Proportion

A proportion is a statement that two ratios are equal. In other words, it expresses the equality of two ratios. The general form of a proportion is:

a/b = c/d

Where a, b, c, and d are numbers or algebraic expressions, and b and d are not zero.

Key Concepts:

Ratios: A ratio is a comparison of two quantities. For example, if there are 3 boys and 5 girls in a class, the ratio of boys to girls is 3:5.
Proportional Relationships: When two ratios are equal, they form a proportional relationship. If a/b = c/d, then a is directly proportional to b and c is directly proportional to d.
Cross Products Property: In a proportion a/b = c/d, the cross products ad and bc are equal. This property can be used to solve proportions.

How to Solve Proportions:

Cross Multiplication: To solve a proportion a/b = c/d, you can cross multiply by multiplying a by d and b by c. Set the resulting products equal to each other and solve for the unknown.
Equivalent Fractions: You can also solve proportions by finding an equivalent fraction. Multiply or divide both parts of the ratio by the same number to find a new ratio that is equivalent to the given one.

Example:

Find the value of x in the proportion 3/5 = x/15.

Solution: Using cross multiplication, we get 3 * 15 = 5x. Simplifying, we have 45 = 5x. Dividing both sides by 5, we find that x = 9.

Study Guide:

Understand the concept of ratios and how they are used to compare quantities.
Learn to recognize proportional relationships and how to express them in the form of a proportion.
Practice solving proportions using cross multiplication and finding equivalent fractions.
Work on real-world problems that involve proportional reasoning, such as scaling and map problems.
Review the cross products property and how it can be used to solve proportions.

Proportions are an important concept in mathematics and have many applications in everyday life. Understanding how to work with proportions will help you solve various types of problems involving comparisons and scaling.

◂Math Worksheets and Study Guides Seventh Grade. Applying Percents

The resources above cover the following skills:

Number and Operations (NCTM)

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

Work flexibly with fractions, decimals, and percents to solve problems.

Grade 7 Curriculum Focal Points (NCTM)

Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity

Students extend their work with ratios to develop an understanding of proportionality that they apply to solve single and multi-step problems in numerous contexts. They use ratio and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease. They also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and identify the unit rate as the slope of the related line. They distinguish proportional relationships (y/x = k, or y = kx) from other relationships, including inverse proportionality (xy = k, or y = k/x).