Math Lesson 4.3.4 - Cross Product in Direct Proportion

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Welcome to our Math lesson on Cross Product in Direct Proportion, this is the fourth 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.

Cross Product in Direct Proportion

As we explained earlier, a proportion is written in the form

a:b = c:d


a/b = c/d

where a and b are similar quantities as well as c and d. This is the ratio form of expressing a proportion, where all units are simplified.

However, another form of writing a proportion is by expressing it as a rate, i.e. as two fractions where the numerator and denominator of each, represent two different quantities (the units cannot be simplified).

For example, if 16 tailors can make 40 shirts in one day, then obviously if the number of tailors doubles (16 × 2 = 32), the number of shirts they can produce doubles too (40 × 2 = 80). Therefore, if we use the above notation, we have

a = 40 shirts
b = 80 shirts
c = 16 workers
d = 32 workers

Hence, the two methods of expressing this proportion is

a/b = c/d


40 shirts/80 shirts = 16 workers/32 workers


a/c = b/d


40 shirts/16 workers = 80 shirts/32 workers

From the two above methods of expressing a proportion, we can see that a and b can switch their position as well as c and d and still the proportion is valid. From chapter 1, we know that the only two operations that possess the commutative property are addition and multiplication. Since ratios are expressed as divisions, which are multiplications by the inverse, we can write the ratio as two products on either side of the expression. Hence, we can write a direct proportion as:

a × d = b × c

This method of expressing a proportion as equality of products is known as cross product.

Considering the numbers written in the above example, we can write

40 shirts × 32 workers = 80 shirts × 16 workers

The result in either side is 1280, which confirms the proportion.

The cross product method of writing a proportion allows us to check the veracity of proportion in an easier way without involving rational numbers, simplifications or GCF calculations.

Example 4

Check the veracity of the following proportions by using cross products

  1. 3:12 = 7:21
  2. 20:15 = 36:27

Solution 4

  1. First, we write the possible proportion in fractional form. We have:

    3/12 = 7/21

    Now, we can write the proportion as cross product:

    3 × 21 = 12 × 7
    63 = 84

    Since the last equality is not true, it is clear that the original proportion is not true either.

  2. Again, we write the possible proportion in the fractional form. We have:

    20/15 = 36/27

    Writing the expression as a cross product yields

    20 × 27 = 15 × 36
    540 = 540

    Therefore, the original proportion is true.

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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