A ratio is a quantitative relationship between two or more values, quantities, or measures. It is used to express how one number or quantity relates to another, often in the form of a fraction or a comparison.
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Ratios can be used to solve percent applications by expressing the relationship between a part and the whole.
Proportions are used to solve similar figure applications by setting up a ratio between corresponding sides of the figures.
Direct variation is a relationship where two quantities are proportional, meaning their ratio remains constant.
Inverse variation is a relationship where two quantities are inversely proportional, meaning their product is constant.
Ratios are fundamental to understanding and solving a variety of mathematical problems, from everyday applications to advanced concepts.
Review Questions
Explain how ratios can be used to solve percent applications.
Ratios can be used to solve percent applications by expressing the relationship between a part and the whole. For example, if a student scored 18 out of 25 points on a test, the ratio of the student's score to the total points would be 18:25. This ratio can then be used to calculate the student's percentage score by setting up a proportion and solving for the unknown value.
Describe how ratios are used to solve proportion and similar figure applications.
Proportions, which are equations that state two ratios are equal, are used to solve for unknown values in similar figure applications. By setting up a proportion between the corresponding sides of two similar figures, you can use the known ratios to determine the lengths of the unknown sides. This is a powerful tool for solving problems involving scale, scale drawings, and other geometric applications that rely on the properties of similar figures.
Analyze how the concept of ratio is fundamental to understanding direct and inverse variation.
$$Direct variation$$ is a relationship where two quantities are proportional, meaning their ratio remains constant. This can be expressed as a ratio equation, such as $\frac{y}{x} = k$, where $k$ is the constant of proportionality. $$Inverse variation$$ is a relationship where two quantities are inversely proportional, meaning their product is constant. This can be expressed as a ratio equation, such as $\frac{x}{y} = k$, where $k$ is the constant of variation. Understanding the ratio relationships between the variables is crucial for identifying, modeling, and solving problems involving direct and inverse variation.