C HAPTER 5 – P ERCENTS Math Skills – Week 6

O UTLINE 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

I NTRODUCTION TO P ERCENTS 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

I NTRODUCTION TO P ERCENTS Percent  Fraction: Steps: 1. Remove the percent sign 2. Multiply by 1/ 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

I NTRODUCTION TO P ERCENTS Percent  Decimal Steps 1. Remove the percent sign 2. Multiply by 0.01 Examples 1. Write 13% as a decimal 13 x 0.01 = Write 120% as a decimal 120 x 0.01 = 1.2 Class Examples 1. Write 125% as a decimal 125 x 0.01 = Write 0.25% as a decimal 0.25 x 0.01 =

I NTRODUCTION TO P ERCENTS Fraction/Decimal  Percentage Steps 1. Multiply the fraction/decimal by 100% Examples 1. Write 3/8 as a percent 3/8 x 100% = 3/8 x 100%/1 = 300/8 % = 37 ½% 2. Write 2.15 as a percent 2.15 x 100% = 215% Class Examples 1. Write 2/3 as a percent. Write any remainder as a fraction 2/3 x 100% = 200/3 % = 66 2/3 % 2. Write 0.37 as a percent 0.37 x 100% = 37%

T HINGS … 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

P ERCENT E QUATIONS – P T. 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?

P ERCENT E QUATIONS – P T. 1 The question: 2% of $10 is what? Percent 2% x Base $10 = Amount n 0.02 x $10 = $0.20 I earn a commission of 20 cents on a sale of $10. Convert to decimal

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

P ERCENT E QUATIONS – P T. 1 We found the solution using the basic percent equation. Examples 1. Find 5.7% of x 160 = n  9.12 = n 2. What is 33 1/3 % of 90? 1/3 x 90 = n  30 = n 3. Discuss Pg. 208 You try it 4 The Basic percent equation Percent x Base = Amount of  X is  = What/Find  n (unknown quantity) Remember

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

P ERCENT E QUATIONS – P T. 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? Percent n x Base 500 = Amount 1 n = 1 ÷ 500 = = 0.2% chance of winning a free ticket P x B = A

P ERCENT E QUATIONS – P T. 2 Examples: 1. 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% is what percent of 75? 25 = n x 75 n = 25 ÷ 75 (Convert to percentage) n = 1/3 x 100% = 33 1/3 % 3. Discussion 1. Pg 212 – You try it 5 n x 518,921 = 6550 n = 6550 ÷ 518,921 n = = 1.26% (Round to nearest tenth %) ≈ 1.3%

P ERCENT E QUATIONS – P T. 2 Class Examples: 1. What percent of 12 is 27 n x 12 = 27 n = 27 ÷ 12 n = 2.25 (Convert to %) n = 225% is what percent of 45? 30 = n x 45 n = 30 ÷ 45 n = 2/3 (Convert to %) n = 66% 3. What percent of 32 is 16? n x 32 = 16 n = 16 ÷ 32 n = ½ (Convert to %) n = 50%

P ERCENT E QUATIONS – P T. 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? Percent 19% x Base n = Amount 538, x n = 538,000  n = 538,000 ÷ 0.19  n ≈ 2,832,000 total population of US in 1780 P x B = A Convert to decimal

P ERCENT E QUATIONS – P T. 3 Examples: 1. 18% of what number is 900? 0.18 x n = 900 n = 900 ÷ 0.18 n = is 1.5% of what? 30 = x n n = 30 ÷ n = Discuss 1. You try it 5 pg x n = $89.60 n = ÷ 0.8 n = $ $ $89.60 = $22.40

P ERCENT E QUATIONS – P T. 3 Class Examples: 1. 86% of what is 215? 0.86 x n = 215  n = 215 ÷ 0.86  n = is 2.5% of what? 15 = x n  n = 15 ÷  n = /3 % of what is 5? 1/6 x n = 5  n = 5 ÷ 1/6  n = 5 x 6/1  n = Discuss 1. You try it 4 pg. 216

I NTEREST – C HAPTER 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.

I NTEREST – C HAPTER 6.3 To calculate the Simple Interest earned, use the Simple Interest Formula for annual interest rates: Example: 1. Calculate the simple interest due on a 2-year loan of $1500 that has an annual interest rate of 7.5% $1500 x x 2 = $225 in interest. 2. 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 x ½ = $ in interest. They owe a total of $75,000 + $ = $ Each month they must pay $ /6 = $12, towards their loan Principal x Annual Interest Rate x time (in years) = Interest

I NTEREST – C HAPTER 6.3 Class Examples: 1. 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 x 5 = $52,500 Owes a total of $120,000 + $52,500 = $172,500 Principal x Annual Interest Rate x time (in years) = interest

I NTEREST – C HAPTER 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

I NTEREST – C HAPTER 6.3 Examples: 1. Pg. 252 Example 4 2. Pg. 252 You try it 4 Principal x Monthly Interest Rate x time (in months) = interest

I NTEREST – C HAPTER 6.3 Calculating compound interest Most common form of earning interest is interest that is compounded after a specific time period. 1. This is different from Simple Interest. 2. 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

I NTEREST – C HAPTER 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 1 st year I have: $ $90 = $1090. This is my new balance. Interest earned for year 2: $1090 x 0.09 x 1 = $98.10 After 2 nd year I have $ $98.10 = $ This is my new balance. Interest earned for year 3: $ x 0.09 x 1 = $ After 3 rd year I have $ $ = $ I earn $ $1000 = $ This is ~$20 more compared to the simple interest case.

I NTEREST – C HAPTER 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. 1.Annually (once a year) 2.Semiannually (twice a year) 3.Quarterly (4 times per year) 4.Monthly (Once a month) 5.Daily (Once a day)

I NTEREST – C HAPTER 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 1. Locate the correct Compound Interest Table which corresponds to the correct compounding period. 2. 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 3. Multiply the Compound Interest Factor x Principal Investment 4. The resulting product is the value of your investment after the given number of years.

I NTEREST – C HAPTER 6.3 Example: Two different investment opportunities 1. (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 = $ after 5 years. How much profit did I make? $15, – $10,000 = $5, (Plan B) Same investment as above, this time compounded semiannually. $10,000 x = $15, after 5 years. How much profit did I make? $15, – $10,000 = $5, Which investment plan was better? Plan B; ~$140 more in profit

I NTEREST – C HAPTER 6.3 Examples: 1. An investment of $650 pays 8% annual interest compounded semiannually. What is the interest earned in 5 years? What is the compound interest factor? What is the value of my investment after 5 years? $650 x = $ How much interest did I earn after 5 years? $ – $650 = $312.16

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