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Proportion

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Suppose that $\alpha$ is inversely proportional to $\beta$. If $\alpha = -3$ when $\beta = -6$, find $\alpha$ when $\beta = 18$. Express your answer as a fraction.

Apr 18, 2022

#1
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$$\frac{-3}{6}=\frac{a}{18}\quad$$

$$\frac{a}{18}=-\frac{1}{2}$$

$$:\quad a=-9$$

-Vinculum

Apr 18, 2022
#2
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You have assumed $$\alpha$$ is directly proportional to $$\beta$$ instead of inversely proportional to $$\beta$$.

MaxWong  Apr 19, 2022
#3
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Since $$\alpha$$ is inversely proportional to $$\beta$$$$\alpha = \dfrac k\beta$$ for some nonzero real number $$k$$, which we call the proportionality constant.

We find k by substituting a pair of values of $$\alpha$$ and $$\beta$$ into the equation.

Substituting $$\alpha = -3$$ and $$\beta = -6$$$$-3 = \dfrac k{-6}$$. Solving gives $$k = 18$$.

Now, substitute $$\beta = 18$$ and $$k = 18$$ to get the corresponding value of $$\alpha$$ when $$\beta = 18$$.

$$\alpha = \dfrac{18}{18} = \boxed{1}$$.

Apr 19, 2022