Is there an equivalent to the distributive law for division over subtraction and/or addition?

I understand that the the distributive law cannot be applied to division over addition/subtraction, but is there an equivalent law to expand it out. For example, I know: $$100 \times (5 + 3) = (100 \times 5) + (100 \times 3),$$ but with $$100 \div (5 + 3),$$ I can't do (that is, is not equal to) $$(100 \div 5) + (100\div 3).$$

Is there a way I can "distribute" when it is division over addition/subtraction?

• Just like addition is associative and commutative but subtraction is not, multiplication distributes over sums, is commutative and associative, but division is not. You can "distribute" if you have a sum followed by a division, $$(a+b)/c = (a/c) + (b/c)$$but not the other way around, because "dividing by $c$" is really the same as "multiplying by the reciprocal of $c$"; but because in general there is no relation between the reciprocal of $a+b$ and the reciprocals of $a$ and of $b$, in general there is no simple relation between dividing by $a+b$ and divisions by $a$ and by $b$. Apr 18, 2011 at 20:56
• Thanks Arturo, it makes sense, I was afraid that would be the answer. I happen to have my division followed by my subtraction/addition, so I guess I am stuck with no distribution :) Apr 18, 2011 at 21:01

$$\dfrac ab + \dfrac ac = a\left( \dfrac 1b + \dfrac 1c \right) = a(b+c) \div (bc)$$