Fundamental Concepts of Ratios

Understanding Ratios and Proportions What Is a Ratio? A ratio is a comparison between two quantities that tells us how many times one quantity contains another. For example, if a sports team wins 4 games for every 3 games it loses, we can express this comparison as a ratio. A ratio can be written in three equivalent ways: Verbal form: "4 to 3" Colon notation: $4:3$ Fraction form: $\frac{4}{3}$ All three of these representations mean exactly the same thing—they're just different ways to write the same comparison. Why ratios matter: Ratios are fundamental because they help us understand relationships and comparisons in everyday life. Whether you're scaling a recipe, understanding map distances, or comparing prices, you're working with ratios. Ratio Notation and Terminology When we write a ratio like $A:B$, we're saying "A is to B" or "the ratio of A to B." It's important to recognize that order matters in a ratio—$4:3$ is different from $3:4$. Each number in a ratio has a specific name: The first number (before the colon) is called the antecedent The second number (after the colon) is called the consequent In the ratio $4:3$, the number 4 is the antecedent and 3 is the consequent. This terminology becomes especially important when you're working with proportions, which are equations that state two ratios are equal. For example: $$4:3 = 8:6$$ In a proportion like $A:B = C:D$: The numbers at the "ends" ($A$ and $D$) are called the extremes The numbers in the "middle" ($B$ and $C$) are called the means This distinction matters because it leads to an important property: in a proportion, the product of the extremes equals the product of the means. For $4:3 = 8:6$, we have $4 \times 6 = 3 \times 8 = 24$. Ratios with the Same Units An important concept is that when both quantities in a ratio share the same unit, the ratio itself is dimensionless. This means it has no unit attached to it. For example: A ratio of 4 meters to 3 meters simplifies to the ratio $4:3$ (no "meters" attached) A ratio of 5 pounds to 2 pounds simplifies to the ratio $5:2$ (no "pounds" attached) This is why a ratio is really just a pure number comparison—we're no longer thinking about what we're measuring, just how the quantities relate to each other. Converting Ratios to Fractions Understanding the relationship between ratios and fractions is crucial. When we express a ratio as a fraction, we're answering the question: "What portion of the whole is this quantity?" The whole-part relationship: If a ratio includes all parts of a situation, we can find what fraction each part represents of the total. For example, suppose you have 2 oranges and 3 apples (total of 5 pieces of fruit): The ratio of oranges to total fruit is $2:5$, which as a fraction is $\frac{2}{5}$ The ratio of apples to total fruit is $3:5$, which as a fraction is $\frac{3}{5}$ To find these fractions: Add all the parts: $2 + 3 = 5$ Put each part over the total: $\frac{2}{5}$ and $\frac{3}{5}$ Notice that these two fractions add up to 1 (the whole): $\frac{2}{5} + \frac{3}{5} = 1$. This is a useful check that you've correctly identified all the parts. <extrainfo> Ratios with more than two terms: If you have a ratio like $2:3:7$, you cannot express it as a single fraction. However, you can still create fractions by comparing any two of its terms. For instance, in $2:3:7$, you could say that the ratio of the first part to the second part is $\frac{2}{3}$, or the second part to the third part is $\frac{3}{7}$. </extrainfo> Equivalent Ratios and Simplifying Two ratios are equal (or equivalent) when their corresponding fractions are equal. This is the key principle behind scaling ratios. Scaling a ratio: Multiplying all terms of a ratio by the same number creates an equivalent ratio. The actual ratio relationship doesn't change. For example: $3:2$ is equivalent to $6:4$ (multiply both by 2) $3:2$ is equivalent to $12:8$ (multiply both by 4) $3:2$ is equivalent to $9:6$ (multiply both by 3) All of these represent the same comparison, just scaled differently. Reducing a ratio to simplest form: A ratio is in simplest form when the two numbers share no common integer factors other than 1. To reduce a ratio: Find the greatest common factor (GCF) of both numbers Divide both numbers by the GCF For example, to simplify $12:8$: The GCF of 12 and 8 is 4 Divide both by 4: $\frac{12}{4}:\frac{8}{4} = 3:2$ So $12:8$ in simplest form is $3:2$. Why simplify? Simplified ratios are easier to understand and work with, just like simplified fractions. They show the most basic comparison between the quantities. <extrainfo> Continued Proportions An equality of three or more ratios is called a continued proportion. For example: $A:B = C:D = E:F$. This represents a chain of equal ratios. While this concept may appear on some exams, it's not as commonly tested as basic ratios and proportions. If your course covered this topic, make sure you can recognize that each ratio in the chain is equivalent to all the others. </extrainfo>