7 4 Percents Fractions decimals and percents as

7 -4 Percents • Fractions, decimals, and percents as representations of rational numbers with conversions from one form to another. • Proportional relationships to solve percent problems. • Techniques to solve problems including discounts, interest, compound interest, and percent increase and percent decrease. • Strategies for percent mental computation and estimation. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 1

Percents are special kinds of fractions, namely, fractions with a denominator of 100. The word percent comes from the Latin phrase per centum, which means per hundred. Definition of Percent where n is any non-negative number. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 2

Example 13 Write each of the following as a percent. a. 0. 03 3% b. 0. 3 33. 3% c. 1. 2 120% d. 0. 00042 0. 042% e. 1 100% f. g. 66. 6% h. ALWAYS LEARNING 60% Copyright © 2020, 2016, 2012 Pearson Education, Inc. 3

Example 14 Write each of the following percents as a decimal. a. 5% 0. 05 b. 6. 3% 0. 063 c. 100% 1 d. 250% 2. 5 e. f. 0. 3% 0. 006% ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 4

Applications Involving Percent Applications involving percent usually fall into one of the following forms: 1. Finding a percent of a number. 2. Finding what percent one number is of another. 3. Finding a number when a percent of that number is known. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 5

Example 15 A house that sells for $92, 000 requires a 20% down payment. What is the amount of the down payment? We need to find 20% of $92, 000 = 0. 20 · $92, 000 = $18, 400 The down payment is $18, 400. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 6

Example 16 If Alberto has 45 correct answers on an 80 -question test, what percent of his answers are correct? We need to find what percent of 80 is 45. 56. 25% of Alberto’s answers are correct. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 7

Example 17 Forty-two percent of the parents of the schoolchildren in the Paxson School District are employed at Di Paloma University. If the number of parents employed by the university is 168, how many parents are in the school district? We need to find a number such that 42% of that number is 168. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 8

Example 17 (continued) Let n be the number of parents in the school district. Then, 42% of n is 168. There are 400 parents in the school district. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 9

Example 18 Skis at a sports store near Humphrey’s Summit are on sale for $476. If the original price was $560, what percent were they discounted? If the original price was $560 and the new price is $476, the skis are discounted $84. The discount would have been taken from the original price, so we need to find what percent $84 is of $560. We can divide 84 ÷ 560 = 0. 15; 0. 15 = 15%. The discount was 15%. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 10

Mental Math with Percents Use fraction equivalents ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 11

Mental Math with Percents Use a known percent Frequently, we may not know a percent of something, but we know a close percent of it. For example, to find 55% of 62, do the following: Adding, we see that 55% of 62 is 31 + 3. 1 = 34. 1. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 12

Estimations with Percent Estimations with percents can be used to determine whether answers are reasonable. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 13

Example 19 Laura wants to buy a blouse originally priced at $26. 50 but now on sale at 40% off. She has $17 in her wallet and wonders if she has enough cash. How can she mentally find out? (Ignore the sales tax. ) It is easier to find 40% of $25 (versus $26. 50) mentally. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 14

Example 19 (continued) Laura estimates that the blouse will cost $26. 50 − $10 = $16. 50 Since the actual discount is greater than $10, Laura will have to pay less than $16. 50 for the blouse, so she has enough cash. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 15

Example 20 Which of the following statements could be true and which are false? a. Leonardo got a 10% raise at the end of his first year on the job and a 10% raise after another year. His total raise was 20% of his original salary. False b. Jung and Dina paid 45% of their first department store bill of $620 and 48% of the second department store bill of $380. They paid 45% + 48% = 93% of the total bill of $1000. False ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 16

Example 20 (continued) c. Bill spent 25% of his salary on food and 40% on housing. Bill spent 25% + 40% = 65% of his salary on food and housing. True d. In a town, 65% of the adult population works in town, 25% works across the border, and 15% is unemployed, and everyone in town is exactly one of these categories. False ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 17

Example 20 (continued) e. In Clean City, the fine for various polluting activities is a certain percentage of one’s monthly income. The fine for smoking in public places is 40%, for driving a polluting car is 50%, and for littering is 30%. Mr. Schmutz committed all three polluting crimes in one day and paid a fine of 120% of his monthly income. True ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 18

Computing Interest Suppose you borrow $6000 from a bank at simple interest at a rate of 5% to be paid off 2 years from now. § The $6000 represents the principal, the amount of the loan. § The 5% represents the interest rate. § The period of the loan is the amount of time you owe the money – two years in this case. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 19

Computing Interest Simple interest means that the interest will only be computed on the principal for the stated time of the loan. The formula for simple interest is I = Prt, where I represents simple interest, P represents principal, r represents the interest rate, and t represents the number of years. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 20

Computing Interest The formula for the total amount, A, to pay off a loan is A = P + I = P + Prt = P(1 + rt). ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 21

Example 21 Vera opened a saving account that pays simple interest at the rate of 5 ¼% per year. If she deposits $2000 and makes no other deposits, find the interest and the final amount for 90 days The interest is approximately $25. 89 and the final amount is approximately $2025. 89. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 22

Example 22 Find the annual interest rate if a principal of $10, 000 increased to $10, 900 at the end of 1 year. Let r = annual interest rate. We know that r% of $10, 000 is the increase. The increase is $10, 900 − $10, 000 = $900, so The annual interest rate is 9%. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 23