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A ratio is a way to compare two quantities. You can think of it as a fraction that shows the relative size of two values.
For example, in the statement '1 inch is to 7 miles', the ratio is \( \frac{1 \, \text{inch}}{7 \, \text{miles}} \). This expresses the relationship between inches and miles in a compact form.
Ratios are used in many real-life scenarios, such as recipes, maps, and scales. Instead of writing out long sentences, you can simply state the ratio. When you see '1:7', it means for every 1 unit of the first quantity, you have 7 units of the second.

To better understand ratios, if you have a recipe needing 2 cups of sugar for every 3 cups of flour, the ratio of sugar to flour is \( \frac{2}{3} \). This ratio tells you how much sugar to use relative to flour. The same idea applies to the inches and miles example in the exercise.

Equivalent fractions are different fractions that represent the same value. The key is that both the numerator and denominator are multiplied or divided by the same number.
For instance, \( \frac{1}{2} = \frac{2}{4} = \frac{3}{6} \). These fractions look different but are equivalent because they all equal 0.5.

In the exercise, we have the fractions \( \frac{1 \, \text{inch}}{7 \, \text{miles}} \) and \( \frac{5 \, \text{inches}}{35 \, \text{miles}} \). To check if they are equivalent, you can simplify \( \frac{5}{35} \). By dividing the numerator and the denominator by 5, you get \( \frac{1}{7} \), which matches the first fraction. Thus, they are equivalent.

Recognizing equivalent fractions helps in understanding and solving problems involving proportions. It simplifies comparisons and makes calculations easier.

Prealgebra is the foundation of algebra, focusing on basic arithmetic and the introduction of variables. It helps students prepare for more complex algebraic concepts.

In prealgebra, we often deal with ratios, proportions, and equivalent fractions. These concepts build the groundwork for understanding relationships between numbers. For example, setting up a proportion like \( \frac{1 \, \text{inch}}{7 \, \text{miles}} = \frac{5 \, \text{inches}}{35 \, \text{miles}} \) is a key prealgebra skill.

Understanding these relationships helps students solve real-world problems involving scaling, conversions, and comparisons. Prealgebra teaches logical thinking and problem-solving skills, laying the foundation for success in higher-level mathematics.

By mastering prealgebra topics, students gain confidence in their math abilities and are better prepared for the challenges of algebra and beyond.