CHAPTER 6 Percents. What is a percent? It is a ratio that compares a number to 100. How are...

CHAPTER 6Percents

What is a percent? It is a ratio that compares a number to 100.

How are percents related to proportional relationships?

PETS Some students are collecting money for a local pet shelter. The model shows they have raised 60% of their $2,000 goal or $1,200.Sketch the model and label using decimals instead of percents. Sketch the model using fractions

instead of percents. Use these models to write two multiplication sentences

that are equivalent to 60% of 2,000 = 1,200.

Chapter 6-1-BPercent of a Number

KEY CONCEPT To find the percent of a number such as 60% of 2,000,

you can use either of the following methods: Write the percent as a fraction and then multiply.

All percents are out of 100. Reduce your fractions to make multiplying easier.

Write the percent as a decimal and then multiply. To change a percent to its decimal form, move the decimal two

spaces left.

FIND THE PERCENT OF A NUMBER

40% of 70


Now you try! You can use the method you prefer. 15% of 100 55% of 160 8% of 125

USE PERCENTS GREATER THAN 100% Percents that are greater than 100% mean that the

percent is greater than the whole. Example: 110% = 1.1 or 11/10

Find 150% of 20 Find 160% of 35 Find 125% of 64


158810

305680

REAL WORLD EXAMPLE Refer to the graph to the right. Suppose 455 students took the survey.

How many can be expected to have more than 4 televisions each in their houses? About 114 students

If 275 students took the survey, howmay can be expected to have 3 televisions each in their houses? About 63 students.


Self Assessment: Complete book page 322 # 1 – 7.

What fraction of women tooklessons at school?

If 200 women were surveyed, how many of them took lessonsat school?

Use a fraction to estimate the number of men who took lessonsat school. Assume 200 menwere surveyed.

Sometimes an exact answer is not needed when using percents.

One way to estimate the percent of a number is to use a fraction.

Chapter 6-1-CPercent and Estimation

REAL WORLD EXAMPLE In a recent year, quarterback Carson Palmer completed 62%

of his passes. Hew threw 520 passes. About how many did he complete? 62% is close to 60%. We can write 60% as the fraction . Then we multiply by 520 to get…

…about 312 passes. REAL WORLD EXAMPLE

Box turtles have been known to live for 120 years. American alligators have been known to live 42% as long as box turtles. About how long can an American alligator live? About 48 years.


Another method for estimating the percent of a number is to first to find 10% of the number and then multiply. REMEMBER: To find 10% of a number, just move the

decimal place to the left once. Example: 10% of 170 is 17. Example: 10% of 15.4 is 1.54.

REAL WORLD EXAMPLE Maria and her friends ordered a pizza that cost $14.72. She

is responsible for 20% of the bill. About how much money will she need to pay?

We can round $14.72 up to $15.00. Then we can find 10% of $15.00 to get $1.50. We can double the $1.50 to find the 20%.

She owes $3.


How could we solve this by using a fraction

to estimate?

Choose which method you’d like to use! Use a fraction or find 10% of the number and multiply!

REAL WORLD EXAMPLE Dante plans to put 80% of his paycheck in to a savings

account. His paycheck this week was $295. About how much money will he put into his savings? About $240.

REAL WORLD EXAMPLE A town sold 440 tickets to a chamber music concert in the

town square. Of the tickets sold, 61% were discounted for senior citizens. About how many senior citizens bought tickets for the concert? About 264


You can also estimate percents of numbers when the percent is greater than 100 or less than 1. Estimate 122% of 50.

122% is close to 120%. 120% of 50 = (100% of 50) + (20% of 50) Or about 60.

Estimate % of 589. 589 is close to 600. % is one-fourth of 1%. 1% of 600 = 0.01 · 600 = 6. One-fourth of 6 is · 6 = 1.5.


Try some on your own! Use the examples to help you!• 174% of 200.

• 0.25% of 789

• 298% of 45.

• % of 898.

About 350.About 2.About 150.About 3.

3

1

REAL WORLD EXAMPLES In a recent year, there were about 200 million people in

the U.S. with cell phones. Of those, about 0.5% used their cell phone as an MP3 player. Estimate the number of people who used their cell phone as an MP3 player. 0.5% is half of 1%. Find one percent, and then half of that.

About 1,000,000 used their cell phone as an MP3 player.

Last year, 639 students attended a summer camp. Of those who attended this year, 0.9% also attended last year. About how many students attended the camp two years in a row? 0.9% is close to 1%. Fine one percent.

About 6 students.


Self Assessment: Complete book page 327 # 1 – 8.

In a percent proportion, one ratio or fraction compares part of a quantity to the whole quantity. The other ratio is the equivalent percent written as a fraction with a denominator of 100. 4 out of 5 is 80%

Chapter 6-2-BThe Percent Proportion

Find the Part How can we solve

this proportion?

So p = 48. 48 is 40% of 120. Now you try!

What number is 5% of 60? 3

12% of 85 is what number? 10.2


NOTICE!The whole usually comes after the word “of”.

Find the Whole How can we solve

this proportion?

So w = 72. 18 is 25% of 72. Now you try!

40% of what number is 26? 65

84 is 75% of what number? 112


Find the Percent How can we solve

this proportion?

So n = 60. $9 is 60% of $15. Now you try:

What percent of 25 is 20? 80%

$12.75 is what percent of $50. 25.5%


REAL WORLD EXAMPLE If 200 of the 550 reptiles in the zoo are on display, what

percent of the reptiles are on display? Round to the nearest whole number. How do we know what we are looking for? Set up a percent proportion and solve.

36% of the reptiles are on display.

Sally read the nutrition facts on a box of her favorite cereal. Each cup of the cereal provides 7% of the recommended daily value of potassium. If a cup of the cereal contains 260 milligrams of potassium, what is the recommended daily value of potassium? How do we know what we are looking for? Set up a proportion and solve.

About 3,714 mg.


Self Assessment: Complete book page 335 # 1 - 7.

Suppose there are 854,000 different species of spiders, insects, crustaceans, millipedes, and centipedes on Earth. The graph shows that 88% of the total number of species of arthropods are insects. Use the percent proportion to find how

many species are insects. Express the percent of insects as a

decimal. Then multiply the decimal by 854,000. What do you notice?

You have used a percent proportion to find the missing part, percent, or whole. You can also use a percent equation.

Chapter 6-2-CThe Percent Equation

Find the Part What number is 12% of 150?

PERCENT EQUATIONpart = percent • whole

p = 0.12 • 150 p = 18 So, 18 is 12% of 150.

Now you try! What is 6% of 200?

p = 0.06 • 200 = 12 Find 72% of 50.

p = 0.72 • 50 = 36 What number is 46% of 200?

p = 0.46 • 200 = 92


A percent must always be converted to a decimal or a fraction when it is used in an equation.

Find the Percent 21 is what percent of 40?


21 = n • 40 Divide both sides by 40. n = 0.525 Since n represent the

decimal form, the percent is 52.5%.

So, 21 is 52.5% of 40.

Now you try! 35 is what percent of 70?

35 = n • 70 50%

What percent of 125 is 75? 75 = n • 125 60%

What percent of 40 is 9? 9 = n • 40 22.5%

27 is what percent of 150? 27 = n • 150 18%


Remember to write

the decimal as a

percent in your final

answer.

To change a decimal into a percent, move the decimal to the right two spaces.

Find the Whole 13 is 26% of what number?


13 = 0.26 • w Divide both sides by 0.26. w = 50 So, 13 is 26% of 50.

Remember, when dividing by decimals, you must move the decimal in the dividend the same number of spaces you move the decimal in the divisor.

Now you try! 39 is 84% of what number?

39 = 0.84 • ww = 46.4 26% of what number is 45?

45 = 0.26 • ww = 173.1 14% of what number is 7?

7 = 0.14 • w w = 50 24 is 32% of what number?

24 = 0.32 • ww = 75


Types of Percent Problems

Type Example Equation

Find the Percent 3 is what percent of 6?

Find the Part What number is 50% of 6?

Find the Whole 3 is 50% of what number?


REAL WORLD EXAMPLE A survey found that 25% of people aged 18-24 gave up their

home phone and only use a cell phone. If 3,264 people only use a cell phone, how many people were surveyed?

About 13,056 peoplewere surveyed.

The Miami-Dade County metropolitan area contains 13.3% of the population in Florida. If the population of Florida is about 18,089,888 people, what is the population of the Miami-Dade Country metropolitan area? What part of the percent equation do we know? About 2,405,955 people



The table shows about how many people attended the home games of a highschool football team each year. How much did the attendance

increase from 2009 to 2010? Write the ratio . Then

write the ratio as a percent. Round to the nearest hundredth.

How much did the attendance increase from 2008 to 2009?

Write the ratio . Then write the ratio as a percent. Round to the nearest hundredth.

MAKE A CONJECTURE: Why are the amounts of increase the same but the percent different?

Chapter 6-3-BPercent of Change

KEY CONCEPT A percent of change is a ratio that compares the

change in quantity to the original amount.

When the percent of change is positive, then it is called a percent of increase. When the percent of change is negative, the it is called percent of decrease.

What is another way we can determine if the change is a decrease or increase?

In the percent of change formula, the decimal representing the percent of change must be written as a percent.


Find each percent of change. Round to the nearest whole percent if necessary. State whether the percent of change is an increase or a decrease. 100 acres to 140 acres

40%, increase 48 notebooks to 14 notebooks

-71%, decrease $15.60 to $11.70

-25%, decrease 624 feet to 702 feet

13%, increase


WATCH OUT!A common error is writing the wrong denominator in the percent of change formula. Make sure that the ORIGINAL AMOUNT is always the denominator!

REAL WORLD EXAMPLE Find the percent of change from 10 yards to 13 yards. Then state

whether the percent of change is an increase or a decrease. 30%, increase

Find the percent of change from $20 to $15. Then state whether the percent of change is an increase or decrease.

-25%, decrease Jonas has been saving for a video game. Last year it cost $28. This

year it costs $36. Find the percent of change in the cost. Round to the nearest whole percent if necessary. Then state whether the percent of change is an increase or a decrease.

29%, increase Last month 349 books were checked out from the school library. This

month, 273 books were checked out. Find the percent of change in the number checked out. Round to the nearest whole percent if necessary. Then state whether the percent of change is an increase or a decrease.

-22%, decrease



Sales tax is an additional amount of money charged on items that people buy. A DVD player costs $140 and the sales tax is 5.75%.

What is the total cost of the DVD player? METHOD 1: Add sales tax to the regular price

First find the sales tax. 0.0575 × 140 = 8.05

Next, add the sales tax to the regular price. $140 + $8.05 = $148.05

METHOD 2: Add the percent of tax to 100% First, add the percents together.

100% + 5.75% = 105.75% Then find the new percent of the regular price.

1.0575 × 140 = $148.05

Chapter 6-3-CSales Tax and Tip

Choose the method you prefer and try this one!

What is the total cost of a sweatshirt if the regular price is $42 and the sales tax is

5.5%?

$44.31

A tip or gratuity is a small amount of money in return for a service. The total price is the regular price of the service plus the tip. A customer wants to tip 15% of the

restaurant bill. What will be the total bill with tip? METHOD 1: Add the top to the regular price.

First, find the tip. 0.15 × 35 = 5.25

Then add the tip to the bill to find the total cost. $35.00 + $5.25 = $40.25

METHOD 2: Add the percent of tip to 100% First, add the percents together.

100% + 15% = 115% Then find the new percent of the bill.

1.15 × 35 = $40.25


Another way to calculate the tip is to find 10% of the bill and double that for a 20% tip!

Scott wants to tip his taxicab driver. If his commute costs $15 and he wants to give the driver a 20% tip, what is the total cost? $18

A haircut costs $20. Sales tax is 4.75%. Is $25 sufficient to cover the haircut with tax and a 15% tip? NOTE: The tax and tip are calculated using the cost of the haircut.

One strategy is to find the percents separately. Another strategy is to add the 15% and 4.75% together to calculate

the total cost with tax and tip. Total cost: $23.95, so $25 is sufficient.

Find the total cost of a spa treatment of $42 including 6% tax and 20% tip. $52.92



Discount is the amount by which the regular price of an item is reduced. The sale price is the regular price minus the discount.

REAL WORLD EXAMPLE A DVD normally costs $22. This week it is on sale for

25% off the original price. What is the sale price of the DVD? METHOD 1: Subtract the discount from the regular price

First, find the amount of the discount. 0.25 × 22 = 5.50

Next, subtract the discount from the original price. $22.00 - $5.50 = $16.50

METHOD 2: Subtract the percent of discount from 100% First, subtract the discount percent from 100%

100%-25% = 75% Then calculate the discounted price.

0.75 × 22 = $16.50

Chapter 6-3-DDiscount

Now you try!

A shirt is regularly

prices at $42. It is on sale for 15% off. What

is the sale price of the

shirt?

$35.70

Find the Sale Price A boogie board that has a regular price of $69 is on sale at a

35% discount. What is the sale price with 7% tax?

STEP 1: Find the amount of the discount. 35% of $69 = 0.35 × 69 = $24.15

STEP 2: Subtract the discount from the regular price. $69 - $24.15 = $44.85

STEP 3: The percent of tax is applied after the discount is taken. 7% of $44.85 = 0.07 × 44.85 = $3.14 the TAX Add the tax to the sale price.

$44.85 + $3.14 = $47.99


Tax and DiscountIf both are represented as percents, sales tax is a percent of increase and discount is a percent of decrease. Which one should be calculated first?

NOW YOU TRY! A CD that has a regular price of $15.50 is on sale at

a 25% discount. What is the sale price with a 6.5% sales tax? $12.38

Miss Holloway had to buy party supplies for her birthday last week. She wanted to buy a set of balloons. The original cost of the balloons was $39, but the store was offering a 25% discount. What was the sale price of the balloons including a 5.75% tax? $30.93


Find the Original Price A cell phone is on sale for 30% off. If the sale price is $239.89,

what is the original price? The sale price is 100% - 30% or 70% of the original price.

We can use the percent equation to solve for the WHOLE.

Solve for p. The original price

is $342.70.

Find the original price is the sale price of the cell phoneis $205.50. $293.57

Mrs. Bybee wants to buy a new cell phone that is on sale for 60% off. If the sale price is $79.98, what is the original price? $199.95



Sami plans to save the $200 she received for her birthday. The table shows the average yearly rates at three different banks.1. Calculate 2.50% of $200 to find the

amount of money Sami can earn in one year at Federal Credit Union.

$5

2. Calculate 2.75% of $200 to find the amount of money Sami can earn in one year at First Bank.

$5.50 Principal is the amount of money deposited or borrowed. Simple interest is the amount of paid or earned for the use

of money. To find simple interest I, use the following formula:

Chapter 6-3-EFinancial Literacy: Simple Interest

When calculating simple interest, it is just a matter of plugging what you know in to the formula and solving for what you don’t know! Arnold has $580 is a savings account that pays 3% interest.

How much interest will he earn in each amount of time? 5 years

I = 580 • 0.03 • 5 I = 87

Arnold will earn $87 in interest in 5 years.

6 months I = 580 •0.05 • 0.5 I = 8.7

Arnold will earn $8.70 in interest in 6 months.


Notice that when we calculated the interest

after 6 months, we used 0.5 for time. Because

time is always in YEARS, we sometimes have to

write our time as a fraction or decimal.

Using the equation, solve the following questions. Jenny has $1,560 in a savings account that pays 2.5%

simple interest. How much interest will she earn in 3 years? What information are you given that should be plugged into

your formula? Interest? Principal? Interest Rate? Time? $117

Phoebe borrowed $2,600 from a bank to help pay for her college tuition. The interest rate is 8% per year. How much simple interest will she pay if it takes her 5 years to repay the loan? What information are you given that should be plugged into

your formula? Interest? Principal? Interest Rate? Time? $1,040


Find Interest Paid on a Loan Mrs. Hanover borrows $1,4oo at a rate of 5.5% per year. How

much simple interest wlil she pay if it takes 8 months to repay the loan?

$51.33 Find Total Paid on a Credit Card

Derrick’s dad bought new tires for $900 using a credit card. His card has an interest rate of 19%. If he has no other charges on his card and does not make a payment, how much money will he owe after one month? Use to calculate the interest.

$14.25 Then, add the interest to the original amount.

$14.25 + $900 = $914.25 TOTAL


Try a few more! An office manager charged $425 worth of office supplies

on a credit card with an interest rate of 9.9%. How much money will he owe at the end of one month if he makes no other charges on the card and does not make a payment? Find the interest. Add the interest to the principal.

$428.51

Dr. Underwood paid for a plane ticket that cost $365 using a credit card. His card has an interest rate of 13.5%. If he has no other charges on his card and does not pay off his balance by the end of the month, how much money will he owe after one month?

$369.11



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