1. 1 Copyright © Cengage Learning. All rights reserved.

2. 2 2.3 Algebra and Problem Solving

3. 3 What You Will Learn  Construct verbal mathematical models from written statements  Translate verbal phrases into algebraic expressions  Identify hidden operations when writing algebraic expressions  Use problem-solving strategies to solve application problems.

4. 4 Constructing Verbal Models

5. 5 Algebra is a problem-solving language that is used to solve real-life problems. It has four basic components, which tend to nest within each other.

6. 6 Constructing Verbal Models In this section you will study ways to construct algebraic expressions from written statements by first constructing a verbal mathematical model. For example, you are paid $9 per hour and your weekly pay can be represented by the verbal model

7. 7 Example 1 – Writing an Algebraic Expressions You are paid 5 cents for each aluminum soda can and 3 cents for each plastic soda bottle you collect. Write an algebraic expression that represents the total weekly income (in cents) for this recycling activity. Solution: Before writing an algebraic expression for the weekly income, it is helpful to construct an informal verbal model. For instance, the following verbal model could be used. Note that the word and in the problem indicates addition. Because both the number of cans and the number of bottle can vary from week to week, you can use the two variables c and b, respectively, to write the following algebraic expression.

8. 8 Translating Phrases

9. 9

10. 10 Example 2 – Translating Verbal Phrases into Algebraic Expressions Translate each phrase into an algebraic expression. a.Three less than m b.y decreased by 10 c.The product of 5 and x d.The quotient of n and 7

11. 11 cont’d a.Three less than m m – 3 b.y decreased by 10 y – 10 c. The product of 5 and x 5x5x d.The quotient of n and 7 “Less than” indicates subtraction. “Decreased by” indicates subtraction. “Product” indicates multiplication. “Quotient” indicates division. Example 2 – Translating Verbal Phrases into Algebraic Expressions

12. 12 Translating Phrases In most applications of algebra, the variables are not specified and it is your task to assign variables to the appropriate quantities. A good way to learn algebra is to do it forward and backward. In the next example, algebraic expressions are translated into verbal phrases. Keep in mind that other key words could be used to describe the operation(s) in each expression. Your goal is to use key words or phrases that keep the verbal descriptions clear and concise.

13. 13 Example 3 – Translating Algebraic Expressions into Verbal Phrases Without using a variable, write a verbal description for each expression. a.x – 12 b.7(x + 12) c. d. e.(3x) 2

14. 14 cont’d a.Algebraic expression:x – 12 Operation:Subtraction Key Phrase:Less than Verbal description: Twelve less than a number b.Algebraic expression:7(x + 12) Operations:Multiplication, addition Key Words:Times, sum Verbal description:Seven times the sum of a number and 12 Example 3 – Translating Algebraic Expressions into Verbal Phrases

15. 15 cont’d c.Algebraic expression: Operations:Addition, division Key Words:Plus, quotient Verbal description:Five plus the quotient of a number and 2 d. Algebraic expression: Operations:Addition, division Key Words:Sum, divided by Verbal description:The sum of 5 and a number, all divided by 2 Example 3 – Translating Algebraic Expressions into Verbal Phrases

16. 16 cont’d e.Algebraic expression:(3x) 2 Operations:Raise to a power, multiplication Key Words: Square, product Verbal description:The square of the product of 3 and x Example 3 – Translating Algebraic Expressions into Verbal Phrases

17. 17 Verbal Models with Hidden Operations

18. 18 Verbal Models with Hidden Operations Most real-life problems do not contain verbal expressions that clearly identify all the arithmetic operations involved. You need to rely on past experience and the physical nature of the problem in order to identify the operations hidden in the problem statement. Multiplication is the operation most commonly hidden in real- life applications. Watch for hidden operations in the next example.

19. 19 Example 4 – Discovering Hidden Operations A cash register contains n nickels and d dimes. Write an expression for this amount of money in cents. Solution: The amount of money is a sum of products. Verbal Model: Labels: Value of nickel = 5 (cents) Number of nickels = n (nickels) Value of dime = 10 (cents) Number of dimes = d (dimes) Expression: 5n + 10d (cents)

20. 20 Example 5 – Discovering Hidden Operations A person riding a bicycle travels at a constant rate of 12 miles per hour. Write an expression showing how far the person can ride in t hours. Solution: The distance traveled is a product. Verbal Model: Labels:Rate of travel = 12 (miles per hour) Time traveled = t (hours) Expression: 12t (miles)

21. 21 Example 6 – Discovering Hidden Operations A person paid x dollars plus 6% sales tax for an automobile. Write an expression for the total cost of the automobile. Solution: The total cost is a sum. Verbal Model: Labels: Percent of sales tax = 0.06 (decimal form) Cost of automobile = x (dollars) Expression: x + 0.06x = (1 + 0.06)x = 1.06x

22. 22 Example 7 – Discovering Hidden Operations A truck travels 100 miles at an average speed of r miles per hour. Write an expression that represents the total travel t ime. Solution: Verbal Model: Labels: Distance = 100 (decimal form) Rate = r (dollars) Expression: (hours)

23. 23 Summary of Additional Problem-Solving Strategies

24. 24 The main tool for removing symbols of grouping is the Distributive Property, as illustrated in this example. Simplify each expression. a. –(3y + 5) b. 5x + (x – 7)2 c. –2(4x – 1) + 3x d. 3(y – 5) – (2y – 7) Example 8 – Removing Symbols of Grouping

25. 25 Additional Problem-Solving Strategies

26. 26 Example 8 – Guess, Check, and Revise You deposit $500 in an account that earns 6% interest compounded annually. The balance A in the account after t years is A = 500(1 + 0.06) t. How long will it take for your investment to double? Solution: You can solve this problem using a guess, check, and revise strategy. For instance, you might guess that it will take 10 years for your investment to double.

27. 27 Example 8 – Guess, Check, and Revise cont’d The balance after 10 years is A = 500(1 + 0.06) 10 ≈ $895.42. Because the amount has not yet doubled, you increase your guess to 15 years. A = 500(1 + 0.06) 15 ≈ $1198.28 Because this amount is greater than double the investment, your next guess should be a number between 10 and 15. After trying several more numbers, you can determine that your balance will double in about 11.9 years.

28. 28 Example 8 – Guess, Check, and Revise cont’d Another strategy that works well for a problem such as Example 7 is to make a table of data values. You can use a calculator to create the following table.

29. 29 The outer dimensions of a rectangular apartment are 25 feet by 40 feet. The combination living room, dining room, and kitchen areas occupy two-fifths of the apartment’s area. Find the total area of the remaining rooms. Solution: For this problem, it helps to draw a diagram. From the figure, you can see that the total area of the apartment is Area = (Length)(Width) = (40)(25) = 1000 square feet. Example 9 – Draw a Diagram

30. 30 The area occupied by the living room, dining room, and kitchen is This implies that the remaining rooms must have a total area of 1000 – 400 = 600 square feet Example 9 – Draw a Diagram cont’d

31. 31 You are driving on an interstate highway at an average speed of 60 miles per hour. How far will you travel in 12.5 hours? Solution: One way to solve this problem is to use the formula that related distance, rate, and time. Suppose however, that you have forgotten the formula. To help you remember, you could solve simpler problems. If you travel 60 miles per hour for 1 hour, you will travel 60 miles. If you travel 60 miles per hour for 2 hours, you will travel 120 miles. If you travel 60 miles per hour for 3 hours, you will travel 180 miles. From the example, it appears that you can find the total miles traveled by multiplying the rate by the time. So, if you travel 60 miles per hour for 12.5 hours, you will travel a distance of (60)(12.5) = 750 miles Example 10 – Solve a Simpler Problem