition for Rational Numbers:

1. Distributivity for multiplication over Addition: 

To understand this, consider the rational numbers `(-3)/4, 2/3, and (-5)/6`.

`(-3)/4 xx {2/3 + ((-5)/6)} = (-3)/4 xx {((4) + (-5))/6}`

`= (-3)/4 xx ((-1)/6) = 3/24 = 1/8`

Also `(- 3)/4 xx 2/3 = (- 3 xx 2 )/(4 xx 3) = (- 6)/12 = (- 1)/2`

And `(- 3)/4 xx (-5)/6 = 5/8`

Therefore `((-3)/4 xx 2/3) + ((-3)/4 xx (-5)/6) = (-1)/2 + 5/8 = 1/8`

Thus, `-3/4 xx {2/3 + (-5)/6} = ((-3)/4 xx 2/3) + ((-3)/4 xx (-5)/6)` 

For all rational numbers a, b, and c, a(b + c) = ab + ac.

2. Distributivity for multiplication over subtraction: 

Consider the rational numbers  `3/8, (- 2)/5, 6/7`.

`3/8 xx ((-2)/5 - 6/7 ) = 3/8 xx (-14 - 30)/35 = 3/8 xx (- 44)/35 = (- 33)/70` 

Also , `3/8 xx (- 2)/5 - 3/8 xx 6/7 = (- 6)/40 - 18/56 = (- 3)/20 - 9/28 = (- 33)/70`

For all rational numbers a, b, and c, a(b – c) = ab – ac.

Example

Find: `2/5 xx (-3)/7 - 1/14 - 3/7 xx 3/5`.

`2/5 xx (-3)/7 - 1/14 - 3/7 xx 3/5`

`= 2/5 xx (-3)/7 - 3/7 xx 3/5 - 1/14`    .......(by commutativity)

`= 2/5 xx (-3)/7 + ((-3)/7) xx 3/5 - 1/14`.

`= (-3)/7(2/5 + 3/5) - 1/14`                .......(by distributivity)

`= (- 3)/7 xx 1 - 1/14`

`= (- 6 - 1)/14`

`= -1/2`.

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