3.3 Problem Solving with Percents. Copyright Cengage Learning. All rights reserved.

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2 What You Will Learn Convert percents decimals and fractions, and vice versa Solve linear equations involving percents Solve problems involving markups and discounts 2

3 Percents 3

4 Percents Cent implies 100, as in the word century. A percent is a number of parts per one hundred. 25% = 0.25 = 4

5 Example 1 Converting Decimals and Fractions to Percents Convert the following to percents: a b. 5

6 Example 2 Converting Percents to Decimals and Fractions a. Convert 3.5% to a decimal. b. Convert 55% to a fraction. Solution: a. 3.5% b. 55% 6

7 Percents In Examples and 2 there is a quick way to convert between percent form and decimal form. To convert from percent form to decimal form, move the decimal point two places to the left. For instance, 3.5% = To convert from decimal form to percent form, move the decimal point two places to the right. For instance, 1.20 = 120% 7

8 The Percent Equations 8

9 Percent Equations The primary use of percents is to compare two numbers. For example, 2 is 50% of 4, and 5 is 25% of 20. The following model is helpful. Verbal Model: Labels: b = base number p = percent (in decimal form) a = number being compared to b Equation: a = p b 9

10 Example 3 Solving Percent Equations for a a. What number is 30% of 70? b. A union negotiates for a cost-of-living raise of 7%. What is the raise for a union member whose salary is $40,240? What is the persons new salary? Solution: a. Verbal Model: Label: a = unknown number Equation: a = (0.3)(70) So, 21 is 30% of

11 Example 3 Solving Percent Equations for a Solution: b. Verbal Model: Label: Raise = a (dollars) Percent = 7% =0.07 Salary = 40,240 Equation: a = 0.07(40,2400) = (decimal form) (dollars) So, the raise is $ and the new salary is 40, = $43,

12 Example 4 Solving Percent Equations for b a. 14 is %25 of what number? b. You missed an A in your chemistry course by only 3 points. Your point total for the course was 402. How many points were possible in the course? (Assume that you needed 90% of the course total for an A.) Solution: a. Verbal Model: Label: Equation: 14 = 0.25b b = unknown number 56 = b So, 21 is 30% of

13 Example 4 Solving Percent Equations for b Solution: b. Verbal Model: cont d Label: Your points = 402 (points) Percent = 90% =0.9 (decimal form) Total points for course = 40,240 (points) Equation: = 9b 405 = 9b 450 = b Write equation Add Divide each side by 0.9 So, there were 450 possible points in the course. 13

14 Example 5 Solving Percent Equations for p a. 135 is what percent of 27? b. A real estate agency receives a commission of $ for the sale of a $124,500 house. What percent commission is this? Solution: a. Verbal Model: Label: p = unknown number (in decimal form) Equation: 135 = p(27) 5 = p So, 135 is 500% of

15 Example 5 Solving Percent Equations for p Solution: b. Verbal Model: cont d Label: Commission = (dollars) Percent = p (decimal form) Sale price= 124,500 (dollars) Equation: = p( ) = p Write equation Divide each side by 124,500 Simplify So, the real estate agency receives a commission of 6.5%. 15

16 Markups and Discounts 16

17 Markups and Discounts You may have had the experience of buying an item at one store and later finding that you could have paid less for the same item at another store. The basic reason for this price difference is markup, which is the difference between the cost (the amount a retailer pays for the item) and the price (the amount at which the retailer sells the item to the consumer). A verbal model for this problem is as follows. 17

18 Markups and Discounts In such a problem, the markup may be known or it may be expressed as a percent of the cost. This percent is called the markup rate. Markup is one of those hidden operations. In business and economics, the terms cost and price do not mean the same thing. The cost of an item is the amount a business pays for the item. The price of an item is the amount for which the business sells the item. 18

19 Example 6 Solving Markup Problems a. The costs is $45. The markup rate if 55%. What is the selling price? b. The selling price is $95. The markup rate is 60%. What is the cost? c. The selling price is $60. The cost is $24. What is the markup rate? Solution: a. Selling price = Cost + Markup = 45 + (0.55)(45) = = $69.75 b. Selling price = Cost + Markup 98 = C + 0.6C Substitute known values Multiply Add Substitute known values 98 = 1.6C = C $61.25 = C Combine like terms Divide each side by 1.6 Simplify 19

20 Example 6 Solving Markup Problems c. Selling price = Cost + Markup rate Cost 60 = 24 + p(24) 36 = 24p = p Substitute known values Subtract 24 from each side Divide each side by 24 cont d 1.5 = p Simplify So, The markup rate is 150%. 20

21 Markups and Discounts The mathematics of discounts is similar to that of markups. The model for this situation is where the discount is given in dollars, and the discount rate is given as a percent of the list price. Notice the hidden operation in the discount. 21

Example 7 Solving Discount Problems During a midsummer sale, a lawn mower listed at $199.95 is on sale for $139.95. What is the discount rate?

22 Example 7 Solving Discount Problems During a midsummer sale, a lawn mower listed at $ is on sale for $ What is the discount rate? Solution: Verbal Model: Labels: Discount = = 60 (dollars) List price = (dollars) Discount rate = p (decimal form) 22

23 Example 7 Solving Discount Problems Equation: 60 = p(199.95) 0.30 p Write equation. Divide each side by cont d Because p 0.30, it follows that the discount rate is approximately 30%. 23

24 HOMEWORK: Page 124 #65-77odd,79-81,89,90 24

Example 1: Cost of Operation. Example 2: Education Pays Off. 3.1: Algebraic Translations of English Phrases. 3.2: Strategy for Solving Word Problems

Example 1: Cost of Operation. Example 2: Education Pays Off. 3.1: Algebraic Translations of English Phrases. 3.2: Strategy for Solving Word Problems 3.1: Algebraic Translations of English Phrases 3.2: Strategy for Solving Word Problems Addition Subtraction Multiplication Division sum more than increased by plus added to Total of sum of 6 and 7 3 more