Proportion Definition Lesson

In our last lesson, we learned how to use a ratio to show the relationship between two quantities. In this lesson, we will expand on this topic and learn about proportions. Before we get into the definition of a proportion, we first need to understand how we can determine if two fractions are equal.

Equality Test for Fractions

The equality test for fractions tells us that two fractions are equal if their cross products are equal. Cross products are formed by multiplying the denominator of one fraction by the numerator of the other. Let's take a look at a few examples.
Example 1: Replace the ? with "=" or "≠".
$$\frac{3}{5}\hspace{.25em}? \hspace{.25em}\frac{21}{35}$$ Form the cross products: 35 x 3 = 105
5 x 21 = 105
Since the cross products are equal, the two fractions are equal. $$\frac{3}{5}\hspace{.25em}=\hspace{.25em}\frac{21}{35}$$ Example 2: Replace the ? with "=" or "≠".
$$\frac{7}{9}\hspace{.25em}? \hspace{.25em}\frac{49}{60}$$ Form the cross products: 60 x 7 = 420
9 x 49 = 441
Since the cross products are not equal, the two fractions are not equal. $$\frac{7}{9}\hspace{.25em}≠ \hspace{.25em}\frac{49}{60}$$

Definition of a Proportion

When two ratios or two rates are equal, they are called a proportion. Proportions have lots of uses. One great example is to scale up or down a recipe. Let’s suppose a cake recipe calls for 5 cups of sugar and 3 cups of flour. We can write the ratio of flour to sugar as » 3:5. If we wanted to be more descriptive, we could use a fraction and write what each number represents: $$\frac{3 \hspace{.25em}\text{cups flour}}{5 \hspace{.25em}\text{cups sugar}}$$ Suppose we want to double the recipe. We can do this by doubling the sugar and the flour. Let's multiply each part of the ratio by 2: $$\frac{3}{5}\cdot \frac{2}{2}=\frac{6}{10}$$ Now we know we would use 6 cups of flour and 10 cups of sugar. $$\frac{6 \hspace{.25em}\text{cups flour}}{10 \hspace{.25em}\text{cups sugar}}$$ The two ratios here » 3:5 and 6:10 are equal and therefore represent a proportion. To determine if two ratios or two rates represent a proportion, we use the equality test for fractions. We only need to work with the number parts. Let's take a look at a few examples:
Example 3: Replace the ? with "=" or "≠". $$\frac{32 \hspace{.2em}\text{shrubs}}{50 \hspace{.2em}\text{acres}}\hspace{.25em}? \hspace{.25em}\frac{160 \hspace{.2em}\text{shrubs}}{250 \hspace{.2em}\text{acres}}$$ We check for proportionality using the equality test for fractions. Let's form the cross products with the number parts only:
50 x 160 = 8000
250 x 32 = 8000
Since the cross products are equal, we have a proportion. $$\frac{32 \hspace{.2em}\text{shrubs}}{50 \hspace{.2em}\text{acres}}\hspace{.25em}=\hspace{.25em}\frac{160 \hspace{.2em}\text{shrubs}}{250 \hspace{.2em}\text{acres}}$$ Example 4: Replace the ? with "=" or "≠". $$\frac{15 \hspace{.2em}\text{yen}}{370 \hspace{.2em}\text{pesos}}\hspace{.25em}? \hspace{.25em}\frac{160 \hspace{.2em}\text{yen}}{2590 \hspace{.2em}\text{pesos}}$$ We check for proportionality using the equality test for fractions. Let's form the cross products with the number parts only:
370 x 160 = 59,200
2590 x 15 = 38,850
Since the cross products are not equal, we do not have a proportion. $$\frac{15 \hspace{.2em}\text{yen}}{370 \hspace{.2em}\text{pesos}}\hspace{.25em}≠ \hspace{.25em}\frac{160 \hspace{.2em}\text{yen}}{2590 \hspace{.2em}\text{pesos}}$$

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