Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 6 Ratio, Proportion, and Percent

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 6.1 Ratio and Proportion

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. A ratio is the quotient of two quantities. Writing Ratios as Fractions For example, a percent can be thought of as a ratio, since it is the quotient of a number and % = or the ratio of 53 to 100

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The ratio of a number a to a number b is their quotient. Ways of writing ratios are and a b a to b,a : b, Ratio

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. A proportion is a statement that two ratios are equal. Solving Proportions If and are two ratios, then is a proportion. a b c d a b c d =

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solving Proportions A proportion contains four numbers. If any three numbers are known, the fourth number can be found by solving the proportion. To solve use cross products. a b c d a b c d = Multiply both sides by the LCD, bd Simplify ad = bc cross product ad bc These are called cross products.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Determining Whether Proportions are True ? True proportion

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Finding Unknown Numbers in Proportions Cross multiply. Simplify the left side. Divide both sides by 28. Check:

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. A 16-oz Cinnamon Mocha Iced Tea at a local coffee shop has 80 calories. How many calories are there in a 28-oz Cinnamon Mocha Iced Tea? Solving Problems by Writing Proportions Solve the proportion. Cross multiply. Simplify the right side. Divide both side by 140. A 28-oz Cinnamon Mocha Iced Tea has 140 calories.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. When writing proportions to solve problems, write the proportions so that the numerators have the same unit measures and the denominators have the same unit measures. For example, Helpful Hint

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 6.2 Percents, Decimals, and Fractions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The word percent comes from the Latin phrase per centum, which means “per 100.” Percent means per one hundred. The “%” symbol is used to denote percent. Understanding Percent

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall  0.65(100%)  65.% or 65% Writing a Decimal as a Percent Multiply by 1 in the form of 100%.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Writing a Percent as a Decimal 43%  43(0.01)  %  100(0.01)  1.00 or 1 Replace the percent symbol with its decimal equivalent, ; then multiply. Replace the percent symbol with its decimal equivalent, 0.01; then multiply.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Writing a Percent as a Fraction Replace the percent symbol with its fraction equivalent, ; then multiply. Don’t forget to simplify the fraction, if possible.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 16 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Writing a Fraction as a Percent Multiply by 1 in the form of 100%.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 17 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. We know that 100% = 1 Recall that when we multiply a number by 1, we are not changing the value of that number. Therefore, when we multiply a number by 100%, we are not changing its value but rather writing the number as an equivalent percent. Helpful Hint

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 18 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall., replace the % symbol with its fraction equivalent, ; then multiply. To write a percent as a fraction, replace the % symbol with its fraction equivalent, ; then multiply. Summary of Converting Percents, Decimals and Fractions To write a percent as a decimal, replace the % symbol with its decimal equivalent, 0.01; then multiply. To write a decimal or fraction as a percent, multiply by 100%.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 6.3 Solving Percent Problems with Equations

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 20 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Key Words of means multiplication (∙) is means equals ()() what (or some equivalent) means the unknown number Let x stand for the unknown number.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 21 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Translating to an Equation Example: Translate to an equation: 60 is what percent of 40? 60 what percentof40 60 = x  40? is

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Translating to an Equation Example: Translate to an equation: What number is 18% of 66? What number 18%of66? x = 18%  66 is

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 23 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Translating to an Equation Example: Translate to an equation: 25% of 68 is what number? 25% 68%iswhat number? 25%  68%=x?x? of

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 24 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Remember that an equation is simply a mathematical statement that contains an equal sign (  ). 6  18x equal sign Helpful Hint

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 25 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 20% of 50  10 20%50  10 percentbaseamount Percent Equation percent ∙ base  amount Solving Percent Problems

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 26 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. When solving a percent equation, write the percent as a decimal or fraction. If your unknown in the percent equation is a percent, don’t forget to convert your answer to a percent. Helpful Hint

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 27 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solving Percent Problems for the Amount Example: What number is 9% of 65? n=9%65 n= n= is 9% of 65. What number is 9% of 65?

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 28 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solving Percent Problems for the Base Example: 36 is 6% of what number? 36=6%x 36 is 6% of what number 36 = 0.06n 36 is 6% of 600.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 29 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solving Percent Problems for the Percent Example: 24 is what percent of 144? 24=x is what percent of 144?

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 30 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Use the following to see if your answers are reasonable. a percent greater than 100% a percent less than 100% a number larger than the original number a number less than the original number   100% of a number  the number Helpful Hint

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 6.4 Solving Percent Problems with Proportions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 32 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To understand the proportion method, recall that 30% means the ratio of 30 to 100, or. Writing Percent Problems as Proportions

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Since the ratio is equal to the ratio, we have the proportion called the percent proportion. Writing Percent Problems as Proportions,

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 34 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. always 100 or percent base amount Percent Proportion

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 35 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. When we translate percent problems to proportions, the percent, p, can be identified by looking for the symbol % or the word percent. The base, b, usually follows the word of. The amount, a, is the part compared to the whole. Symbols and Key Words

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 36 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Part of ProportionHow It’s Identified Percent% or percent Base Amount Appears after of Part compared to whole Helpful Hints

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 37 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. What number is 20% of 8? amountpercentbase amount base percent Solving Percent Proportions for the Amount

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 38 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 20 is 40% of what number? amountpercentbase amount base percent Solving Percent Proportions for the Base

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 39 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. What percent of 40 is 8? amountpercentbase amount base percent Helpful Hint Recall from our percent proportion that this number, p already is a percent. Just keep the number the same and attach a % symbol. Solving Percent Proportions for the Percent

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 40 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. A ratio in a proportion may be simplified before solving the proportion. The unknown number in both and is 20. Helpful Hint

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 6.5 Applications of Percent

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 42 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The freshman class of 450 students is 36% of all students at State College. How many students go to State College? State the problem in words, then translate to an equation. Method 1 In words: In words: 450 is 36% of what number? Translate: 450  36%x Solving Applications Involving Percent Continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 43 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall students go to State College. Solve: 450  0.36x 450  0.36x 450  0.36x 0.36 Divide both sides by  1250 = x Continued Solving Applications Involving Percent

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The freshman class of 450 students is 36% of all students at State College. How many students go to State College? the problem in words, then translate to a proportion. State the problem in words, then translate to a proportion. Method 2 In words: 450 is 36% of what number? Translate: amountbasepercent Solving Applications Involving Percent Continued

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 45 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Solving Applications Involving Percent Continued 1250 students go to State College. b = 1250 Solve: = b b = = 36b 36

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 46 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Percent Increase and Percent Decrease percent increase percent increase  percent decrease percent decrease  In each case write the quotient as a percent. Helpful Hint Make sure that this number in the denominator is the original number and not the new number.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 47 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Percent Increase Example: Nancy’s salary increased from $16,000 last year to $17,280 this year. What was the percent increase in her salary? Procedure: First, find the amount of increase. Then compare that amount to the previous amount, last year’s salary. Amount of increase = original amount – new amount Nancy’s salary increased by 8%. = 17,280 – 16,000 = 1280

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 48 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Percent Decrease Example: Mark weighed 285 pounds two years ago. After dieting, he reduced his weight to 171 pounds. What was the percent decrease in his weight? Procedure: First, find the amount of decrease. Then compare that amount to Mark’s previous weight. Amount of decrease = original amount – new amount Mark’s weight decreased by 40%. = 285 – 171 = 114

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 6.6 Percent and Problem Solving: Sales Tax, Commission, and Discount

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 50 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Percents are frequently used in the retail trade. For example, most states charge a tax on certain items when purchased. This tax is called a sales tax, and retail stores collect it for the state. Sales tax is almost always stated as a percent of the purchase price. Sales Tax and Total Price A 5% sales tax rate on a purchase of a $10.00 item gives a sales tax of sales tax  5% of $10  0.05 ∙ $10.00  $0.50

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 51 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The total price to the customer would be purchase price plus sales tax $10.00  $0.50  $10.50 Sales Tax and Total Price

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 52 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. sales tax  tax rate ∙ purchase price total price  purchase price  sales tax Sales Tax and Total Price

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 53 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Finding Sales Tax and Total Price Example: If the sales tax rate is 7.75%, find the amount of sales tax due on a purchase of $ Also find the total that the customer must pay. Solution: Write the percent as a decimal and multiply it by the price. Round that amount to the nearest penny. Add the sales tax to the price to find the total amount to be paid. 7.75% = , so the sales tax is   $5.03. The total amount paid is $ = $69.98.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 54 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. commission  commission rate sales Calculating Commissions A wage is payment for performing work. Hourly wage, commissions, and salary are some of the ways wages can be paid. Many people who work in sales are paid a commission. An employee who is paid a commission is paid a percent of his or her total sales.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Finding a Commission Rate Example: Adam is a salesperson at a furniture store and earns a 4.5% commission on everything he sells. Last week, he sold furniture totaling $38,957. How much did Adam earn in commissions last week? Solution: Multiply the percent by the total amount sold and round that amount to the nearest penny. 4.5% = 0.045, so the commission is ,957 = $ The amount of commission Adam earned last week was $

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 56 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Discount and Sale Price` Suppose that an item that normally sells for $40 is on sale for 25% off. This means that the original price of $40 is reduced, or discounted, by 25% of $40, or $10. The discount rate is 25%, the amount of discount is $10, and the sale price is $40  $10, or $30. amount of discount  discount rate ∙ original price sale price  original price  amount of discount

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 57 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Finding a Discount and a Sale Price Example: This week at Brown’s Department Store, purses are discounted 15%. Find the new price of a purse that costs $ Solution: Amount of discount = = $ 7.03 Sales price = $46.90 – 7.03 = $39.87 The new price of the purse is $39.87.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 6.7 Percent and Problem Solving: Interest

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 59 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Interest is money charged for using other people’s money. Money borrowed, loaned, or invested is called the principal amount, or simply principal. The interest rate is the percent used in computing the interest (usually per year). Simple interest is interest computed on the original principal. Calculating Simple Interest

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 60 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simple Interest  Principal Rate Time or I  P R T where the rate is understood to be per year and time is in years. Simple Interest

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 61 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. total amount (paid or received)  principal  interest Finding the Total Amount of a Loan

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 62 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Compound interest is computed on not only the principal, but also on the interest already earned in previous compounding periods. If interest is compounded annually on an investment, this means that interest is added to the principal at the end of each year and next year’s interest is computed on this new amount. Calculating Compound Interest

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 63 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Compound Interest Formula The total amount A in an account is given by where P is the principal, r is the interest rate written as a decimal, t is the length of time in years, and n is the number of times compounded per year.