Fractions, Decimals, Fractions, Decimals, and Percentsand Percents

Fractions, Decimals, Fractions, Decimals, and Percentsand Percents

Parts of the wholeParts of the whole

Let’s watch this clip to see how some people can be confused on how fractions and percents can be used as examples.

Percent comes from the Latin per centum, or “per hundred”Consequently, a number such as 32% can be written as “32 per hundred” or the fraction

32/100. This fraction is equivalent to the decimal

0.32.Percent – is a ratio of a

number to 100.

The word “percent” meaning “per hundred” is used to show parts of a whole, the same as a fraction is used to represent part of a whole.

If you had a pizza that was cut into 100 pieces, 25% of the pizza would be 25 pieces!

Let’s begin with a simple concept. Consider the blue square below. Let’s think of this blue

square as One Whole Square. How let’s divide it into 100 pieces—every piece just the same size as every other piece. We can easily see that every one of the 100 pieces is shaded blue. So we say 100% of the square is shaded

blue. So 100% and 1 Whole are the same thing.

Since percent means “per hundred” it tells us how many for each hundred, 25% means 25 for each hundred, or 25 out of each hundred.

Here is our One Whole Square with a portion shaded green. What percent is shaded green.

In percent, every whole is divide into 100 pieces. Now count the pieces shaded green. There are 50 pieces out of 100 shaded green,

so 50% is green.

Once again our One Whole Square has a portion shaded, this time it’s blue. What percent is shaded blue. Remember, in

percents, every quantity is divided into 100 pieces. Now count the pieces shaded blue.

There are 86 pieces out of 100 shaded blue, so 86% of the pieces are shaded

blue.

Can you calculate what percent of our “One Whole” that is shaded red?

The Relationship The Relationship Between Fractions Between Fractions

Decimals and Decimals and PercentsPercents

The Relationship The Relationship Between Fractions Between Fractions

Decimals and Decimals and PercentsPercents

All represent part of a wholeAll represent part of a whole

How do we get from one form to another?

A percent is based on the number in terms of 100 or “per hundred”.

12%; 4%; 0.05%...

A fraction is based on the number into

which the whole is divided (the

denominator).The numerator (the top)

is the PART, the denominator (the

bottom) is the whole.½; ¼; ⅝…

A decimal is based on the number in terms of tenth, hundredths, thousandths, etc…0.5; 0.05; 0.005

Fraction to Decimal Divide the denominator (the bottom of the fraction) into the

numerator (the top of the fraction). Place a

decimal after the number inside the division “box” and

attach as many zeros as necessary to

complete the division. If the quotient does not

come out evenly, follow the rules for

“rounding off” numbers.

numerator

denominator

Decimal to percent

.50 = 50%

(0.50 x 100 = 50.0) Attach the % sign

Move the decimal point two (2) places

to the right (this multiplies the

number by 100)

Percent to decimal

50% = .5050 ÷ 100 = .50

Move the decimal point two (2) places to the left (this divides the

number by 100)

Percent to fraction

Place the number over 100 and reduce.

Fraction to percent

Multiply the number by 100, reduce and

attach a percent (%) sign.

Decimal to fraction

You will be using place value to do this! Count the decimal places of the decimal starting from the

decimal point. If there is one decimal point, place the number over 10 and reduce. If there are two decimal places, place the number over

100, and reduce. If there are three decimal places, place the number over 1000, and

reduce…Etc.(This is really just using your knowledge of place value to

name the denominator.)

1 decimal place = tenths,2 decimal places =

hundredths,3 decimal places =

thousandths

Remember that fractions,

decimals, and percents are

discussing parts of a whole, not how large the whole is.

Fractions, decimals, and percents are part of our world.

They show up constantly when you least expect them. Don’t let them catch you off guard. Learn

to master these numbers.

Percents to Remember

Problem Solving with Problem Solving with PercentsPercents

Problem Solving with Problem Solving with PercentsPercents

When solving a problem with a percent When solving a problem with a percent greater than 100%, the greater than 100%, the partpart will be greater will be greater

than the than the wholewhole..

There are three types of percent problems: 1) finding a percent of a number,

2) finding a number when a percent of it is known, and 3) finding the percent when the

part and whole are known

1) what is 60% of 30?

2) what number is 25% of 160?

3) 45 is what percent of 90?

Solving Equations Containing Percents

Most percent problems are word problems and deal

with data. Percents are used

to describe relationships or

compare a part to a whole.

Sloths may seen lazy, but their extremely slow movement helps make them almost invisible to predators. Sloths sleep an average of 16.5 hours per day. What percent of the day do they sleep?

Solution

Equation Method

What percent of 24 is 16.5.

n · 24 = 16.5n = 0.6875n = 68.75%

Proportional method

Part Part

Whole Whole

24

5.16

100

n

Solve the following percent problems

1) 27 is what percent of 30?2) 45 is 20% of what number?3) What percent of 80 is 10?4) 12 is what percent of 19?5) 18 is 15% of what number?6) 27 is what percent of 30?7) 20% of 40 is what number?8) 4 is what percent of 5?

9) The warehouse of the Alpha Distribution Company measures 450 feet by 300 feet. If 65% of the floor space is covered, how

many square feet are NOT covered?

10) A computer that normally costs $562.00 is on sale for 30% off. If the sales tax is 7%, what will be the total cost of the computer? Round to the nearest dollar.

11) Teddy saved $63.00 when he bought a CD player on sale at his local electronics store. If the sale price is 35% off the regular price, what was the regular price of the CD player?

Percent of ChangePercent of ChangePercent of ChangePercent of Change

Markup or DiscountMarkup or Discount

One place percents are used frequently is in the

retail business. Sales are advertised on

television, in newspapers, in store displays, etc. Stores

purchase merchandise at wholesale prices, then markup the price to get the retail price.

To sell merchandise quickly, stores may

decide to have a sale and discount retail

prices.

Percent of change = amount of change ÷ original

amount

When you go to the store to purchase

items, the price marked on the merchandise is the retail price (price you pay). The retail

price is the wholesale price from the

manufacturer plus the amount of markup

(increase). Markup is how the store makes a profit on merchandise.

Using percent of change

• The regular price of a portable CD player at Edwin’s Electronics is $31.99. This week the CD player is on sale at 25% off. Find the amount of discount, then find the sale price.

25% · 31.99 = d Think: 25% of $31.99 is what number?

0.25 · 31.99 = d Write the percent as a decimal.

7.9975 = d Multiply.

$8.00 = d Round to the nearest cent.

The discount is $8.00. To find the sale price subtract the discount from the retail price.

$31.99 - $8.00 = $23.99The sale price is $23.99

When solving percent problems there are two ways to solve these problems. Take a look at the problem below and see the two

solutions.

A water tank holds 45 gallons of water. A new water tank can hold 25% (+) more water. What is the capacity of the new

water tank?

25% · 45 = g 25% of 45 gallons

0.25 · 45 = g Write percent as a decimal

11.25 = g Multiply

Add increase to original amount

45 + 11.25 = 56.25 gallons

125% · 45 = g 125% of 45 gallons

1.25 · 45 = g Write percent as a decimal

56.25 = gallons

The original tank holds 100% and the new tank holds 25% more, so together they hold;

100% + 25% = 125%

Find percent of increase or decrease

1) from 40 to 552) from 85 to 303) from 75 to 1504) from 9 to 55) from $575 to $4056) An automobile dealer agrees to reduce the

sticker price of a car priced at $10,288 by 5% for a customer. What is the price of the car for the customer?

Remember,Percent of change is the

difference of the two numbers divided by the

original amount

Simple InterestSimple InterestSimple InterestSimple Interest

II = = PP · · rr · · tt

When you keep money in a savings account, your money earns interest.

Interest – the amount that is collected or paid for the use of money.

For example, the bank pays you interest to use your money to conduct its business.

Likewise, when you borrow money from the bank, the bank charges interest on its loans

to you.One type of interest, called simple interest,

is money paid only on the principal (the amount saved or borrowed). To solve

problems involving simple interest, you use the simple interest formula. I = Prt

Most loans and savings accounts today use compound interest. This means that

interest is paid not only on the principal but also on all the interest earned up to

that time.

Interest rate of interest per year(as a decimal)

I = P · r · t

Principal time in years that themoney earns interest

Using the simple Interest Formula

I = ?, P = $225, r = 3%, t = 2 yearsI = P · r · t Substitute. Use 0.03 for 3%

I = 225 · 0.03 · 2 Multiply

I = 13.50The simple interest is $13.50I = $300, P = $1,000, r = ?, t = 5 yearsI = P · r · t Substitute

300 = 1,000 · r · 5 Multiply

300 = 5000r300/5000 = 5000r/5000 Divide by 5,000 to isolate

variable r = 0.06 Interest rate is 6%

Solve the following

Find the interest and total amount1) $225 at 5% for 3 years.2) $775 at 8% for 1 year.3) $700 at 6.25% for 2 years.4) $550 at 9% for 3 months.5) $4250 at 7% for 1.5 years.6) A bank offers an annual simple interest rate

of 7% on home improvement loans. How much would Nick owe if he borrows $18,500 over a period of 3.5 years.

Compound Interest Formula

A = Amount (new balance)P = Principal (original amountr = rate of annual interestn = number of years, andk = number of compounding periods per year (quarterly)

Amount Principal rate number of years number of compounding

periods

kn

k

rPA

1

Since simple interest is rarely used in real-world situations today, it is important to understand how compound interest is

used.

The contrast between simple interest and compound interest does not become very evident until the length of time increase. Look at the comparison below using simple versus compound interest.

$1000 at 8% for 1 year $1000 at 8% for 30 yearsSimple interest $1,080.00Simple $2,400.00Compound interest $1,082.43 Compound

$10,765.16

Formula explained

A = P(1 + r/k)n · kWrite down formula

A =1000(1 + .08/4)30 · 4 Substitute values

A =1000(1 + .02)30 · 4 Evaluate parenthesis

A = 1000(1.02)120 Evaluate parenthesis and exponents

A =1000(10.76516303…) Evaluate the power

A = 10765.16303 = $10,765.16

Remember, compound interest is computed on the principal plus all interest earned in previous

periods.Compound interest is used for loans,

investments, bank accounts, and in almost all other real-world applications.

Using Percents to Find Using Percents to Find Commissions, Sales Commissions, Sales Tax, and other taxes.Tax, and other taxes.

Using Percents to Find Using Percents to Find Commissions, Sales Commissions, Sales Tax, and other taxes.Tax, and other taxes.

Percent of MoneyPercent of Money

PercentsPercents are used everyday to

compute sales tax, withholding tax, commissions, and many

other types of monies.

Think about this, you go to Wal-Mart to buy a new CD or video game. You make your selections and go to the check out counter. This happens when you make your purchase. You pay for your CD,

along with your purchase you pay sales tax on what you bought, your Wal-Mart associate that takes your money is paid

money to work there, they also may make a commission on what they sell. From

her salary, withholding taxes are taken out to pay to the state and federal

government.

Using Percents to Find Commissions

A real-estate agent is paid a monthly salary of $900 plus commission. Last month she sold one condo for $65,000, earning a 4% commission on the sale How much was her commission? What was her total pay last month?

First find her commission.4% · $65,000 = c0.04 · 65,000 = c Change percent to

decimal.

$2,600 = c

She earned $2,600 on the sale.Now find her total pay.$2,600 + $900 = $3,500 Total pay

Total commission earned

Monthly salary

Oct 1, 2009, NC Sales Tax increased

to 7.75%. Use percents to find

sales tax.

If the sales tax rate is 7.75%, how much tax would Daniel pay if he bought two CD’s at $16.99 each and one DV D for $36.29? What would his total purchase cost him?

2 CD’s @ $16.99 each $33.98

1 DVD @ $36.29

$33.98

$36.98

$67.96Sales tax · total purchase = total tax

.0775 · 67.96 = $5.2669Total purchase + sales tax = total

due

$67.96 + $5.27 = $73.23

Use percent to find withholding

tax

Anna earns $1,500 monthly. Of that, $114.75 is withheld for Social Security and Medicare. What percent of Anna’s earnings are withheld for Social Security and Medicare?

Write an equation.114.75 is what % of $1,500114.75 = x · 1,500114.75 = 1500x 1500 1500 Divide both sides by 1500

0.0765 = x Change to percent

7.65% = xAnna pays 7.65%

withholding tax on her salary

Think!!!$114.75 is what % of $1,500

orWhat % of $1,500 is

$114.75

Remember, when changing a decimal to percent, move the

decimal two places to the right and add the percent sign %

Note:

Commissions and sales tax are based on the price of an item.

Withholding taxes are also called income tax. This tax is taken before you get your

paycheck. This is where the terms gross pay and net pay comes from.

Gross pay is the amount of salary you earn before taxes are remove.

Net pay is the amount of your actual check you receive after the taxes are remove.

When you get a job, which would you prefer, a job

that pays commission or one that pays a straight

salary?