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11.1 Problem Solving Using Ratios and Proportions. A ratio is the comparison of two numbers written as a fraction. For example:. Your school’s basketball team has won 7 games and lost 3 games. What is the ratio of wins to losses?.
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11.1 Problem Solving Using Ratios and Proportions A ratio is the comparison of two numbers written as a fraction. For example: Your school’s basketball team has won 7 games and lost 3 games. What is the ratio of wins to losses? Because we are comparing wins to losses the first number in our ratio should be the number of wins and the second number is the number of losses. The ratio is ___________ games won _______ 7 games __ 7 = = games lost 3 games 3
11.1 Problem Solving Using Ratios and Proportions In a ratio, if the numerator and denominator are measured in different units then the ratio is called a rate. A unit rate is a rate per one given unit, like 60 miles per 1 hour. Example: You can travel 120 miles on 60 gallons of gas. What is your fuel efficiency in miles per gallon? 120 miles ________ ________ 20 miles Rate = = 60 gallons 1 gallon Your fuel efficiency is 20 miles per gallon.
11.1 Problem Solving Using Ratios and Proportions An equation in which two ratios are equal is called a proportion. A proportion can be written using colon notation like this a:b::c:d or as the more recognizable (and useable) equivalence of two fractions. ___ ___ a c = b d
11.1 Problem Solving Using Ratios and Proportions When Ratios are written in this order, a and d are the extremes, or outside values, of the proportion, and b and c are the means, or middle values, of the proportion. ___ ___ a c a:b::c:d = b d Extremes Means
11.1 Problem Solving Using Ratios and Proportions To solve problems which require the use of a proportion we can use one of two properties. The reciprocal property of proportions. If two ratios are equal, then their reciprocals are equal. The cross product property of proportions. The product of the extremes equals the product of the means
11.1 Problem Solving Using Ratios and Proportions Solve: x – 1 = 2x x = –1
11.1 Problem Solving Using Ratios and Proportions Solve: x2 = -2x - 1 x2 +2x + 1= 0 (x + 1)(x + 1)= 0 (x + 1) = 0 or (x + 1)= 0 x = -1
11.2 Problem Solving Using Percents • Percent means per hundred, or parts of 100 • When solving percent problems, convert the percents to decimals before performing the arithmetic operations
11.2 Problem Solving Using Percents • What is 20% of 50? • x = .20 * 50 • x = 10 • 30 is what percent of 80? • 30 = x * 50 • x = 30/50 = .6 = 60%
11.2 Problem Solving Using Percents • 12 is 60% of what? • 12 = .6x • x = 12/.6 = 20 • 40 is what percent of 300? • 40 = x * 300 • x = 40/300 = .133… = 13.33%
60 10 40 10 11.2 Problem Solving Using Percents • What percent of the region is shaded? 100 is what percent of 2400? 100 = x * 2400? x = 100/2400 x = 4.17%
11.3 Direct and Inverse Variation Direct Variation The following statements are equivalent: • yvaries directly as x. • y is directlyproportional to x. • y=kx for some nonzero constant k. k is the constant of variation or the constant of proportionality
11.3 Direct and Inverse Variation Inverse Variation The following statements are equivalent: • yvaries inversely as x. • y is inverselyproportional to x. • y=k/x for some nonzero constant k.
11.3 Direct and Inverse Variation If yvaries directly asx, then y = kx. If y = 10 when x = 2 , then what is the value of y when x = 8? x and y go together. Therefore, by substitution 10 = k(2). What is the value of k? 10 = 2k 10 = 2k 5 = k
11.3 Direct and Inverse Variation k = 5 Replacing k with 5 gives us y = 5x What is y when x = 8 ? y = 5(8) y = 40
11.3 Direct and Inverse Variation If yvaries inversely asx, then xy = k. If y = 6 when x = 4 , then what is the value of y when x = 8? x and y go together. Therefore, by substitution (6)(4) = k. What is the value of k? 24 = k
11.3 Direct and Inverse Variation k = 24 Replacing k with 24 gives us xy = 24 What is y when x = 8 ? 8y = 24 y = 3
11.3 Direct and Inverse Variation 15 10 5 0 15 20 0 5 10 Direct variation • • y = kx • y = 2x •
11.3 Direct and Inverse Variation 15 10 5 0 15 20 0 5 10 • Inverse Variation xy= k • xy= 16 • • •
11.5 Simplifying Rational Expressions • Define a rational expression. • Determine the domain of a rational function. • Simplify rational expressions.
11.5 Simplifying Rational Expressions • Rational numbers are numbers that can be written as fractions. • Rational expressions are algebraic fractions of the form P(x) , where P(x)and Q(x) Q(x) are polynomials and Q(x) does not equal zero. • Example:
11.5 Simplifying Rational Expressions • P(x) ; Since division by zero is not Q(x) possible, Q(x) cannot equal zero. • The domain of a function is all possible values of x. • For the example , 4x + 1 ≠ 0 so x ≠ -1/4.
11.5 Simplifying Rational Expressions • The domain of is all real numbers except -1/4. • Domain = {x|x ≠ -1/4}
11.5 Simplifying Rational Expressions • Find domain of • Solve: x2 –5x – 6 =0 (x – 6)(x + 1) = 0 The excluded values are x = 6, -1 • Domain = {x|x ≠-1, 6}
11.5 Simplifying Rational Expressions • To simplify rational expressions, factor the numerator and denominator completely. Then reduce. • Simplify:
11.5 Simplifying Rational Expressions Factor: Reduce: 2
11.5 Simplifying Rational Expressions Simplify: Factor –1 out of the denominator:
11.5 Simplifying Rational Expressions • Multiply rational expressions. • Divide rational expressions
11.5 Simplifying Rational Expressions • To multiply, factor each numerator and denominator completely. • Reduce • Multiply the numerators and multiply the denominators. • Multiply:
11.5 Simplifying Rational Expressions Factor: 5 Reduce:
11.5 Multiplying and Dividing Multiply:
11.6 Multiplying and Dividing • To divide, change the problem to multiplication by writing the reciprocal of the divisor. • (Change to multiplication and flip the second fraction.) • Divide:
11.5 Multiplying and Dividing Change to multiplication: Factor completely:
11.5 Multiplying and Dividing Reduce: Multiply:
11.7 Dividing Polynomials Dividing a Polynomial by a Monomial Let u, v, and w be real numbers, variables or algebraic expressions such that w≠ 0.
11.7 Dividing Polynomials Use Long Division x + 6 Note: (x + 6) (x – 2) = x2 + 4x - 12 x2 -2x 6x - 12 6x - 12 0
11.7 Dividing Polynomials Use Long Division x + 5 x2 - x 5x - 1 5x - 5 4
11.7 Dividing Polynomials Note: x2 term is missing x2 - 2x + 6 x3 + 2x2 - -2x2 + 2x -2x2 – 4x 6x - 1 6x + 12 - -13
11.8 Solving Rational Equations LCD: 2x Multiply each fraction through by the LCD Check your solution!
Solve. 11.8 Solving Rational Equations LCD: ? LCD: (x+1) ? No Solution! Check your solution!
Solve. 11.8 Solving Rational Equations LCD: (x + 2)(x - 2) Factor 1st! Check your solutions!
Short Cut! 11.8 Solving Rational Equations When there is only fraction on each side of the =, just cross multiply as if you are solving a proportion.
Example: Solve. 11.8 Solving Rational Equations Check your solutions!
Solve. 11.8 Solving Rational Equations Check your solutions!
