Math Lesson 4.3.1 - Definition of Proportion

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Welcome to our Math lesson on Definition of Proportion, this is the first lesson of our suite of math lessons covering the topic of Proportion, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Definition of Proportion

If two quantities involved in a situation (we call them variables) are related in such a way that if one quantity changes, the other quantity changes at the same (or opposite) degree, we say that these quantities are in proportion with (or proportional to) each other. In simpler words, a proportion indicates to us how one thing changes in relation to another.

For example, if a car travels 240 km with 8L of gasoline, it can travel other for other 720 km with 24 L of gasoline if moving at the same rate (speed). The unit rate u.r. of gasoline consumption therefore is

u.r. = 240 km/8 L = 30 km/L


u.r. = 720 km/24 L = 30 km/L

Since there two unit rates are equal, we say they are in proportion. Hence, it is clear that a proportion includes two equal fractions, i.e. if

a/b = c/d

then a and c are proportional (as well as b and d).

In other words, if a/b and c/d are two equal ratios or rates, they are in proportion.

We can also write the above proportion in the form

a:b = c:d

where a is the first term, b is the second term, c is the third term and d is the fourth term of the proportion. The terms a and d are called outer terms or extremes, while b and c are called inner terms or means of the proportion.

Example 1

Which of the following pairs of fractions are in proportion?

  1. 4/18 and 3/15
  2. 21/7 and 12/4
  3. 24/36 and 9/15

Solution 1

First, we express each fraction in the simplest terms. In a certain sense, a fraction expressed in the simplest terms is similar to the unit rate (in fact, one of the two associated unit rates) we discussed earlier. Thus,

  1. 4/18 = 4 ÷ 2/18 ÷ 2 = 2/9


    3/15 = 3 ÷ 3/15 ÷ 3 = 1/5
    Since these two fractions are not equal, the quantities they represent are not in proportion.
  2. 21/7 = 3/1 = 3


    12/4 = 3/1 = 3
    These fractions are equal, so the quantities they represent are in proportion.
  3. 24/36 = 24 ÷ 12/36 ÷ 12 = 2/3


    9/15 = 9 ÷ 3/15 ÷ 3 = 3/5
    Since these two fractions are not equal, the quantities they represent are not in proportion.

More Proportion Lessons and Learning Resources

Ratio and Proportion Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Lesson IDMath Lesson TitleLessonVideo
4.3.1Definition of Proportion
4.3.2Direct Proportion
4.3.3Graph of Direct Proportion
4.3.4Cross Product in Direct Proportion
4.3.5Inverse (Indirect) Proportion
4.3.6Graph of Inverse Proportion

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  7. Continuing learning ratio and proportion - read our next math tutorial: Properties of Proportion. Geometric Mean

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