College Prep Mathematics Mrs. Barnett

College Prep Mathematics Mrs. Barnett

3-1 Percent and Number Equivalents Goals: Write any number as a percent equivalent Write any percent as a numerical equivalent

Writing numbers as percents Remember that 100% = 1 Write a number as a percent by multiplying by 100% (which is the same as multiplying by 1 Examples: 0.27 = 0.27(100%) = 27% 0.375 = 0.375(100%) = 37.5% 1.85 = 1.85(100%) = 185% 2 = 2(100%) = 200%

More examples Fractions to percents examples: 33 100 3 5 2 7 20 1 3 100% = 33% 100% = 3 100 5 1 = 3 20 1 1 = 60% 100% = 47 20 100 1 = 47 5 1 1 = 235% 100% = 1 100 3 1 = 100 3 = 33 1 3 %

Writing a Percent as a Number Divide the number by 100% or multiply the number by 1 100 % The quotient does not have a % symbol because the % cancels out Examples: 38% 100% = 38% 1 = 38 = 19 100% 100 50 5 % 100% = 5 % 1 = 5 1 = 1 1 = 1 6 6 100% 6 100 6 20 120 275% 100% = 2.75 = 2 3 4 66 2 3 % 100% = 66 2 3 % 1 100% = 200 3 1 100 = 2 3

Word Problems

3-2 Percentage problems Identify the portion, base, and rate in percent problems Solve percent problems using the percentage formula Solve percentage problems using the percentage proportion Solve application problems involving percent

Identify the portion, base, and rate in percent problems The percentage formula is Portion = Rate x Base, or P = RB Base (B) represents the original number or one entire quantity Portion (P) represents the part of the base Rate (R) is the percent that relates the portion and the base Example: 20 is 50% of 40 40 is the base, 20 is the portion, and 50% shows how the two are related Portion is also called percentage - however, the rate can also be called the percentage

Solve percent problems using the percentage Three forms for the percentage formula The shaded region represents the part of the percentage you are trying to find Identify the three quantities in the problem Choose the form of the percentage formula that best fits the problem Substitute and solve the formula for the missing value formula

Solve percentage problems using the percentage proportion The percentage formula, R = P B R 100 R can be written as a proportion = P 100 B represents the fractional form of the rate or percent, P = portion, and B = base or total What is 30% of 270? portion is the missing value 30 = P 100 270 the rate is 30%, the total is 270, the This method is useful since it works for all formats of percentage problems and involves fractions which can be worked out mentally.

Recall that a proportion is an equation of two ratios, and the crossproducts property applies Solving a proportion Cross-product property: The cross-products of a proportion are equal

3-3 Increase and Decrease Find the amount of increase or decrease in percentage problems Find the amount directly in increase or decrease applications Find the rate or base in increase or decrease applications

Find the amount of increase or decrease in percentage problems Use subtraction to find the amount of increase or decrease when the original amount and new amount are known. E.g. The shirt at Kohl s was marked down from $45.99 to $18.99 what is the amount of decrease? E.g. A worn brake lining is measured to be 3 in. thick. If the 32 original thickness was ¼ in., what is the change in thickness of the brake lining?

Find the amount of increase or decrease in percentage problems (cont) Finding the amount when given a percent of change The amount of change (portion) is a percentage of the original amount (base) Identify the original amount and the percent of change Use P=RB to solve for the amount of change

Find the amount directly in increase or decrease applications If there is a percent of increase given, then the rate of the new amount = 100% + percent of increase If there is a percent of decrease given, then the rate of the new amount = 100% - percent of decrease The new amount is the portion of the original amount (base) Use P=RB to solve for the new amount

Examples Joey s boss tells him he is getting a 2% raise on his hourly wage. If Joey was making $12.60 an hour, what will be Joey s new hourly wage? Since he is adding 2% to his original amount, the new rate is 100% + 2% = 102% So New Wage = 102% of $12.60 1 102 New Wage = 102% 12.60 = 100% 1 1.02 12.60 = 12.852 = $12.85 1 100 12.60 =

Complement of a % Complement typically means the amount that completes a set Since the total amount is represented by 100%, the complement of a % is the amount that adds to it to reach 100% In other words, complement = 100% - given percent E.g. Find the complement of: 55% 35% 9% 25%

Use Complement to find a new amount The sale flyer says save 70%! The original price of the TV was $299. What is the sale price? Since the savings is 70%, the new price will be 30% of the original price. sale price = 30% 299 = 30 1 100 299 =.3 299 = $89.70 Use complement to estimate a new amount Round the original price and the percent to make numbers easier to work with mentally Find the complement of the rounded percent Relate the complement to 10% by dividing it by 10% Find 10% of the rounded original price (10% is the same as dividing by 10 move the decimal left 1 place) Multiply results of the last two steps Eg. Refer to above example: the original price was $299 so round to $300. use the complement, 30%. Divide 30% by 10% to get 3. 10% of 300 is 30. multiply 30 by 3 to find $90. The new price of the TV is about $90

Find the rate or base in increase or decrease applications Identify the amount of change (increase or decrease) Determine what you need to find: To find rate of increase or decrease, use R = P B amount as B) (always use the original To find the original amount or B, use B = P R so B = amount of change rate of change If you don t know the amount of change, you can use this alternative process: Use the complement of the rate of change as R and use the reduced amount in place of P B = reduced amount complement of rate of change

Examples An engine that has a 4% loss of power has an output of 336 hp. What is the input(base) horsepower of the engine? B = 100% 1 reduced amount complement of rate of change = 350 hp = 336 hp 96% = 336 1 96% = 336 1 96% A chicken farmer bought 2575 baby chicks. Of this number, 2060 lived to maturity. What percent loss was experienced by the farmer? Amount of decrease = 2575 2060 = 515 R = P B = 515 2575 = 0.2 100% = 20% loss