Proportions and Problem Solving with Rational Equations

Slides:



Advertisements
Similar presentations
Chapter 14 Rational Expressions.
Advertisements

Using Rational Equations
6.7 Applications of Rational Expressions. Objective 1 Solve problems about numbers. Slide
Slide 7- 1 Copyright © 2012 Pearson Education, Inc.
Algebra 7.3 Solving Linear Systems by Linear Combinations.
Equations, Inequalities and Problem Solving
Chapter 11 Systems of Equations.
Chapter 15 Roots and Radicals.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 6 Ratio, Proportion, and Percent.
Solving Multiplication and Division Equations. EXAMPLE 1 Solving a Multiplication Equation Solve the equation 3x = x = 15 Check 3x = 45 3 (15)
Chapter 7 Section 7.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 3 Solving Equations and Problem Solving.
Any questions on the Section 6.5 homework?. Section 6.6 Rational Expressions and Problem Solving.
Rational Expressions Simplifying Rational Expressions.
Chapter 14 Rational Expressions.
Test 3 Review Session Monday night at 6:30 in this room Attendance at this optional review session is highly encouraged due to the difficulty many students.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 9 Equations, Inequalities, and Problem Solving.
Ratio and Proportion.
Proportions Section 6.2. A proportion is a statement that two ratios or rates are equal. Solving Proportions If and are two ratios, then If and are two.
Chapter 8 Roots and Radicals.
Copyright © Cengage Learning. All rights reserved. Rational Expressions and Equations; Ratio and Proportion 6.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 12 Rational Expressions.
Chapter 6 Section 7 Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Applications of Rational Expressions Solve story problems about.
Section 9.6 Inverse and Joint Variation and Other Applications Yielding Equations with Fractions.
§ 2.8 Solving Linear Inequalities. Martin-Gay, Beginning and Intermediate Algebra, 4ed 22 Linear Inequalities in One Variable A linear inequality in one.
Section 5Chapter 7. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives Applications of Rational Expressions Find the value of an.
§ 2.7 Ratios and Proportions. Angel, Elementary Algebra, 7ed 2 Ratios A is a quotient of two quantities. Ratios provide a way to compare two numbers.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 11 Systems of Equations.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Systems of Linear Equations and Problem Solving.
Chapter 1 Review Advanced Algebra 1.
Ratios, Proportions and Similar Figures Ratios, proportions and scale drawings.
4.5 Systems of Linear Equations and Problem Solving.
MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Sections 7.6 – Applications and Variations  Motion (rate – time – distance)  Shared Work.
Unit Goals – 1. Solve proportions and simplify ratios. 2. Apply ratios and proportions to solve word problems. 3. Recognize, determine, and apply scale.
Chapter 6 Rational Expressions § 6.1 Rational Functions and Multiplying and Dividing Rational Expressions.
Solving Equations Containing First, we will look at solving these problems algebraically. Here is an example that we will do together using two different.
Example 1 Solve a Rational Equation The LCD for the terms is 24(3 – x). Original equation Solve. Check your solution. Multiply each side by 24(3 – x).
Systems of Linear Equations and Problem Solving. Steps in Solving Problems 1)Understand the problem. Read and reread the problem Choose a variable to.
Unit 6 Similarity.
Rational Expressions Simplifying Rational Expressions.
Ratios and Proportions Notes. Ratios A ratio compares two numbers or two quantities. The two numbers being compared are called terms. A ratio can be written.
Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.
§ 2.3 Solving Linear Equations. Martin-Gay, Beginning and Intermediate Algebra, 4ed 22 Solving Linear Equations Solving Linear Equations in One Variable.
Ratios, Proportions and Similar Figures
Solving equations that involve formulas.
Chapter 7 Rational Expressions
Ratio is the quotient of two numbers or two quantities.
Rational Expressions and Equations
Systems of Linear Equations and Problem Solving
Ratios, Proportions and Similar Figures
Chapter 14 Rational Expressions.
8.1 Ratio and Proportion.
8.1 Exploring Ratio and Proportion
Finding a Solution by Graphing
6.2 Proportions.
Solving Equations Containing
Ratios, Proportions and Similar Figures
Ratio, Proportion, and Other Applied Problems
February 1, Math 201 OBJECTIVE: Students will be able to calculate the length of corresponding sides of similar triangles, using proportional.
Ratios, Proportions and Similar Figures
Solving Rational Equations
Adding and Subtracting Rational Expressions
Quadratic Equations, Inequalities, and Functions
Solving Equations Containing Rational Expressions § 6.5 Solving Equations Containing Rational Expressions.
Ratios, Proportions and Similar Figures
A rational equation is an equation that contains one or more rational expressions. The time t in hours that it takes to travel d miles can be determined.
Dear Power point User, This power point will be best viewed as a slideshow. At the top of the page click on slideshow, then click from the beginning.
4 Chapter Chapter 2 Solving Systems of Linear Equations.
Ratios, Proportions and Similar Figures
Presentation transcript:

Proportions and Problem Solving with Rational Equations § 7.6 Proportions and Problem Solving with Rational Equations

Ratios and Rates A ratio is the quotient of two numbers or two quantities. The ratio of the numbers a and b can also be written as a:b, or . The units associated with the ratio are important. The units should match. If the units do not match, it is called a rate, rather than a ratio.

Proportions A proportion is two ratios (or rates) that are equal to each other. We can rewrite the proportion by multiplying by the LCD, bd. This simplifies the proportion to ad = bc. This is commonly referred to as the cross product.

Solving Proportions Example: Solve the proportion for x. Continued.

Solving Proportions Example continued: Substitute the value for x into the original equation, to check the solution. true So the solution is

Solving Proportions Example: If a 170-pound person weighs approximately 65 pounds on Mars, how much does a 9000-pound satellite weigh?

Solving Proportions Example: Given the following prices charged for various sizes of picante sauce, find the best buy. 10 ounces for $0.99 16 ounces for $1.69 30 ounces for $3.29 Continued.

Solving Proportions Example continued: Size Price Unit Price 10 ounces $0.99 $0.99/10 = $0.099 16 ounces $1.69 $1.69/16 = $0.105625 30 ounces $3.29 $3.29/30  $0.10967 The 10 ounce size has the lower unit price, so it is the best buy.

Similar Triangles In similar triangles, the measures of corresponding angles are equal, and corresponding sides are in proportion. Given information about two similar triangles, you can often set up a proportion that will allow you to solve for the missing lengths of sides.

Similar Triangles Example: Given the following similar triangles, find the unknown length y. 10 m 12 m 5 m y Continued

Similar Triangles Example: 1.) Understand Read and reread the problem. We look for the corresponding sides in the 2 triangles. Then set up a proportion that relates the unknown side, as well. 2.) Translate By setting up a proportion relating lengths of corresponding sides of the two triangles, we get Continued

Similar Triangles Example continued: 3.) Solve meters Continued

Similar Triangles Example continued: true 4.) Interpret Check: We substitute the value we found from the proportion calculation back into the problem. true State: The missing length of the triangle is meters

Finding an Unknown Number Example: The quotient of a number and 9 times its reciprocal is 1. Find the number. 1.) Understand Read and reread the problem. If we let n = the number, then = the reciprocal of the number Continued

Finding an Unknown Number Example continued: 2.) Translate The quotient of  is = a number n and 9 times its reciprocal 1 Continued

Finding an Unknown Number Example continued: 3.) Solve Continued

Finding an Unknown Number Example continued: 4.) Interpret Check: We substitute the values we found from the equation back into the problem. Note that nothing in the problem indicates that we are restricted to positive values. true true State: The missing number is 3 or –3.

Solving a Work Problem Example: An experienced roofer can roof a house in 26 hours. A beginner needs 39 hours to do the same job. How long will it take if the two roofers work together? 1.) Understand Read and reread the problem. By using the times for each roofer to complete the job alone, we can figure out their corresponding work rates in portion of the job done per hour. Experienced roofer 26 1/26 Beginner roofer 39 1/39 Together t 1/t Time in hrs Portion job/hr Continued

Solving a Work Problem Example continued: 2.) Translate Since the rate of the two roofers working together would be equal to the sum of the rates of the two roofers working independently, Continued

Solving a Work Problem Example continued: 3.) Solve Continued

Solving a Work Problem Example continued: true 4.) Interpret Check: We substitute the value we found from the proportion calculation back into the problem. true State: The roofers would take 15.6 hours working together to finish the job.

Solving a Rate Problem Example: The speed of Lazy River’s current is 5 mph. A boat travels 20 miles downstream in the same time as traveling 10 miles upstream. Find the speed of the boat in still water. 1.) Understand Read and reread the problem. By using the formula d = rt, we can rewrite the formula to find that t = d/r. We note that the rate of the boat downstream would be the rate in still water + the water current and the rate of the boat upstream would be the rate in still water – the water current. Down 20 r + 5 20/(r + 5) Up 10 r – 5 10/(r – 5) Distance rate time = d/r Continued

Solving a Rate Problem Example continued: 2.) Translate Since the problem states that the time to travel downstream was the same as the time to travel upstream, we get the equation Continued

Solving a Rate Problem Example continued: 3.) Solve Continued

Solving a Rate Problem Example continued: 4.) Interpret Check: We substitute the value we found from the proportion calculation back into the problem. true State: The speed of the boat in still water is 15 mph.