7.1 Ratio and proportion

Ratios and proportions form the foundation for nearly everything you'll study in this unit on similarity. A ratio gives you a way to compare two quantities, and a proportion tells you that two ratios are equal. Once you can set up and solve proportions fluently, you'll be ready to work with similar figures, scale factors, and indirect measurement.

Ratios and Proportions

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Definition of ratios and proportions

A ratio compares two quantities. You can write it as $a:b$ or $\frac{a}{b}$. For example, $3:4$ and $\frac{3}{4}$ mean the same thing.

There are two types of ratios to keep straight:

• Part-to-part ratios compare two parts of a whole. If a class has 12 boys and 16 girls, the ratio of boys to girls is $12:16$, which simplifies to $3:4$.
• Part-to-whole ratios compare one part to the total. In that same class, the ratio of boys to all students is $12:28$, which simplifies to $3:7$.

To simplify any ratio, divide both terms by their greatest common factor. For $12:18$, the GCF is 6, so it simplifies to $2:3$.

A proportion is an equation stating that two ratios are equal:

$\frac{a}{b} = \frac{c}{d}$

Two key properties make proportions useful:

• Cross Products Property: If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$. For example, $\frac{2}{3} = \frac{4}{6}$ because $2 \cdot 6 = 3 \cdot 4 = 12$.
• Reciprocal Property: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{b}{a} = \frac{d}{c}$. Flipping both fractions preserves the equality.

The Cross Products Property is also how you check whether a proportion is true. If the cross products aren't equal, the ratios aren't proportional.

Problem-solving with ratios

To solve for a missing value in a proportion, use cross multiplication:

1. Set up the proportion with the unknown as a variable: $\frac{2}{3} = \frac{x}{12}$
2. Cross multiply: $2 \cdot 12 = 3 \cdot x$, giving you $24 = 3x$
3. Solve for the variable: $x = 8$

Scaling problems rely on this same process. Suppose a map has a scale of $1:50{,}000$ and two cities are 4 cm apart on the map. Set up the proportion:

$\frac{1}{50{,}000} = \frac{4}{d}$

Cross multiply to get $d = 200{,}000$ cm, which converts to 2 km.

Proportions also come up constantly with similar figures. When two triangles (or rectangles, or any polygons) are similar, their corresponding sides are proportional. If triangle ABC is similar to triangle DEF, then:

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$

You'll use this relationship throughout the rest of the unit.

Real-world applications of proportions

• Recipes: A bread recipe calls for flour and water in a $5:3$ ratio. If you use 10 cups of flour, you need 6 cups of water.
• Scale models: An architect's model built at $1:200$ scale means a wall that's 3 cm on the model represents $3 \times 200 = 600$ cm (6 m) in real life.
• Finance: Exchange rates are ratios. If 1 USD = 0.92 EUR, then converting 50 USD means solving $\frac{1}{0.92} = \frac{50}{x}$, giving $x = 46$ EUR.

Relationship between ratios and fractions

The ratio $a:b$ can be written as the fraction $\frac{a}{b}$, and from there you can convert to a decimal or percentage. These are all different ways of expressing the same comparison:

• $3:4 = \frac{3}{4} = 0.75 = 75\%$
• $2:5 = \frac{2}{5} = 0.4 = 40\%$

One thing to watch: a ratio like $3:4$ doesn't always mean "three-fourths of something." If it's a part-to-part ratio (say, 3 red marbles to 4 blue marbles), the fraction of red marbles out of the total is actually $\frac{3}{7}$, not $\frac{3}{4}$. Pay attention to whether a problem gives you a part-to-part or part-to-whole ratio before converting to a fraction.