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Proportions: Introduction (page 2 of 7)

Sections: Ratios, Proportions, Checking proportionality, Solving proportions

A ratio is one thing compared to or related to another thing; it is just a statement or an expression. A proportion is two ratios that have been set equal to each other; a proportion is an equation that can be solved. When I say that a proportion is two ratios that are equal to each other, I mean this in the sense of two fractions being equal to each other. For instance, 5/10 equals 1/2. Solving a proportion means that you are missing one part of one of the fractions, and you need to solve for that missing value. For instance, suppose you were given the following equation:

    x / 10 = 1 / 2

You already know, by just looking at this equation and comparing the two fractions, that x must be 5, but suppose you hadn't noticed this. You can solve the equation by multiplying through on both sides by 10 to clear the denominators:

    10 ( x / 10 ) = 10 ( 1 / 2 )

              x = 5 Copyright © Elizabeth Stapel 2001-2011 All Rights Reserved

Verifying what we already knew, we get that x = 5.

Often times, students are asked to solve proportions before they've learned how to solve rational equations, which can be a bit of a problem. If you haven't yet learned about rational expressions (that is, polynomial fractions), then you will need to "get by" with "cross-multiplication".

To cross-multiply, you take each denominator aCROSS the "equals" sign and MULTIPLY it on the other fraction's numerator. The cross-multiplication solution of the above exercise looks like this:

    2 (x) = 10 (1)

Then you would solve the resulting linear equation by dividing through by 2.

Proportions wouldn't be of much use if you only used them for reducing fractions. A more typical use would be something like the following:

  • Consider those ducks and geese we counted back at the park. Their ratio was 16 ducks to 9 geese. Suppose that there are 192 ducks. How many geese are there?
  • I'll let "G" stand for the unknown number of geese. Then I'll clearly label the orientation of my ratios, and set up my proportional equation:

      .(ducks : geese) : 16/9 = 192/G




    I'll multiply through on both sides by the G to get it up to the left-hand side, out of the denominator, and then I'll solve for the value of G:

      .16/9 = 192/G, 16G = 9(192) = 1728, G = 108

    Then there are 108 geese.

To solve the propertion above with cross-multiplication, you would do the following:

    16/9 = 192/G then 16G = 192×6

"Cross-multiplying" is standard language, in that it is very commonly used, but it is not technically a mathematical term. You might not see it in your book, but you will almost certainly hear it in your class or study group.

Notice how, in my equation at the beginning of my solution above, I wrote out the ratio in words:

    (ducks) / (geese)

This is not standard notation, but it can be very useful for setting up your proportion. Clearly labelling what values are represented by the numerators and denominators will help you keep track of what each number stands for. In other words, it will help you set up your proportion correctly. If you do not set up the ratios consistently (if, in the above example, you mix up where the "ducks" and the "geese" go in the various fractions), you will get an incorrect answer. Clarity can be very important.

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Cite this article as:

Stapel, Elizabeth. "Proportions: Introduction." Purplemath. Available from Accessed



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