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**Fractions, Decimals, and Percents**

Parts of the whole

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

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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 Percent – is a ratio of a number to 100.

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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!

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**Let’s begin with a simple concept. Consider the blue square below**

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.

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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.

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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.

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

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**The Relationship Between Fractions Decimals and Percents**

All represent part of a whole

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**How do we get from one form to another?**

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

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;

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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

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Decimal to percent Move the decimal point two (2) places to the right (this multiplies the number by 100) .50 = 50% (0.50 x 100 = 50.0) Attach the % sign

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Percent to decimal 50% = .50 50 ÷ 100 = .50 Move the decimal point two (2) places to the left (this divides the number by 100)

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**Place the number over 100 and reduce.**

Percent to fraction Place the number over 100 and reduce.

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**Multiply the number by 100, reduce and attach a percent (%) sign.**

Fraction to percent Multiply the number by 100, reduce and attach a percent (%) sign.

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Decimal to fraction 1 decimal place = tenths, 2 decimal places = hundredths, 3 decimal places = thousandths 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.)

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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.

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Percents to Remember

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**Problem Solving with Percents**

When solving a problem with a percent greater than 100%, the part will be greater than the whole.

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**1) what is 60% of 30? 2) what number is 25% of 160?**

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?

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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

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Proportional method Part Part Whole Whole Equation Method What percent of 24 is n · 24 = n = n = %

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**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?

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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 $ 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?

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Percent of Change Markup or Discount

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**amount of change ÷ original amount**

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

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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.

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**Using percent of change**

The regular price of a portable CD player at Edwin’s Electronics is $ This week the CD player is on sale at 25% off. Find the amount of discount, then find the sale price. 25% · = d Think: 25% of $31.99 is what number? 0.25 · = d Write the percent as a decimal. = d Multiply. $8.00 = d Round to the nearest cent. The discount is $ To find the sale price subtract the discount from the retail price. $ $8.00 = $23.99 The sale price is $23.99

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**0.25 · 45 = g Write percent as a decimal 11.25 = g Multiply **

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 = 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%

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**Find percent of increase or decrease**

Remember, Percent of change is the difference of the two numbers divided by the original amount 1) from 40 to 55 2) from 85 to 30 3) from 75 to 150 4) from 9 to 5 5) from $575 to $405 6) 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?

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Simple Interest I = P · r · t

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

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

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**Most loans and savings accounts today use compound interest**

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 the money earns interest

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**Using the simple Interest Formula**

I = ?, P = $225, r = 3%, t = 2 years I = P · r · t Substitute. Use 0.03 for 3% I = 225 · 0.03 · 2 Multiply I = 13.50 The simple interest is $13.50 I = $300, P = $1,000, r = ?, t = 5 years I = P · r · t Substitute 300 = 1,000 · r · 5 Multiply 300 = 5000r 300/5000 = 5000r/5000 Divide by 5,000 to isolate variable r = 0.06 Interest rate is 6%

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Solve the following Find the interest and total amount 1) $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 years.

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**Amount Principal rate number of years**

number of compounding periods Compound Interest Formula A = Amount (new balance) P = Principal (original amount r = rate of annual interest n = number of years, and k = number of compounding periods per year (quarterly)

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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 years Simple interest $1, Simple $2,400.00 Compound interest $1, Compound $10,765.16

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**Formula explained A = P(1 + r/k)n · k. Write down formula A =1000(1 +**

Formula explained A = P(1 + r/k)n · k Write down formula A =1000( /4)30 · 4 Substitute values A =1000( )30 · Evaluate parenthesis A = 1000(1.02) Evaluate parenthesis and exponents A =1000( …) Evaluate the power A = = $10, 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.

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

Percent of Money

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Percents Percents 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.

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**Using Percents to Find Commissions**

First find her commission. 4% · $65,000 = c 0.04 · 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 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? Monthly salary Total commission earned

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**Oct 1, 2009, NC Sales Tax increased to 7. 75%**

Oct 1, 2009, NC Sales Tax increased to 7.75%. Use percents to find sales tax. 2 $16.99 each $33.98 1 $36.29 $33.98 $36.98 $67.96 Sales tax · total purchase = total tax .0775 · = $5.2669 Total purchase + sales tax = total due $ $5.27 = $73.23 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?

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**Use percent to find withholding tax**

Think!!! $ is what % of $1,500 or What % of $1,500 is $114.75 Anna earns $1,500 monthly. Of that, $ is withheld for Social Security and Medicare. What percent of Anna’s earnings are withheld for Social Security and Medicare? Write an equation. is what % of $1,500 = x · 1,500 = 1500x Divide both sides by 1500 = x Change to percent 7.65% = x Anna pays 7.65% withholding tax on her salary Remember, when changing a decimal to percent, move the decimal two places to the right and add the percent sign %

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

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?

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Being able to use fractions, decimals and percents interchangeably

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