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	Slopes of Perpendicular Lines
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	Roots - Radicals 1
	Graph of a Line
	Sum of the Roots of a Quadratic
	Writing Linear Equations Using Slope and Point
	Factoring Trinomials with Leading Coefficient 1
	Writing Linear Equations Using Slope and Point
	Simplifying Expressions with Negative Exponents
	Solving Equations 3
	Solving Quadratic Equations
	Parent and Family Graphs
	Collecting Like Terms
	nth Roots
	Power of a Quotient Property of Exponents
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	Percents
	Solving Linear Systems of Equations by Elimination
	The Quadratic Formula
	Fractions and Mixed Numbers
	Solving Rational Equations
	Multiplying Special Binomials
	Rounding Numbers
	Factoring by Grouping
	Polar Form of a Complex Number
	Solving Quadratic Equations
	Simplifying Complex Fractions
	Algebra
	Common Logs
	Operations on Signed Numbers
	Multiplying Fractions in General
	Dividing Polynomials
	Polynomials
	Higher Degrees and Variable Exponents
	Solving Quadratic Inequalities with a Sign Graph
	Writing a Rational Expression in Lowest Terms
	Solving Quadratic Inequalities with a Sign Graph
	Solving Linear Equations
	The Square of a Binomial
	Properties of Negative Exponents
	Inverse Functions
	fractions
	Rotating an Ellipse
	Multiplying Numbers
	Linear Equations
	Solving Equations with One Log Term
	Combining Operations
	The Ellipse
	Straight Lines
	Graphing Inequalities in Two Variables
	Solving Trigonometric Equations
	Adding and Subtracting Fractions
	Simple Trinomials as Products of Binomials
	Ratios and Proportions
	Solving Equations
	Multiplying and Dividing Fractions 2
	Rational Numbers
	Difference of Two Squares
	Factoring Polynomials by Grouping
	Solving Equations That Contain Rational Expressions
	Solving Quadratic Equations
	Dividing and Subtracting Rational Expressions
	Square Roots and Real Numbers
	Order of Operations
	Solving Nonlinear Equations by Substitution
	The Distance and Midpoint Formulas
	Linear Equations
	Graphing Using x- and y- Intercepts
	Properties of Exponents
	Solving Quadratic Equations
	Solving One-Step Equations Using Algebra
	Relatively Prime Numbers
	Solving a Quadratic Inequality with Two Solutions
	Quadratics
	Operations on Radicals
	Factoring a Difference of Two Squares
	Straight Lines
	Solving Quadratic Equations by Factoring
	Graphing Logarithmic Functions
	Simplifying Expressions Involving Variables
	Adding Integers
	Decimals
	Factoring Completely General Quadratic Trinomials
	Using Patterns to Multiply Two Binomials
	Adding and Subtracting Rational Expressions With Unlike Denominators
	Rational Exponents
	Horizontal and Vertical Lines

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Ratios and Proportions

Objective Learn the concepts of ratio and proportion and to solve proportion problems.

Ratios

Mathematically, a ratio is simply a fraction viewed as a division of two numbers. Its importance is that it is used to compare two numbers (the numerator and the denominator of the fraction). For example, the ratio of x to y is simply the fraction . A ratio can also be written as x to y or x:y.

In the following example you can see how a ratio compares two numbers.

Example 1

In Ms. Cunningham's class there are 18 girls and 14 boys. Write the ratio of boys to girls.

Solution

The ratio of boys to girls in the class can be written as 14 to 18 or 14:18 or When simplified, this ratio can also be expressed as 7 to 9 or 7:9 or

Proportions

An equation that states that two ratios are equal is called a proportion. The equation is a proportion. This proportion can also be written as 14:18 = 7:9

More generally a proportion will often involve variables. Solving these problems usually involves elementary algebra, because they involve solving for the value of a variable. To solve these kinds of problems we use a process called cross multiplying. The cross multiplying fact should be explained carefully to your students.

Cross Multiplication Fact Suppose 0 y and 0 b are not zero. Then occurs exactly when xb = ya.

Example 1

Solve for a .

Solution

Use the cross multiplication fact and solve the resulting equation.

3(15) = 5( a )

45 = 5a

9 = a

Why is the cross multiplication fact true?

Notice that occurs exactly when . Subtract these fractions by finding a common denominator.

occurs exactly when .

Remember that a fraction equals 0 only when its numerator equals 0. So for this equation, when xb - ya = 0 or when xb = ya. This is the result of cross multiplying .

The above discussion about why the cross multiplication fact is true is a very important example of mathematical reasoning. Namely, it uses a computational technique (subtraction of fractions) to derive a general principle (cross multiplication fact). In order to solidify this concept you should do many examples of how this technique is applied.