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 Finance Charges Compound Interest

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

INTRODUCTION TO PERCENTS

Percent Fraction: Steps:

1. Remove the percent sign 2. Multiply by 1/1003. 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

INTRODUCTION TO PERCENTS Percent Decimal

Steps1. Remove the percent sign2. Multiply by 0.01

Examples1. Write 13% as a decimal

13 x 0.01 = 0.132. Write 120% as a decimal

120 x 0.01 = 1.2 Class Examples

1. Write 125% as a decimal 125 x 0.01 = 1.25

2. Write 0.25% as a decimal 0.25 x 0.01 = 0.0025

INTRODUCTION TO PERCENTS

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 Examples1. 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%

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

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?

PERCENT EQUATIONS – PT. 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

PERCENT EQUATIONS – PT. 1

The question:

2% of $10 is what?

Percent

2% x Base

$10 = Amount

n

Note relationship/translation between English and math

of Xis =

What (Find) n (unknown quantity)

PERCENT EQUATIONS – PT. 1 We found the solution using the basic percent

equation.

Examples1. Find 5.7% of 160

0.057 x 160 = n 9.12 = n2. What is 33 1/3 % of 90?

1/3 x 90 = n 30 = n3. Discuss

Pg. 208 You try it 4

The Basic percent equation

Percent x Base = Amount

of Xis =

What/Find n (unknown quantity)

Remember

PERCENT EQUATIONS – PT. 1

Class Examples1. Find 6.3% of 150

0.063 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 425 0.12 x 425 = n 51 = n

The Basic percent equation

Percent x Base = Amount

of XIs =

What/Find n (unknown quantity

“Amount”)

Remember

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?Percent

n x Base

500 = Amount

1

n = 1 ÷ 500 = 0.002 = 0.2% chance of winning a free ticket

P x B = A

PERCENT EQUATIONS – PT. 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% 2. 25 is what percent of 75?

25 = n x 75 n = 25 ÷ 75 (Convert to percentage) n = 1/3 x 100% = 33 1/3 %

3. Discussion1. 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%

PERCENT EQUATIONS – PT. 2 Class Examples:

1. What percent of 12 is 27 n x 12 = 27 n = 27 ÷ 12 n = 2.25 (Convert to %) n = 225%

2. 30 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%

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?Percent

19% x Base

n = Amount

538,0000.19 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

PERCENT EQUATIONS – PT. 3 Examples:

1. 18% of what number is 900? 0.18 x n = 900 n = 900 ÷ 0.18 n = 5000

2. 30 is 1.5% of what? 30 = 0.015 x n n = 30 ÷ 0.015 n = 2000

3. Discuss1. 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

PERCENT EQUATIONS – PT. 3

Class Examples:1. 86% of what is 215?

0.86 x n = 215 n = 215 ÷ 0.86 n = 250

2. 15 is 2.5% of what? 15 = 0.025 x n n = 15 ÷ 0.025 n =

600

3. 16 2/3 % of what is 5? 1/6 x n = 5 n = 5 ÷ 1/6 n = 5 x 6/1

n = 30

4. Discuss 1. You try it 4 pg. 216

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.

INTEREST – CHAPTER 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 0.075 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 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

INTEREST – CHAPTER 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 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

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

INTEREST – CHAPTER 6.3

Examples:1. Pg. 252 Example 42. Pg. 252 You try it 4

Principal x Monthly Interest Rate x time (in months) = interest

INTEREST – CHAPTER 6.3

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.

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.

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)

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 investment1. 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.

INTEREST – CHAPTER 6.3 Example: Two different investment

opportunities1. (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

2. (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

INTEREST – CHAPTER 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? 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

INTEREST – CHAPTER 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? 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