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

Chapter 7

The Basic Concepts of Algebra

© 2008 Pearson Addison-Wesley.All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved

7-3-2

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

7.5 Properties of Exponents and Scientific Notation

7.6 Polynomials and Factoring

7.7 Quadratic Equations and Applications

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Chapter 1

Section 7-3Ratio, Proportion, and Variation

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Ratio, Proportion, and Variation

• Writing Ratios

• Unit Pricing

• Solving Proportions

• Direct Variation

• Inverse Variation

• Joint and Combined Variation

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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.

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Ratio

A ratio is a quotient of two quantities. The ratio of the number a to the number b is written

to , , or : .a

a b a bb

When ratios are used in comparing units of measure, the units should be the same.

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Example: Writing Ratios

Write a ratio for the word phrase: 8 hours to 2 days.

Solution

8 hr 8 hr 8 1.

2 days 48 hr 48 6

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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.

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Example: Finding Price per Unit

Find the unit price of a 16-oz box of cereal that has a price of $3.36.

Solution

$3.36$.21 per ounce.

16

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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.

1 3

2 6

a c

b d ( 0, 0),b d

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Cross Products

If then the cross products ad and bc

are equal.

,a c

b d

Also, if ad = bc, then (as long as 0, 0).a c

b db d

The product of the extremes equals the product of the means.

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Example: Solving Proportions

Solve the proportion4 22

.5 x

Solution4 22 5x

4 110x

110 55

4 2x

Set the cross products equal.

Multiply.

Divide by 4.

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Direct Variation

y varies directly as x, or y is directly proportional to x, if there exists a nonzero constant k such that

, or equivalently, .y

y kx kx

The constant k is a numerical value called the constant of variation.

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Solving a Variation Problem

Step 1 Write the variation equation.

Step 2 Substitute the initial values and solve for k.

Step 3 Rewrite the variation equation with the value of k from Step 2.

Step 4 Substitute the remaining values, solve for the unknown, and find the required answer.

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Example: Direct Variation

Suppose y varies directly as x, and y = 45 when x = 30, find y when x = 12.

y kxSolution

45 30k 3

2k

3

2y x

312 18.

2y

y = 18 when x = 12.

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Direct Variation as a Power

y varies directly as the nth power of x if there exists a nonzero real number k such that

.ny kx

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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

2.A kr

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Inverse Variation

y varies inversely as x if there exists a nonzero real number k such that

, or equivalently, .k

xy k yx

y varies inversely as the nth power of x if there exists a nonzero real number k such that

.n

ky

x

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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

2

kw

dSet up:

w is the weight and d is the distance to the center of Earth.

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Example: Inverse Variation

Solution (continued)

257

6700

k

257(6700)k

2

2

57(6700)153 pounds.

(4090)w

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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.

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Example: Joint and Combined Variation

m varies jointly as x and y can be expressed as .m kxy

m varies directly as x and inversely as the square of y can be expressed as

2.

kxm

y

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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,

2.S kwd