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Chapter 5 – Percents

Chapter 5 – Percents. Math Skills – Week 6. Outline . Introduction to Percents – Section 5.1 Percent Equations Part I– Section 5.2 Percent Equations Part II – Section 5.3 Percent Equations Part III – Section 5.4 Interest – Section 6.3 Applications of percents Simple Interest

austin-cummings
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Chapter 5 – Percents

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  1. Chapter 5 – Percents Math Skills – Week 6

  2. Outline • Introduction to Percents – Section 5.1 • Percent Equations Part I– Section 5.2 • Percent Equations Part II – Section 5.3 • Percent Equations Part III – Section 5.4 • Interest – Section 6.3 • Applications of percents • Simple Interest • Finance Charges • Compound Interest

  3. Introduction to Percents • Percent means “Parts of 100” • (See page 203) • 13 parts of 100 means 13% • 20 parts of 100 means 20% • Percents can be written as fractions and decimals • We will need to: • Rewrite a percent as a fraction or a decimal • Rewrite a fraction or decimal as a percent

  4. Introduction to Percents • Percent Fraction: • Steps: • Remove the percent sign • Multiply by 1/100 • Simplify the fraction (if needed) • Examples • Write 13% as a fraction • = 13 x 1/100 = 13/100 • Write 120% as a fraction • = 120 x 1/100 = 120/100 = 1 1/5 • Class examples • Write 33 1/3% as a fraction • = 100/3 x 1/100 = 1/3

  5. Introduction to Percents • Percent  Decimal • Steps • Remove the percent sign • Multiply by 0.01 • Examples • Write 13% as a decimal • 13 x 0.01 = 0.13 • Write 120% as a decimal • 120 x 0.01 = 1.2 • Class Examples • Write 125% as a decimal • 125 x 0.01 = 1.25 • Write 0.25% as a decimal • 0.25 x 0.01 = 0.0025

  6. Introduction to Percents • Fraction/Decimal  Percentage • Steps • Multiply the fraction/decimal by 100% • Examples • Write 3/8 as a percent • 3/8 x 100% = 3/8 x 100%/1 = 300/8 % = 37 ½% • Write 2.15 as a percent • 2.15 x 100% = 215% • Class Examples • Write 2/3 as a percent. Write any remainder as a fraction • 2/3 x 100% = 200/3 % = 66 2/3 % • Write 0.37 as a percent • 0.37 x 100% = 37%

  7. Things… • Practice Final Exam on Website • Major focus on being able to solve these problems • Second practice final exam available later today • Sample Projects • Extra help • Tutoring in the IDEA center • REMEMBER: Only 1 late quiz and 1 late HW for the entire class • Check MyInfo page for (late) indicator next to quiz/hw assignment • Homework Grades • Early Final Candidates

  8. Percent Equations – Pt. 1 • Real estate brokers, retail sales, car salesmen, etc. make the majority of their money on commission. • When they make a sale, they get a percentage of the total sale. • For example: I sell a scarf to a customer for $10. My commission says I earn 2% (commission) of the total of each sale that I make. How much commission do I earn for this sale?

  9. Percent Equations – Pt. 1 • The question: • 2% of $10 is what? Amount n Percent 2% Base $10 x = Convert to decimal 0.02 x $10 = $0.20 I earn a commission of 20 cents on a sale of $10.

  10. Percent Equations – Pt. 1 • The question: • 2% of $10 is what? Amount n Percent 2% Base $10 x = Note relationship/translation between English and math of  X is  = What (Find)  n (unknown quantity)

  11. Percent Equations – Pt. 1 • We found the solution using the basic percent equation. • Examples • Find 5.7% of 160 • 0.057 x 160 = n  9.12 = n • What is 33 1/3 % of 90? • 1/3 x 90 = n 30 = n • Discuss • Pg. 208 You try it 4 The Basic percent equation Percent x Base = Amount Remember of  X is  = What/Find  n (unknown quantity)

  12. Percent Equations – Pt. 1 • Class Examples • Find 6.3% of 150 • 0.063 x 150 = n  9.45 = n • What is 16 2/3% of 66? • 1/6 x 66 = n 11 = n • Find 12% of 425 • 0.12 x 425 = n  51 = n The Basic percent equation Percent x Base = Amount Remember of  X Is  = What/Find  n (unknown quantity “Amount”)

  13. Percent Equations – Pt. 2 • What if we are given the base and the amount and we want to find the corresponding percent? • Example: A lottery scratcher game advertises that there is a 1 in 500 chance of winning a free ticket. What is our percent chance of winning a free ticket? • The question: • What percent of 500 is 1? P x B = A Percent n Base 500 Amount 1 x = n = 1 ÷ 500 = 0.002 = 0.2% chance of winning a free ticket

  14. Percent Equations – Pt. 2 • Examples: • What percent of 40 is 30? • n x 40 = 30 • n = 30 ÷ 40 • n = 0.75 • (Convert to percentage) n = 0.75 x 100%  n = 75% • 25 is what percent of 75? • 25 = n x 75 • n = 25 ÷ 75 • (Convert to percentage) n = 1/3 x 100% = 33 1/3 % • Discussion • Pg 212 – You try it 5 • n x 518,921 = 6550 • n = 6550 ÷ 518,921 • n = 0.0126 = 1.26% • (Round to nearest tenth %) ≈ 1.3%

  15. Percent Equations – Pt. 2 • Class Examples: • What percent of 12 is 27 • n x 12 = 27 • n = 27 ÷ 12 • n = 2.25 • (Convert to %) n = 225% • 30 is what percent of 45? • 30 = n x 45 • n = 30 ÷ 45 • n = 2/3 • (Convert to %) n = 66% • What percent of 32 is 16? • n x 32 = 16 • n = 16 ÷ 32 • n = ½ • (Convert to %) n= 50%

  16. Percent Equations – Pt. 3 • What if we are given the percent and the amount and we want to find the corresponding base? • Example: In 1780, the population of Virginia was 538,000; this accounted for 19% of the total population. Find the total population of the USA. • Question: • 19% of what number is 538,000? P x B = A Base n Amount 538,000 Percent 19% x = Convert to decimal 0.19 x n = 538,000  n = 538,000 ÷ 0.19  n ≈ 2,832,000 total population of US in 1780

  17. Percent Equations – Pt. 3 • Examples: • 18% of what number is 900? • 0.18 x n = 900 • n = 900 ÷ 0.18 • n = 5000 • 30 is 1.5% of what? • 30 = 0.015 x n • n = 30 ÷ 0.015 • n = 2000 • Discuss • You try it 5 pg. 216 • 0.8 x n = $89.60 • n = 89.60 ÷ 0.8 • n = $112.00 • $112 .00 - $89.60 = $22.40

  18. Percent Equations – Pt. 3 • Class Examples: • 86% of what is 215? • 0.86 x n = 215  n = 215 ÷ 0.86  n = 250 • 15 is 2.5% of what? • 15 = 0.025 x n  n = 15 ÷ 0.025  n = 600 • 16 2/3 % of what is 5? • 1/6 x n = 5  n = 5 ÷ 1/6  n = 5 x 6/1  n = 30 • Discuss • You try it 4 pg. 216

  19. Interest – Chapter 6.3 • When we deposit money into a bank, they pay us interest. Why? • They use our money to loan out to other customers. • When we borrow money from the bank, we must pay interest to the bank. • Definitions • The original amount we deposited is called the principal (or principal balance). • The amount we earn from interest is based on the interest rate the bank gives us. • Given as a percent (i.e annual percentage rate) • Interest paid on the original amount we deposited (principal) is called simple interest.

  20. Interest – Chapter 6.3 • To calculate the Simple Interest earned, use the Simple Interest Formula for annual interest rates: • Example: • Calculate the simple interest due on a 2-year loan of $1500 that has an annual interest rate of 7.5% • $1500 x 0.075 x 2 = $225 in interest. • A software company borrowed $75,000 for 6 months at an annual interest rate of 7.25%. Find the monthly payment on the loan • $75,000 x 0.0725 x ½ = $2178.75 in interest. • They owe a total of $75,000 + $2178.75 = $77178.75 • Each month they must pay $77178.75/6 = $12,953.13 towards their loan Principal x Annual Interest Rate x time (in years) = Interest

  21. Interest – Chapter 6.3 • Class Examples: • A rancher borrowed $120,000 for 5 years at an annual interest rate of 8.75%. What is the simple interest due on the loan? • $120,000 x 0.0875 x 5 = $52,500 • Owes a total of $120,000 + $52,500 = $172,500 Principal x Annual Interest Rate x time (in years) = interest

  22. Interest – Chapter 6.3 • Finance Charges on a Credit Card • When you buy things with your credit card, you are borrowing money from a credit institution • In borrowing the money, you are subject to paying interest charges. • Interest charges on purchases are called finance charges. • To calculate the monthly finance charge use the Simple Interest Formula. Principal x Monthly Interest Rate x time (in months) = interest

  23. Interest – Chapter 6.3 • Examples: • Pg. 252 Example 4 • Pg. 252 You try it 4 Principal x Monthly Interest Rate x time (in months) = interest

  24. Interest – Chapter 6.3 • Calculating compound interest • Most common form of earning interest is interest that is compounded after a specific time period. • This is different from Simple Interest. • Given interest based on the amount in your account; NOT based on your principal balance. • Example: • I invest $1000 in a Certificate of Deposit (CD) which is locked up for 3 years. The CD has an annual interest rate of 9% compounded annually. What does this mean? • First Simple interest case • (Principal amount) x (Annual Percentage Rate) x (#Years) = Simple interest • $1000 x 0.09 x 3 = $270

  25. Interest – Chapter 6.3 • Calculating Compound Interest • Example: • I invest $1000 in a CD which is locked up for 3 years. The CD has an annual interest rate of 9% compounded annually. What does this mean? • Compounded (yearly) interest: • Interest earned for year 1: $1000 x 0.09 x 1 = $90 • After 1st year I have: $1000 + $90 = $1090. This is my new balance. • Interest earned for year 2: $1090 x 0.09 x 1 = $98.10 • After 2nd year I have $1090 + $98.10 = $1188.10. This is my new balance. • Interest earned for year 3: $1188.10 x 0.09 x 1 = $106.93 • After 3rd year I have $1188.10 + $106.93 = $1295.03 • I earn $1295.03 - $1000 = $295.03. This is ~$20 more compared to the simple interest case.

  26. Interest – Chapter 6.3 • The Compounding Period defines how often an interest payment is made on your account • The compounding periods can vary as shown below: • NOTE: The more frequent the compounding occurs, the more interest you earn over any given period of time. • Annually (once a year) • Semiannually (twice a year) • Quarterly (4 times per year) • Monthly (Once a month) • Daily (Once a day)

  27. Interest – Chapter 6.3 • Example: Calculate the interest earned on an initial investment of $2,500 that earns 5% interest compounded annually over 15 years. • Very tedious. • A little help please • Compound Interest Table • Pg. 584 – 585 • Using the Compound Interest table • Steps to determine the compound interest earned on a principal investment • Locate the correct Compound Interest Table which corresponds to the correct compounding period. • Look at number in the table where the Interest rate and number of years for the investment meet. This is called the Compound Interest Factor • Multiply the Compound Interest Factor x Principal Investment • The resulting product is the value of your investment after the given number of years.

  28. Interest – Chapter 6.3 • Example: Two different investment opportunities • (Plan A) I invest $10,000 in a CD which is locked up for 5 years. The CD has an annual interest rate of 9% compounded annually. Use a Compound Interest chart to determine the value of my investment after 5 years. • $10,000 x 1.53862 = $15380.62 after 5 years. How much profit did I make? • $15,380.62 – $10,000 = $5,380.62 • (Plan B) Same investment as above, this time compounded semiannually. • $10,000 x 1.55297 = $15,520.97 after 5 years. How much profit did I make? • $15,520.97 – $10,000 = $5,520.97 • Which investment plan was better? • Plan B; ~$140 more in profit

  29. Interest – Chapter 6.3 • Examples: • An investment of $650 pays 8% annual interest compounded semiannually. What is the interest earned in 5 years? • What is the compound interest factor? • 1.48024 • What is the value of my investment after 5 years? • $650 x 1.48024 = $962. 16 • How much interest did I earn after 5 years? • $962.16 – $650 = $312.16

  30. Interest – Chapter 6.3 • Class Example: • An investment of $1000 pays 6% annual interest compounded quarterly. What is the interest earned in 20 years? • What is the compound interest factor? • 3.29066 • What is the value of my investment after 20 years? • $1,000 x 3.29066 = $3,290.66 • How much interest did I earn after the 20 years? • $3,290.66 – $1,000 = $2,290.66

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