UPSKILL MATH PLUS

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Distributive property of multiplication over addition: $$a(b + c) = ab + ac$$ where $$a$$, $$b$$ and $$c$$ are rational numbers.

Distributive property of multiplication over subtraction: $$a(b - c) = ab - ac$$ where $$a$$, $$b$$ and $$c$$ are rational numbers.
Now let's see the distributive property of multiplication over addition.
Distribution of multiplication over addition:
If $$a$$, $$b$$ and $$c$$ are three integers, then:

$$a ×$$ ($$b + c$$) $$=$$ ($$a × b$$) $$+$$ ($$a × c$$)

Let $$a = \frac{-9}{2}$$, $$b = \frac{-5}{4}$$ and $$c = \frac{1}{3}$$.

Then, $$\frac{-9}{2} \times \left(\frac{-5}{4} + \frac{1}{3} \right)$$ $$=$$ $$\left(\frac{-9}{2} \times \frac{-5}{4} \right) +$$ $$\left(\frac{-9}{2} \times \frac{1}{3} \right)$$

Let us verify the property.

Consider the LHS.

LHS $$= \frac{-9}{2} \times \left(\frac{-5}{4} + \frac{1}{3} \right)$$

$$= \frac{-9}{2} \times \left(\frac{-15 + 4}{12} \right)$$ [LCM of $$3$$ and $$4$$ is $$12$$]

$$= \frac{-9}{2} \times \left(\frac{-11}{12} \right)$$

$$= \frac{99}{24}$$

$$= \frac{33}{8}$$ ---- ($$1$$)

Now, consider the RHS.

RHS $$=$$ $$\left(\frac{-9}{2} \times \frac{-5}{4} \right) +$$ $$\left(\frac{-9}{2} \times \frac{1}{3} \right)$$

$$= \frac{45}{8} + \left(\frac{-9}{6} \right)$$

$$= \frac{45}{8} - \frac{3}{2}$$

$$= \frac{45 - 12}{8}$$ [LCM of $$2$$ and $$8$$ is $$8$$]

$$= \frac{33}{8}$$ ---- ($$2$$)

From equations ($$1$$) and ($$2$$), we have:

$$\frac{-9}{2} \times \left(\frac{-5}{4} + \frac{1}{3} \right)$$ $$=$$ $$\left(\frac{-9}{2} \times \frac{-5}{4} \right) +$$ $$\left(\frac{-9}{2} \times \frac{1}{3} \right)$$

This implies that $$a ×$$ ($$b + c$$) $$=$$ ($$a × b$$) $$+$$ ($$a × c$$).
Distribution of multiplication over subtraction:
If $$a$$, $$b$$ and $$c$$ are three integers, then:

$$a ×$$ ($$b - c$$) $$=$$ ($$a × b$$) $$-$$ ($$a × c$$)

Let $$a = \frac{-9}{2}$$, $$b = \frac{-5}{4}$$ and $$c = \frac{1}{3}$$.

Then, $$\frac{-9}{2} \times \left(\frac{-5}{4} - \frac{1}{3} \right)$$ $$=$$ $$\left(\frac{-9}{2} \times \frac{-5}{4} \right) -$$ $$\left(\frac{-9}{2} \times \frac{1}{3} \right)$$

Let us verify the property.

Consider the LHS.

LHS $$= \frac{-9}{2} \times \left(\frac{-5}{4} - \frac{1}{3} \right)$$

$$= \frac{-9}{2} \times \left(\frac{-15 - 4}{12} \right)$$ [LCM of $$3$$ and $$4$$ is $$12$$]

$$= \frac{-9}{2} \times \left(\frac{-19}{12} \right)$$

$$= \frac{171}{24}$$

$$= \frac{57}{8}$$ ---- ($$1$$)

Now, consider the RHS.

RHS $$=$$ $$\left(\frac{-9}{2} \times \frac{-5}{4} \right) -$$ $$\left(\frac{-9}{2} \times \frac{1}{3} \right)$$

$$= \frac{45}{8} - \left(\frac{-9}{6} \right)$$

$$= \frac{45}{8} + \frac{3}{2}$$

$$= \frac{45 + 12}{8}$$ [LCM of $$2$$ and $$8$$ is $$8$$]

$$= \frac{57}{8}$$ ---- ($$2$$)

From equations ($$1$$) and ($$2$$), we have:

$$\frac{-9}{2} \times \left(\frac{-5}{4} - \frac{1}{3} \right)$$ $$=$$ $$\left(\frac{-9}{2} \times \frac{-5}{4} \right) -$$ $$\left(\frac{-9}{2} \times \frac{1}{3} \right)$$

This implies that $$a ×$$ ($$b - c$$) $$=$$ ($$a × b$$) $$-$$ ($$a × c$$).