troduction to Algebra

Transcription

1 Chapter Six Percent Percents, Decimals, and Fractions Understanding Percent 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. 1% = 1 =

2 Writing a Percent as a Decimal Replace the percent symbol with its decimal equivalent, 0.01; then multiply. 43% = 43(0.01) = % = 100(0.01) = 1.00 or 1 4 Writing a Decimal as a Percent Multiply by 1 in the form of 100%. 065= 0.65 = 0.65(100%) = 65.% or 65% 5 Writing a Percent as a Fraction Replace the percent symbol with its fraction equivalent, 1 ; then 100 multiply. Don t forget to simplify the fraction, if possible. 1 43% = 43 =

3 Writing a Fraction as a Percent Multiply by 1 in the form of 100% = 100% = % = 300 % = 60% Helpful Hint We know that 100% = 1 Recall that t when we multiply l 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. 8 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 percent as a fraction,, replace the % symbol with its fraction equivalent, 1 ; then multiply. 100 To write a decimal or fraction as a percent, multiply by 100%. 9

4 Solving Percent Problems with Equations Key Words of means multiplication ( ) is means equals (=) what (or some equivalent) means the unknown number Let x stand for the unknown number. 11 Helpful Hint Remember that an equation is simply a mathematical statement that contains an equal sign (=). 6 = 18x equal sign 12

5 Solving Percent Problems 20% of 50 = 10 20% 50 = 10 percent base amount Percent Equation percent base = amount 13 Helpful Hint When solving a percent equation, write the percent as a decimal or fraction. If your unknown in the percent equation is percent, don t forget to convert your answer to a percent. 14 Helpful Hint Use the following to see if your answers are reasonable. 100% of a number = the number a percent greater than 100% a percent less than 100% = = a number larger than the original number a number less than the original number 15

6 Solving Percent Problems with Proportions Writing Percent Problems as Proportions To understand the proportion method, recall that 30% means the 30 ratioof30to100 of to 100, or % = 30 = Writing Percent Problems as Proportions... Since the ratio 30 is equal to the 100 ratio 3, we have the proportion =, called the percent proportion. 18

7 Percent Proportion amount percent = base 100 always 100 amount base or a p b = 100 percent 19 When we translate percent problems to proportions, the percent can be identified by looking for the symbol % or the word percent.. The base usually follows the word of.. The amount is the part compared to the whole. 20 Helpful Hint Part of Proportion How It s Identified Percent Base Amount % or percent Appears after of Part compared to whole 21

8 Solving Percent Proportions for the Amount What number is 20% of 8? amount percent base amount base a 20 = percent 22 Solving Percent Proportions for the Base 20 is 40% of what number? amount percent base amount base b = 100 percent 23 Solving Percent Proportions for the Percent What percent of 40 is 8? percent base amount amount base Helpful Hint 8 p = percent Recall from our percent proportion that this number, p already is a percent. Just keep the number the same and attach a % symbol. 24

9 Helpful Hint A ratio in a proportion may be simplified before solving the proportion. The unknown number in both 6 30 = 4 b and is = 2 b 25 Applications of Percent The freshman class of 450 students is 36% of all students at State College. How many students go to State College? Equation Method State the problem in words, then translate to an equation. In words: Solve: 450 = 0.36x 450 is 36% of what number? Translate: 450 = 36% x 27

10 The freshman class of 450 students is 36% of all students at State College. How many students go to State College? Proportion Equation Method State the problem in words, then translate to a proportion. In words: 450 is 36% of what number? amount percent Translate and Solve: base b = Percent Increase Percent Decrease percent increase = percent decrease = amount of increase original amount amount of decrease original amount 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. 29 Percent and Problem Solving: Sales Tax, Commission, and Discount

11 Calculating Sales Tax and Total Price Most states charge a tax on certain items when purchased called a sales tax. 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 = $ The total price to the customer would be purchase price sales tax plus $ $0.50 = $ Sales Tax and Total Price sales tax = tax rate purchase price total t price = purchase price + sales tax 33

12 Calculating Commissions A wage is payment for performing work. An employee who is paid a commission as a wage is paid a percent of his or her total sales. Commission commission = commission rate sales 34 Discount and Sale Price amount of discount = discount rate original price sale price = original price amount of discount 35 Percent and Problem Solving: Interest

13 Calculating Simple Interest 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. 37 Simple Interest simple interest = principal rate time or I = P r t where the rate is understood to be per year and time is in years. 38 Simple Interest App. If $100 was borrowed for 2 years at a 10% interest rate, the interest would be? $100*10/100*2 = $20 The total amount that would be due would be? $100+$20=$

14 Finding the Total Amount of a Loan or Investment total amount (paid or received) = principal + interest 40 Simple Interest: Example Example: Ray put $1,000 into a savings account. The interest on the account is 3.5%. He wants to put the money away for 18 months. How much will Ray have at the end of that time period? I = $1,000 x.035 x 1.5 (divide the number of months by 12) I = $52.50 Adding the interest back on to the principle, Ray now has $1, Simple Interest: Example Beth owes $38,000 in student loans. The interest rate on her loans is 8.25%. She will be paying these loans off for 20 years. How much will Beth pay altogether? I = $62,700 Adding the interest back on to the principle, Beth has to pay $100,

15 Calculating Compound Interest 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. 43 Finding Total Amounts with Compound Interest total amount = original principal compound interest factor The compound interest factor comes from the compound interest table found in Appendix C of the textbook. 44 Example of Compound Interest An amount of $ is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6 years. A. Using the formula above, with P = 1500, r = 4.3/100 = 0.043, n = 4, and t = 6: So, the balance after 6 years is approximately $1,

16 Finding the Monthly Payment of a Loan monthly payment = principal + interest total number of payments You borrow $500 and will pay me back in 6 months. We agree a 7% interest rate is fair. What will you pay me per month? 46 Finding the Monthly Payment of a Home Loan monthly payment = principal + interest total number of payments You purchased a new home for $150, at 6.5% for 30 years. What is your monthly payment? 47