How to Solve Proportions in Rational Expressions

Solve proportions by cross-multiplying.

Rational expressions refer to polynomial fractions, or fractions that have constants, variables or exponents in the numerator and denominator. Proportions refer to rational expressions that are equal to each other. Proportions are usually written either with colons, such as 2:x = 4:8, or as fractions, such as 2/x = 4/8. When you solve a proportion, you find the value of x, allowing you to complete one or both fractions. Cross-multiplying to solve proportions is straightforward, but polynomials with many terms can make the process more complicated.

Instructions

1. Set the two proportions as fractions, if necessary. For example, write x+3:4 = 4x:10 as (x+3)/4 = (4x)/10.

2. Multiply the denominator of one fraction by the numerator of the other fraction, and vice versa. Set these expressions as equal to each other. For example, (x+3)*10 = 4*4x.

3. Simplify the cross-multiplied expressions. For example, (x+3)*10 = 4*4x simplifies to 10x+30=16x.

4. Solve the equation. For example, 10x+30=16x simplifies to 30=6x, which simplifies to x=5.

5. Plug the solution back into the equation to check your work. For example, (5+3)/4 = (4*5)/10 simplifies to 8/4 = 20/10, which is correct.

Tips & Warnings

• You may need to factor complex polynomial expressions before cross-multiplying.

References

• "Introductory Algebra"; Richard N. Aufmann, et al.; 2010
• "Pre-Algebra"; Michael Buckley; 2006