 UPSKILL MATH PLUS

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### Theory:

The distributive property states that multiplying a sum by a number gives the same result as multiplying each added by the number and then adding the products together.

For any number $$a$$, $$b$$ and $$c$$, we have $$a × b + c = a × b + a × c$$.

Consider the number $$4$$, $$3$$ and $$2$$ we have

$$4 × 3 + 2 = 4 × 3 + 4 × 2$$. (LHS $$=$$ RHS)

LHS: $$4 × 2 + 3 = 4 × 5$$

$$= 20$$.

RHS: $$4 × 3 + 4 × 2$$

$$= 12 + 8$$

$$= 20$$.

Thus, $$4 × 3 + 2 = 4 × 3 + 4 × 2$$. (LHS $$=$$ RHS)

For any three rational numbers $\frac{a}{b}$, $\frac{c}{d}$ and $\frac{e}{f}$, we have: $\frac{a}{b}$ $$×$$ $\left(\frac{c}{d}+\frac{e}{f}\right)$ $$=$$ $\left(\frac{a}{b}×\frac{c}{d}\right)$ $$+$$ $\left(\frac{a}{b}×\frac{e}{f}\right)$.

Consider the rational numbers $\frac{-3}{4}$$\frac{2}{3}$ and $\frac{-5}{6}$ we have

$\frac{-3}{4}$ $$×$$ $\left\{\frac{2}{3}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{-5}{6}\right\}$ $$=$$ $\left(\frac{-3}{4}×\frac{2}{3}\right)$ $$+$$ $\left(\frac{-3}{4}×\frac{-5}{6}\right)$. (LHS $$=$$ RHS)

LHS: $\frac{-3}{4}$ $$×$$ $\left\{\frac{2}{3}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{-5}{6}\right\}$

$\frac{-3}{4}$ $$×$$ $\left\{\frac{2}{3}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{-5}{6}\right\}$

$$=$$ $\frac{-3}{4}$ $$×$$ $\left\{\frac{2}{3}\phantom{\rule{0.147em}{0ex}}×\frac{2}{2}+\phantom{\rule{0.147em}{0ex}}\frac{-5}{6}\right\}$ (LCM was taken to get the same denominator)

$$=$$ $\frac{-3}{4}$ $$×$$ $\left\{\frac{\left(4+\left(-5\right)\right)}{6}\right\}$

$$=$$ $\frac{-3}{4}$ $$×$$ $\frac{-1}{6}$ $$=$$ $\frac{3}{24}$ $$=$$ $\frac{1}{8}$.

RHS: $\left(\frac{-3}{4}×\frac{2}{3}\right)$ $$+$$ $\left(\frac{-3}{4}×\frac{-5}{6}\right)$:

$\left(\frac{-3}{4}×\frac{2}{3}\right)$ $$=$$ $\frac{-3×2}{4×3}$ $$=$$ $-\frac{6}{12}$ $$=$$ $-\frac{1}{2}$.

And $\left(\frac{-3}{4}×\frac{-5}{6}\right)$ $$=$$ $\frac{5}{8}$.

Therefore $\left(\frac{-3}{4}×\frac{2}{3}\right)$ $$+$$ $\left(\frac{-3}{4}×\frac{-5}{6}\right)$

$$=$$ $-\frac{1}{2}$ $$+$$ $\frac{5}{8}$

$$=$$$-\frac{1}{2}×\frac{4}{4}$ $$+$$ $\frac{5}{8}×\frac{1}{1}$ (LCM was taken to get the same denominator)

$$=$$ $\frac{\left(-4+5\right)}{8}$

$$=$$ $\frac{1}{8}$.

Thus, $\frac{-3}{4}$ $$×$$ $\left\{\frac{2}{3}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{-5}{6}\right\}$ $$=$$ $\left(\frac{-3}{4}×\frac{2}{3}\right)$ $$+$$ $\left(\frac{-3}{4}×\frac{-5}{6}\right)$. (LHS $$=$$ RHS)