A ratio is a comparison of two quantities. It is often written as a fraction or using the colon symbol (:). For example, the ratio of apples to oranges can be written as 2:3 or 2/3.

Ratios can be used to compare different quantities and can be simplified just like fractions. For example, the ratio 6:9 can be simplified to 2:3 by dividing both numbers by their greatest common divisor which is 3.

Ratios are commonly used in problem-solving situations, such as mixing ingredients in a recipe, determining distances on a map, and comparing quantities in financial or business contexts.

There are different types of ratios, including:

- Simple Ratio: A comparison of two quantities using the colon symbol (:).
- Rates: Ratios that compare quantities with different units, such as miles per hour or cost per item.
- Proportions: Equations that show that two ratios are equal.

When solving proportions, cross-multiplication is a useful method to find the missing value. Cross-multiplication involves multiplying the numerator of one ratio by the denominator of the other ratio, and vice versa.

Here are some practice problems to test your understanding of ratios:

- If the ratio of boys to girls in a class is 3:5 and there are 24 students in the class, how many boys are there?
- A recipe calls for a ratio of 2 cups of flour to 3 cups of sugar. If you want to use 6 cups of sugar, how much flour should you use?
- If a car travels 180 miles in 3 hours, what is its average speed in miles per hour?

Ratios are a fundamental concept in mathematics that are used to compare and relate quantities. Understanding ratios and how to work with them is important for solving various real-world problems.

Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.