MATH 125 Section 3.1 - Pasadena City College | MATH125 Section 3.1 - Pasadena City College

MATH 125 Section 3.1 - Pasadena City College | MATH125 Section 3.1 - Pasadena City College Figure 3.1 Sophisticated mathematical models are used to predict traffic patterns on our nation’s highways. Chapter Outline 3.1 Use a Problem-Solving Strategy 3.2 Solve Percent Applications 3.3 Solve Mixture Applications 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem 3.5 Solve Uniform Motion Applications 3.6 Solve Applications with Linear Inequalities Introduction Mathematical formulas model phenomena in every facet of our lives. They are used to explain events and predict outcomes in fields such as transportation, business, economics, medicine, chemistry, engineering, and many more. In this chapter, we will apply our skills in solving equations to solve problems in a variety of situations. 3.1 Use a Problem-Solving Strategy Learning Objectives By the end of this section, you will be able to: Approach word problems with a positive attitude Use a problem-solving strategy for word problems Solve number problems Be Prepared! Before you get started, take this readiness quiz. 1. Translate “6 less than twice x” into an algebraic expression. If you missed this problem, review Example 1.26. 2. Solve: 2 3x = 24. If you missed this problem, review Example 2.16. 3. Solve: 3x + 8 = 14. If you missed this problem, review Example 2.27. Approach Word Problems with a Positive Attitude “If you think you can… or think you can’t… you’re right.”—Henry Ford The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to 3 MATH MODELS Chapter 3 Math Models 295make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion? How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay? Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below? -- - - - - - - - - - - -- - Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were 11 girls in the study group. How many boys were in the study group? Solution Step 1. Read the problem. Step 2. Identify what we are looking for. How many boys were in the study group? Step 3. Name. Choose a variable to represent the number of boys. Let n = the number of boys. 298 Chapter 3 Math Models This OpenStax book is available for free at all the important information. Translate into an equation. Step 5. Solve the equation. Subtract 3 from each side. Simplify. Divide each side by 2. Simplify. Step 6. Check. First, is our answer reasonable? Yes, having 4 boys in a study group seems OK. The problem says the number of girls was 3 more than twice the number of boys. If there are four boys, does that make eleven girls? Twice 4 boys is 8. Three more than 8 is 11. Step 7. Answer the question. There were 4 boys in the study group. TRY IT : : 3.3 Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was 3 more than twice the number of notebooks. He bought 7 textbooks. How many notebooks did he buy? TRY IT : : 3.4 Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do? Solve Number Problems Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above. EXAMPLE 3.3 The difference of a number and six is 13. Find the number. Solution Step 1. Read the problem. Are all the words familiar? Step 2. Identify what we are looking for. the number Step 3. Name. Choose a variable to represent the number. Let n = the number. Step 4. Translate. Remember to look for clue words like "difference... of... and..." Restate the problem as one sentence. Translate into an equation. Chapter 3 Math Models 299Step 5. Solve the equation. Simplify. Step 6. Check. The difference of 19 and 6 is 13. It checks! Step 7. Answer the question. The number is 19. TRY IT : : 3.5 The difference of a number and eight is 17. Find the number. TRY IT : : 3.6 The difference of a number and eleven is −7. Find the number. EXAMPLE 3.4 The sum of twice a number and seven is 15. Find the number. Solution Step 1. Read the problem. Step 2. Identify what we are looking for. the number Step 3. Name. Choose a variable to represent the number. Let n = the number. Step 4. Translate. Restate the problem as one sentence. Translate into an equation. Step 5. Solve the equation. Subtract 7 from each side and simplify. Divide each side by 2 and simplify. Step 6. Check. Is the sum of twice 4 and 7 equal to 15? 2 ⋅ 4 + 7 ≟ 15 15 = 15 Step 7. Answer the question. The number is 4. Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need. TRY IT : : 3.7 The sum of four times a number and two is 14. Find the number. TRY IT : : 3.8 The sum of three times a number and seven is 25. Find the number. Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other. 300 Chapter 3 Math Models This OpenStax book is available for free at One number is five more than another. The sum of the numbers is 21. Find the numbers. Solution Step 1. Read the problem. Step 2. Identify what we are looking for. We are looking for two numbers. Step 3. Name. We have two numbers to name and need a name for each. Choose a variable to represent the first number. Let n = 1st number. What do we know about the second number? One number is five more than another. n + 5 = 2nd number Step 4. Translate. Restate the problem as one sentence with all the important information. The sum of the 1st number and the 2nd number is 21. Translate into an equation. Substitute the variable expressions. Step 5. Solve the equation. Combine like terms. Subtract 5 from both sides and simplify. Divide by 2 and simplify. Find the second number, too. Step 6. Check. Do these numbers check in the problem? Is one number 5 more than the other? 13 ≟ 8 + 5 Is thirteen 5 more than 8? Yes. 13 = 13 Is the sum of the two numbers 21? 8 + 13 ≟ 21 21 = 21 Step 7. Answer the question. The numbers are 8 and 13. TRY IT : : 3.9 One number is six more than another. The sum of the numbers is twenty-four. Find the numbers. TRY IT : : 3.10 The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers. Chapter 3 Math Models 301EXAMPLE 3.6