Fractions, Decimals, and Percents Parts of the whole

Percent comes from the Latin per centum, or “per hundred” 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 think of this blue square as One Whole Square Let’s think of this blue square as One Whole Square. Now 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?

One Whole Square has a portion shaded, this time it’s blue One Whole Square has a portion shaded, this time it’s blue. What percent is shaded blue?

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

The Relationship Between Fractions Decimals and Percents All represent part of a whole

Writing percentages as decimals We can write percentages as decimals by dividing by 100. For example, 130% = 46% = 7% = 0.2 100 Explain that to convert a percentage to a decimal we simply divide the percentage by 100. 0.2% =

.50 = 50% (0.50 x 100 = 50.0) Attach the % sign 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

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

Place the number over 100 and reduce. Percent to fraction Place the number over 100 and reduce.

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.

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

Using a calculator We can also convert fractions to decimals and percentages using a calculator. For example, 5 16 = 4 7 = 5 8 = 1 Explain to pupils that when the denominator of a fraction is not a factor of 100 (i.e. 2, 4, 5, 10, 20, 25 or 50) it is more difficult to find an equivalent fraction out of 100. Ask pupils if they can think of a way to convert 5/16, for example, to a decimal. 5/16 means 5 ÷ 16 so we can simply enter this into the calculator. Ask pupils to work this out on their calculators. To write the decimal as a percentage we multiply by 100% (multiply by 100 and write % on the end). Stress that this does not change the value of the decimal because 100% is equal to one and multiplying by one has no effect. Ask pupils to use their calculators to convert 4/7 to a decimal. We have 0.571428 recurring. Multiplying by 100 gives us 57.14 % (to two decimal places). To convert a mixed number to a percentage we can write it as an improper fraction first. Alternatively, we can convert the fractional part to a percentage and then add it to the whole number. In this example we would have 5/8 = 62.5% plus 100% to give 162.5%. Give pupils further examples to convert using their calculators. 17/40 = 0.425 9/11 = 0.36 recurring (we can write this with a dot above the 3 and the 6 or write it as 0.36 to 2 decimal places). Encourage pupils to to check the answer given by the calculator by estimating the given fraction as a fraction of 100. For example, if the fraction is less than 1/2 we would expect the corresponding percentage to be less than 50%.

Table of equivalences Ask pupils to work out each fraction, decimal and percentage conversion mentally. Fill in the table by clicking on each empty cell.

Table of equivalences Ask pupils to work out each fraction, decimal and percentage conversion mentally. Fill in the table by clicking on each empty cell.

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

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 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

Proportional method Part Part Whole Whole Equation Method What percent of 24 is 16.5. n · 24 = 16.5 n = 0.6875 n = 68.75%

Solve the following percent problems 27 is what percent of 30? 45 is 20% of what number? What percent of 80 is 10? 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? 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?

Using percentages to compare proportions Michael sat tests in English, Maths and Science. His results were: Science Maths English 74 80 17 20 66 70 Which test did he do best in? Start by asking pupils to discuss which subjects Matthew did best in giving their reasons. Some pupils may argue that he did best in Maths because he got the least number wrong. Others may reason that English is his best mark because it is his highest mark. Stress to pupils that to tell how well he did in each subject we have to compare the number of marks he got in that subject compared to the total number of marks available in that subject altogether. If we write the marks as percentages then we can easily compare them. An alternative would be to write all the marks as decimals or to write all the marks as fractions over a common denominator. (It is most common to use percentages).

Using percentages to compare proportions English Maths Science 17 20 74 80 66 70 This time let’s use a calculator. Remind pupils that 74 over 80 is equivalent to 74 ÷ 80. Ask a volunteer to work out 74 ÷ 80 × 100 on a calculator. Click to reveal the answer on the board. Repeat the process for Maths and Science.

Using percentages to compare proportions Nutrition Information Chocolate Cookies Typical Value Per 10g biscuit Energy Protein Carbohydrate Fat Fibre Sodium 233kj 0.6g 6.7g 2.2g 0.2g <0.05g Nutrition Information Cheesy Crisps Typical Value Per 23 g bag Energy Protein Carbohydrate Fat Fibre Sodium 504kj 1.6g 13g 7g 0.3g 0.2g Discuss the proportions of each nutrient. Stress that although one biscuit has less carbohydrate than a bag of crisps, the biscuit may or may not contain a smaller proportion of carbohydrate. To compare the two we need to either find the amount of carbohydrate for the same weight of product or we need to find the amount of carbohydrate as a percentage of the weight. Which product contains the smallest percentage of carbohydrate?

Percent of Change Markup or Discount

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

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.

A water tank holds 45 gallons of water 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? 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 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?

Simple Interest I = P · r · t

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. 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 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

Using the simple Interest Formula I = ?, P = $225, r = 3%, t = 2 years I = $300, P = $1,000, r = ?, t = 5 years

You Try It! 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 3.5 years.

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)

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 $1,080.00 Simple $2,400.00 Compound interest $1,082.43 Compound $10,765.16

Formula explained A = P(1 + r/k)n · k Formula explained A = P(1 + r/k)n · k 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 Commissions, Sales Tax, and other taxes. Percent of Money

Percents Percents are used everyday to compute sales tax, withholding tax, commissions, and many other types of monies.

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?

Percentage increase To increase an amount by a 20%, for example, we can find 20% of the amount and then add it on to the original amount. We can represent the original amount as 100% like this: 100% 20% Remind pupils that 100% of the original amount is like one times the original amount: it remains unchanged. 100% means ‘all of it’ When we add on 20% we are adding on 20% to the original amount, 100%, to make 120% of the original amount. Why might it be better to find 120% than to find 20% and add it on? Establish that it’s quicker because we can do the calculation in one step. What is 120% as a decimal? Conclude that to increase an amount by 20% we multiply it by 1.2. What would we multiply by to increase an amount by 40%? (140% or 1.4) What would we multiply by to increase an amount by 63%? (163% or 1.63) What would we multiply by to increase an amount by 17½% for VAT? (117½% or 1.175) When we add on 20%, we have 120% of the original amount. Finding 120% of the original amount is equivalent to finding 20% and adding it on.

Percentage increase What happens if we increase an amount by 100%? We take the original amount and we add on 100%. 100% 100% To increase an amount by 100% we add on the same amount again. This is like finding 200% of the original amount or doubling the original amount. We now have 200% of the original amount. This is equivalent to 2 times the original amount.

Percentage increase What happens if we increase an amount by 200%? We take the original amount and we add on 200%. 100% 200% To increase an amount by 200% we add on 2 times the original amount. This gives us 300% of the original amount or 3 times the original amount. What would be the effect of increasing an amount by 500%? Establish that we would have 6 times the original amount. What would be the effect of increasing an amount by 900%? Establish that we would have 10 times the original amount. Give more examples as necessary. We now have 300% of the original amount. This is equivalent to 3 times the original amount.

Percentage decrease To decrease an amount by 30%, for example, we can find 30% of the amount and then subtract it from the original amount. We can represent the original amount as 100% like this: 100% 70% 30% Remind pupils again that 100% of the original amount is like one times the original amount, it remains unchanged. 100% is ‘all of it’. When we subtract 30% we are subtracting 30% from the original amount, 100%, to make 70% of the original amount. Why might it be better to find 70% than to find 30% and take it away? Again, establish that it’s quicker. We can do the calculation in one step. What is 70% as a decimal? Conclude that to decrease an amount by 30% we multiply it by 0.7. What is a quick way to decrease an amount by 15%? Establish that we can find 85% (100% – 15%) or multiply by 0.85. What is a quick way to decrease an amount by 37%? Establish that we can find 63% (100% – 37%) or multiply by 0.63” Give more verbal examples as necessary. When we subtract 30% we have 70% of the original amount. Finding 70% of the original amount is equivalent to finding 30% and subtracting it.

Percentage decrease For example, Decrease £56 by 34% Ask pupils how we can decrease £75 by 20% in one step. Establish that we can find 80% of £75. Ask pupils how we can write 80% as a decimal and reveal the required calculation on the board. Discuss mental methods to multiply 0.8 and 75. For example, 8 × 75 is 600, so 0.8 × 75 is 60. Talk through the second example, using a calculator if necessary. Impress upon pupils that it would take much longer to work out 34% of 56 and then subtract that amount from £56. However, we would still get the same answer.

Shares problem Alex bought $200 worth of shares. In the first week the shares went up 12%. In the second week, however, the shares went down 12%. “Oh well,” said Alex, “at least I’m back to the amount I started with.” Discuss the problem on the board. At first glance most pupils will assume that if an amount increases by 12% and then decreases by 12% it will return to its original value. Demonstrate that this is untrue by asking pupils to calculate the relevant increase and decrease. If the shares went up 12% the new value of Jason’s shares would be £224. We then need to decrease £224 by 12%. Since 12% of £224 is more than 12% of £200, the new value of the shares in now less than the original. The shares are now worth 88% of £224 or £197.12. Tell pupils to be careful when calculating a percentage increase or decrease followed by a second percentage increase or decrease. The two percentages are percentages of different amounts and so you cannot simply add or subtract them.

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. 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?

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?

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 removed. Net pay is the amount of your actual check you receive after the taxes are removed. When you get a job, which would you prefer, a job that pays commission or one that pays a straight salary?