Mathematics I (Math 1132)

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Durham College, Mathematics
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Mathematics I (Math 1132)

Ratios and Proportions

As mentioned at the beginning of this unit, there are several ways to represent a fraction, one of which was in ratio form. Just like a fraction, let’s say a over b, a ratio is a comparison of two or more quantities (called terms) that are often expressed in the same units, making the ratio dimensionless – a/b would also be written as a : b or a to b in ratio form. Dimensionless ratios are handy because you do not have to worry about units; some common examples used in mathematics include:

  • Trigonometric ratios (sine, cosine, tangent)
  • Radian measure of angles
  • Gear ratios

Ratios are reduced the same way fractions are. The video below shows two examples of ratios being reduced.

Introduction to Proportions

A proportion is an equation obtained when two ratios are set equal to each other. If the ratio a : b is equal to the ratio c : d, we have the proportion:

a : b  =  c : d

This reads “the ratio of a to b equals the ratio of c to d” or “a is to b as c is to d.” In fraction form, this is written as:


The two inside terms of a proportion are called the means, and the two outside terms are the extremes. Therefore, b and c are the means, and a and d are the extremes.

Mean Proportional

When the means of two ratios are the same, let’s say:

a : b  =  b : d

… b is referred to as the mean proportional or geometric mean. b can be found by rearranging the ratio into:


Once you begin learning algebra, rearranging this ratio on your own will become less of a mystery. For now, all you’re expected to know is how to use it. Use the comment section below to show your proof if you already know how it’s done. Note that the phrase geometric mean is used in place of mean proportional.

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