Overview:
Ratios and proportions are common ways to compare numbers. They have many mathematical applications, such as scale models, percentages, and interest. Ratios and proportions can be set up with variables, and solved for those variables.
What Is a Ratio?
A ratio is a comparison of two numbers by division, and can be expressed in three different ways. Suppose there are 2 blue marbles in a jar to every 3 red marbles. That ratio of blue marbles to red marbles could be expressed as 2 to 3, 2:3, or the fraction 2/3. The fractional form is the one most often used in equations.
What Is a Rate?
In a ratio, the units must be the same, as the ratio of 6 grams to 11 grams. In a rate, the units are different, such as 40 miles per gallon, or 13 feet per second. Many problems that involve rates can be set up comparing proportions.
What Is a Proportion?
A proportion states that two ratios are equal, such as 2/3 = 4/6. Proportions have a useful property that can be used to solve equations when one of the values is unknown, called the means-extremes property of proportions. In algebraic terms, if a/b equals c/d, then ad = bc. That can be tested in the above example, as 2(6) = 4(3).
Solving Proportions with Algebra
If a set of proportions contains a variable, it can be turned into an equation and solved using the means-extremes property and cross multiplying. Suppose the problem were 3/15 = b/45. Then it could be turned into an equation by cross multiplying, so that 3(45) =15b, or 135 = 15b, or 135/15 = b =9. If the problem were x/5 = (x + 3)/10, then 10x = 5(x + 3), or 10x = 5x + 15. Subtracting 5x from both sides, 10x-5x = 5x-5x +15, or 5x = 15, or 5x/5 =15/5, or x =3.
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