# Rational Equations Using Proportions

## Key Questions

Solving proportions is like solving fractions:

$a : b \to c : d$ can be rewritten as $\frac{a}{b} = \frac{c}{d}$ and now you can solve for any of the variables.

See an example below:

#### Explanation:

3 to 4 is like what to 16?

This can be rewritten as a proportion:

$3 : 4 \to c : 16$

Which can be rewritten as:

$\frac{3}{4} = \frac{c}{16}$

Which can be solved as:

$\textcolor{red}{16} \times \frac{3}{4} = \textcolor{red}{16} \times \frac{c}{16}$

$\cancel{\textcolor{red}{16}} \textcolor{red}{4} \times \frac{3}{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}}} = \cancel{\textcolor{red}{16}} \times \frac{c}{\textcolor{red}{\cancel{\textcolor{b l a c k}{16}}}}$

$12 = c$

$c = 12$

3 to 4 is like 12 to 16?

• We multiply the numerator of each (or one) side by the denominator of the other side. For example, if I have want to solve for $x$ for the following equation:

$\frac{x}{5} = \frac{3}{4}$

I can use cross-multiplication, and the equation becomes:

$x \cdot 4 = 3 \cdot 5$

$4 x = 15$

$x = \frac{15}{4} = 3.75$

• A proportion is a statement that two ratios are equal to each other.
For example $\frac{3}{6} = \frac{5}{10}$ (We sometimes read this "3 is to 6 as 5 is to 10".)

There are $4$ 'numbers' (really number places) involved. If one or more of those 'numbers' is a polynomial, then the proportion becomes a rational equation.

For example: $\frac{x - 2}{2} = \frac{7}{x + 3}$ ("x-2 is to 2 as 7 is to x+3").

Typically, once they show up, we want to solve them. (Find the values of $x$ that make them true.)

In the example we would "cross multiply" or multiply both sides by the common denominator (either description applies) to get:
$\left(x - 2\right) \left(x + 3\right) = 2 \cdot 7$. Which is true exactly when
${x}^{2} + x - 6 = 14$ Which in turn, is equivalent to
${x}^{2} + x - 20 = 0$ (Subtract 14 on both sides of the equation.)
Solve by factoring $\left(x + 5\right) \left(x - 4\right) = 0$
so we need $x + 5 = 0$ or $x - 4 = 0$ the first requires
$x = - 5$ and the second $x = 4$.

Notice that we can check our answer:
$\frac{- 5 - 2}{2} = - \frac{7}{2}$ and $\frac{7}{- 5 + 3} = \frac{7}{-} 2 = - \frac{7}{2}$. So the ratios on both sides are equal and the statement is true.