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12.8: Rational Equations Using Proportions

Difficulty Level: Basic Created by: CK-12
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Suppose you were traveling on a paddle boat at a constant speed. In 6 minutes, you traveled \begin{align*}x\end{align*} meters, and in 10 minutes, you traveled \begin{align*}x+4\end{align*} meters. Could you find the value of \begin{align*}x\end{align*} in this scenario? If so, how would you do it? 

Rational Equations Using Proportions

You are now ready to solve rational equations! There are two main methods you will learn to solve rational equations:

  • Cross-products
  • Lowest common denominators

This Concept will focus on cross-products. When two rational expressions are equal, a proportion is created and can be solved using its cross-products.

For example, to solve \begin{align*}\frac{x}{5}=\frac{(x+1)}{2}\end{align*}, cross multiply and set the products to be equal.

\begin{align*}\frac{x}{5} = \frac{(x+1)}{2} \rightarrow 2(x)=5(x+1)\end{align*}

Solve for \begin{align*}x\end{align*}:

\begin{align*}2(x) &= 5(x+1) \rightarrow 2x=5x+5\\ 2x-5x &= 5x-5x+5\\ -3x &= 5\\ x &= -\frac{5}{3}\end{align*}

Let's solve the following rational equations:

  1. \begin{align*}\frac{2x}{x+4}=\frac{5}{x}\end{align*}.

\begin{align*}\frac{2x}{x+4} &= \frac{5}{x} \rightarrow 2x^2=5(x+4)\\ 2x^2 &= 5(x+4) \rightarrow 2x^2=5x+20\\ &2x^2-5x-20 = 0\end{align*}

Notice that this equation has a degree of two; that is, it is a quadratic equation. We can solve it using the quadratic formula.

\begin{align*}x=\frac{5 \pm \sqrt{185}}{4} \Rightarrow x \approx -2.15 \ \text{or} \ x \approx 4.65\end{align*}

  1. \begin{align*}\frac{3x}{5x+2}=\frac{1}{x}\end{align*}

Start by cross multiplying:

\begin{align*}\frac{3x}{5x+2}=\frac{1}{x} \Rightarrow 3x^2=5x+2 \Rightarrow 3x^2-5x-2=0\end{align*}

Since this equation has a squared term as its highest power, it is a quadratic equation. We can solve this by using the quadratic formula, or by factoring.

Since there are no common factors, start by finding the product of the coefficient in front of the squared term and the constant:

\begin{align*}3\cdot -2=-6\end{align*}

What factors of -6 add up to 5?

That would be -6 and 1, since -6+1=-5

Factor, beginning by breaking up the middle term, \begin{align*}-5x\end{align*}, as above:

\begin{align*} 0&=3x^2-5x-2=3x^2-6x+1x-2\\ &=3x(x-2)+1(x-2)= (3x+1)(x-2) \end{align*}

Use the Zero Product Principle:

\begin{align*}(3x+1)(x-2)=0 \Rightarrow 3x+1=0 \text{ or } x-2=0 \Rightarrow x=-\frac{1}{3} \text{ or } x=2\end{align*}


Example 1

Earlier, were asked to find the value of \begin{align*}x\end{align*} given that you were traveling on a paddle boat at a constant speed. You traveled \begin{align*}x\end{align*} meters in 6 minutes, and \begin{align*}x+4\end{align*} meters in 10 minutes.

You can use cross-products to find the value of \begin{align*}x\end{align*} in this scenario.

First, let's set up an equation using the given information. The distance traveled in a specific amount of time can be written as a fraction of the distance over the time.

\begin{align*}\frac{x}{6}=\frac{x+4}{10} \end{align*} 

Now, we can cross multiply. 

\begin{align*}\text{Cross multiply:} && \frac{x}{6}=\frac{x+4}{10} \Rightarrow 10x=6(x+4)\\ && 10x=6x+24\\ && 4x=24\\ && x=6\\ \end{align*}  

Example 2

Solve \begin{align*} -\frac{x}{2}=\frac{3x-8}{x}\end{align*}.

\begin{align*} \text{Cross multiply:} && -\frac{x}{2}=\frac{3x-8}{x} \Rightarrow x^2&=-2(3x-8)\\ \text{Set one side equal to zero to get a quadratic equation:} && 0&=x^2+2(3x-8)\\ \text{Simplify by distributing:} && 0&=x^2+6x-16\\ \text{Factor by determining } -16=8\cdot -2 \text{ and } 6=8+(-2): && 0&=x^2-2x+8x-16=x(x-2)+8(x-2)=(x+8)(x-2)\\ \text{Use the zero product principle:} && 0&=(x+8)(x-2) \Rightarrow x+8=0 \text{ or } x-2=0 \Rightarrow x=-8 \text{ or } x=2 \end{align*}


Solve the following equations.

  1. \begin{align*}\frac{2x+1}{4}=\frac{x-3}{10}\end{align*}
  2. \begin{align*}\frac{4x}{x+2}=\frac{5}{9}\end{align*}
  3. \begin{align*}\frac{5}{3x-4}=\frac{2}{x+1}\end{align*}
  4. \begin{align*}\frac{7x}{x-5}=\frac{x+3}{x}\end{align*}
  5. \begin{align*}\frac{2}{x+3}-\frac{1}{x+4}=0\end{align*}
  6. \begin{align*}\frac{3x^2+2x-1}{x^2-1}=-2\end{align*}

Mixed Review

  1. Divide: \begin{align*}-2 \frac{9}{10} \div - \frac{15}{8}\end{align*}.
  2. Solve for \begin{align*}g: -1.5 \left(-3 \frac{4}{5}+g \right)=\frac{201}{20}\end{align*}.
  3. Find the discriminant of \begin{align*}6x^2+3x+4=0\end{align*} and determine the nature of the roots.
  4. Simplify \begin{align*}\frac{6b}{2b+2}+3\end{align*}.
  5. Simplify \begin{align*}\frac{8}{2x-4}- \frac{5x}{x-5}\end{align*}.
  6. Divide: \begin{align*}(7x^2+16x-10) \div (x+3)\end{align*}.
  7. Simplify \begin{align*}(n-1)*(3n+2)(n-4)\end{align*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.8. 

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proportion A statement in which two fractions are equal: \frac{a}{b} = \frac{c}{d}.
Zero Product Property The only way a product is zero is if one or more of the terms are equal to zero: a\cdot b=0 \Rightarrow a=0 \text{ or } b=0.
Rational Expression A rational expression is a fraction with polynomials in the numerator and the denominator.

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