Solve Rational Equations Using Proportions

By Mary Jane Sterling

Proportions are nice to work with when you’re solving rational equations, because you can eliminate the fractions or change them so that they feature better denominators.

A proportion is an equation in which one fraction is set equal to another. For example, the equation

image0.png

is a proportion.

When you have the proportion

image1.png

the following are also true:

  • the cross-products, ad and bc, are equal, giving you ad = bc.

  • the reciprocals,

    image2.png

    are equal (you can flip the proportion).

Another wonderful feature of proportions is that you can reduce the fractions in a proportion by finding common factors in four different directions: top, bottom, left, and right. The ability to reduce a proportion comes in handy when you have large numbers in the equation.

Here are the rules for reducing proportions across the top (numerators), bottom (denominators), left, and right, and an example for each:

image3.png

The reduced forms of the proportions make cross-multiplication much easier and more manageable. Take the following proportion, for example. You first reduce across the numerators, and then you reduce the left fractions.

image4.png

reduces

image5.png

which reduces again

image6.png

which is then written

image7.png

Now cross multiply and solve the quadratic equation.

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When x – 6 = 0, x = 6; and when x + 1 = 0, x = –1. Both solutions check out.

To solve an equation such as

image9.png

you can find a common denominator and then multiply each side by the common denominator, but here’s a quicker, easier way:

  1. Add

    image10.png

    to each side and add the terms with the same denominator to form a proportion.

    image11.png

  2. Cross multiply.

    (x + 5)x = 24

  3. Simplify the quadratic equation, and set it equal to zero.

    x2 + 5x = 24 becomes x2 + 5x – 24 = 0

  4. Solve for the solutions by factoring.

    (x + 8)(x – 3) = 0

    The solutions are x = –8 or x = 3. Both work.