How to Calculate Ratios and Proportions in Math

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The concept of proportion is probably familiar to you, but you might not be able to write a strict mathematical definition for it. For example, you might recognize that a 10-year-old is smaller than a normal-sized adult in the same "way" that same adult is smaller than a professional basketball player, even though the three sizes are different.

Similarly, you're probably no stranger to the notion of a ratio. For example, if you are at a sports contest and know that the ratio of opposing fans to friendly fans is high, you might be inclined to be less demonstrative when your favored club scores a goal than you would if this ratio were reversed.

In math and statistics, proportion, percentage and ratio questions abound. Fortunately, a brief explanation of the underlying concepts and a few examples should be enough to make you a proportionally better math student.

Ratios and Proportions

A ratio is fundamentally a fraction, or two numbers expressed as a quotient, such as 3/4 or 179/2,385. But it is a special kind of fraction, one that is used to compare related quantities. For example, if there are 11 boys and 13 girls in a room, the ratio of boys to girls is 11 to 13, which may be written 11/13 or 11:13.

Ratio is the Latin word for "reason." The definition of a rational number is one that can be expressed as a fraction; some numbers, like the value of π in geometry, are irrational and cannot be expressed in such a way, instead being expressed as a never-ending decimal number. Perhaps mathematicians of antiquity found this situation "unreasonable."

A proportion is just an expression setting two ratios equal to each other, using different absolute numbers in the fractions. Proportions are written like ratios are, for example, a/b = c/d or a:b = c:d.

How to Solve Ratios

You don't need a fancy ratio calculator function to solve most simple ratio problems. For example, say you go to the gym 17 times in a 30-day month. What is you ratio of gym days to non-gym days in this month?

The answer is not (gym days/total days), so don't be seduced into thinking the answer is 17:30. Instead, subtract gym days from total days to get non-gym days, the required second part of your ratio. The answer is therefore 17:13 (or 17/13).

How to Calculate Proportion

Sometimes, it is evident without doing any calculations that two ratios are proportional to each other. If you and your dog are the only two animals in a room, and you are told that the adjoining gymnasium contains 457 people and 457 dogs, then you know the proportion of people to dogs is the same in both spaces.

But what about ratios that are not easily compared at a glance? For example, is 17/52 proportional to 3/9? If not, which is greater?

One way to do this would be to compute the decimal numbers of each fraction and see which is greater. But if you understand proportions, you can use cross-multiplication instead, multiplying opposite denominators and numerators:

(17/52) =?= (3/9)

(17)(9) = 153; (3)(52) = 156

Thus the ratios are not quite equal (3/9 is slightly greater), and the fractions are not proportional.

What Is a Proportionality Constant?

A proportionality constant represents the constant difference between proportional ratios. If a is proportional to b, then in the expression a = kb, k is the constant of proportionality. Two variables a and b are said to be inversely proportional when their product ab is a constant for all a and b, that is, when a = C/b and b = C/a.

Example: The number of archery fans is proportional to the number of baseball fans in a given coffee shop. At first, there are 6 archery fans and 9 baseball fans. If the number of baseball fans increases to 24, how many archery fans must there be?

Solve for k, where a = kb, a = 6 and b = 9:
k = 6/9 = 2/3 = 0.667

Now, solve the equation a = (0.667)(24) to get 16 archery fans in the now-more-crowded cafe.

References

About the Author

Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with ScienceBlogs.com and the editor of "Run Strong," he has written for Runner's World, Men's Fitness, Competitor, and a variety of other publications. More about Kevin and links to his professional work can be found at www.kemibe.com.