Chapter 7 The Basic Concepts of Algebra © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 7: The Basic Concepts of Algebra 7.1 Linear Equations 7.2 Applications of Linear Equations 7.3 Ratio, Proportion, and Variation 7.4 Linear Inequalities 7.5Properties of Exponents and Scientific Notation 7.6 Polynomials and Factoring 7.7Quadratic Equations and Applications

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 7-3 Ratio, Proportion, and Variation

© 2008 Pearson Addison-Wesley. All rights reserved Ratio, Proportion, and Variation Writing Ratios Unit Pricing Solving Proportions Direct Variation Inverse Variation Joint and Combined Variation

© 2008 Pearson Addison-Wesley. All rights reserved Writing Ratios A commonly used mathematical concept is a ratio. A baseball player’s batting average is a ratio. The slope, or pitch of a roof on a building may be expressed as a ratio. Ratios provide a way of comparing two numbers or quantities.

© 2008 Pearson Addison-Wesley. All rights reserved Ratio A ratio is a quotient of two quantities. The ratio of the number a to the number b is written When ratios are used in comparing units of measure, the units should be the same.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Writing Ratios Write a ratio for the word phrase: 8 hours to 2 days. Solution

© 2008 Pearson Addison-Wesley. All rights reserved Unit Pricing Ratios can be applied in unit pricing, to see which size of an item offered in different sizes produces the best price per unit. To do this, set up the ratio of the price of the item to the number of units on the label. Then divide to obtain the price per unit.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Finding Price per Unit Find the unit price of a 16-oz box of cereal that has a price of $3.36. Solution

© 2008 Pearson Addison-Wesley. All rights reserved Proportion A proportion is a statement that says that two ratios are equal. Example: In the proportion a, b, c, and d are the terms of the proportion. The a and d terms are called the extremes and the b and c terms are called the means.

© 2008 Pearson Addison-Wesley. All rights reserved Cross Products If then the cross products ad and bc are equal. Also, if ad = bc, then The product of the extremes equals the product of the means.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Solving Proportions Solve the proportion Solution Set the cross products equal. Multiply. Divide by 4.

© 2008 Pearson Addison-Wesley. All rights reserved Direct Variation y varies directly as x, or y is directly proportional to x, if there exists a nonzero constant k such that The constant k is a numerical value called the constant of variation.

© 2008 Pearson Addison-Wesley. All rights reserved Solving a Variation Problem Step 1Write the variation equation. Step 2Substitute the initial values and solve for k. Step 3Rewrite the variation equation with the value of k from Step 2. Step 4Substitute the remaining values, solve for the unknown, and find the required answer.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Direct Variation Suppose y varies directly as x, and y = 45 when x = 30, find y when x = 12. Solution y = 18 when x = 12.

© 2008 Pearson Addison-Wesley. All rights reserved Direct Variation as a Power y varies directly as the nth power of x if there exists a nonzero real number k such that

© 2008 Pearson Addison-Wesley. All rights reserved Example: Direct Variation as a Power The area of a circle varies directly as the square of its radius. If A represents the area and r the radius, then for some constant k

© 2008 Pearson Addison-Wesley. All rights reserved Inverse Variation y varies inversely as x if there exists a nonzero real number k such that y varies inversely as the nth power of x if there exists a nonzero real number k such that

© 2008 Pearson Addison-Wesley. All rights reserved Example: Inverse Variation The weight of an object above Earth varies inversely as the square of its distance from the center of Earth. If an astronaut in a space vehicle weighs 57 pounds when 6700 miles from the center of Earth, what does the astronaut weigh when 4090 miles from the center? Solution Set up: w is the weight and d is the distance to the center of Earth.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Inverse Variation Solution (continued)

© 2008 Pearson Addison-Wesley. All rights reserved Joint and Combined Variation If one variable varies as the product of several other variables, the first is said to vary jointly as the others. Situations that involve combinations of direct and inverse variation are combined variation problems.

© 2008 Pearson Addison-Wesley. All rights reserved Example: Joint and Combined Variation m varies jointly as x and y can be expressed as m varies directly as x and inversely as the square of y can be expressed as

© 2008 Pearson Addison-Wesley. All rights reserved Example: Joint and Combined Variation The strength of a rectangular beam varies jointly as its width and the square of its depth. If S represents the strength, w the width, and d the depth, then for some constant k,