Lesson Objectives

- Demonstrate an understanding of ratios
- Learn how to use the equality test for fractions
- Learn the definition of a proportion
- Learn how to determine if two ratios or two rates represent a proportion

## What is a Proportion

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.

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. $$\frac{7}{9}\hspace{.25em}≠ \hspace{.25em}\frac{49}{60}$$

Example 3: Replace the ? with "=" or "≠" $$\frac{32 \hspace{.2em}shrubs}{50 \hspace{.2em}acres}\hspace{.25em}? \hspace{.25em}\frac{160 \hspace{.2em}shrubs}{250 \hspace{.2em}acres}$$ 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. $$\frac{32 \hspace{.2em}shrubs}{50 \hspace{.2em}acres}\hspace{.25em}=\hspace{.25em}\frac{160 \hspace{.2em}shrubs}{250 \hspace{.2em}acres}$$ Example 4: Replace the ? with "=" or "≠" $$\frac{15 \hspace{.2em}yen}{370 \hspace{.2em}pesos}\hspace{.25em}? \hspace{.25em}\frac{160 \hspace{.2em}yen}{2590 \hspace{.2em}pesos}$$ 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. $$\frac{15 \hspace{.2em}yen}{370 \hspace{.2em}pesos}\hspace{.25em}≠ \hspace{.25em}\frac{160 \hspace{.2em}yen}{2590 \hspace{.2em}pesos}$$

### 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: $$\frac{3 \hspace{.25em}cups \hspace{.25em}flour}{5 \hspace{.25em}cups \hspace{.25em}sugar}$$ 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. $$\frac{6 \hspace{.25em}cups \hspace{.25em}flour}{10 \hspace{.25em}cups \hspace{.25em}sugar}$$ 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 "≠" $$\frac{32 \hspace{.2em}shrubs}{50 \hspace{.2em}acres}\hspace{.25em}? \hspace{.25em}\frac{160 \hspace{.2em}shrubs}{250 \hspace{.2em}acres}$$ 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. $$\frac{32 \hspace{.2em}shrubs}{50 \hspace{.2em}acres}\hspace{.25em}=\hspace{.25em}\frac{160 \hspace{.2em}shrubs}{250 \hspace{.2em}acres}$$ Example 4: Replace the ? with "=" or "≠" $$\frac{15 \hspace{.2em}yen}{370 \hspace{.2em}pesos}\hspace{.25em}? \hspace{.25em}\frac{160 \hspace{.2em}yen}{2590 \hspace{.2em}pesos}$$ 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. $$\frac{15 \hspace{.2em}yen}{370 \hspace{.2em}pesos}\hspace{.25em}≠ \hspace{.25em}\frac{160 \hspace{.2em}yen}{2590 \hspace{.2em}pesos}$$

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