DEV 085 Unit 3 Notes Decimals Percents Proportions

Decimal Place Value: Decimal points are read as the word “and” Place values to the right of the decimal point represent part of a whole Read the numbers in groups of three then read the place value name Place values to the right of the decimal point end with “ths” Place values to the right of the decimal point “mirror” place values to the left of the decimal point

___ , ___ ___ ___ ___ ___ ___ Decimal Place Value: ___ , ___ ___ ___ ___ ___ ___ Thousands Hundreds Tens Ones Tenths Hundredths Thousandths

Rounding Decimals: Steps for Rounding: Step 1: Identify the place value you are rounding to and underline it Step 2: Circle the number to the right Step 3: Determine whether to “round up” or to “round down” If the circled number is 0-4, the underlined number stays the same and all the digits to the right of the circled number fall off If the circled number is 5-9, the underlined number goes up one and all the digits to the right of the circled number fall off

Rounding Practice Problems: Nearest Tenth Nearest Hundredth 4 . 5 7 6 1 3 . 8 0 4 1 7 9.8 5 6 4.6 4.58 13.8 13.80 179.86 179.9

Comparing Decimals: Steps for Comparing Decimals Values Step 1: List the numbers vertically “Stack” the decimal points Add zeros as place holders as needed Step 2: Compare the whole number part then compare the decimal parts moving to the right (as you would if you were alphabetizing words) Step 3: Put in the correct order (from least to greatest or greatest to least)

Comparing Decimals Practice: Practice Problems: Arrange each group of numbers in order from least to greatest. 0.342 0.304 0.324 0.340 2.37 2.7 2.3 2.73 0.304 0.324 0.340 0.342 2.3 2.37 2.7 2.73

Comparing Decimals Practice: Practice Problems: Arrange each group of numbers in order from least to greatest. 5.23 5.023 5.203 5.032 1.010 1.101 1.011 1.110 5.023 5.032 5.203 5.23 1.010 1.011 1.101 1.110

Basic Operations with Decimals: Addition and Subtraction Step 1: Write the numbers vertically “Stack” the decimal points Add zeros as place holders Step 2: Move the decimal point straight down into your answer Step 3: Add or subtract

Adding and Subtracting Decimals Practice: Practice Problems: Find the sum or difference for each. 2.3 + 3.71 + 27 = 3.14 + 2.073 + 8.9 = 4.023 + 24.311 = 33.01 14.113 28.334

Adding and Subtracting Decimals Practice: Practice Problems: Find the sum or difference for each. 31.73 – 12.07 = 9 – 8.185 = 23.5 – 17.097 = 19.66 0.815 8.593

Adding and Subtracting Decimals Practice: Practice Problems: Find the sum or difference for each. 2.45 – 4.66 = 3 + 5.76 + 0.11 = 25 – 0.14 + 2.36 = -2.21 8.87 27.22

Multiplying Decimals: Steps for Multiplication Step 1: Write the problem vertically (just as you would a regular multiplication problem) Step 2: Ignore the decimal point(s) and multiply as if you were multiplying whole numbers Step 3: Determine where the decimal point goes in the product However many digits are to the right of the decimal point(s) in the problem… that’s how many digits are to be to the right of the decimal point in the product.

Multiplying Decimals Practice: Practice Problems: Find the product of each. 2 x 3.14 = 8.097 x .05 = 1.042 • 2.3 = 6.28 0.40485 2.3966

Multiplying Decimals Practice: Practice Problems: Find the product of each. 4.7 x 1000 = 3 x 0.567 = 0.27 • 15 = 4,700 1.701 4.05

Multiplying Decimals Practice: Practice Problems: Find the product of each. (2.5)(1.02) = (1.003)(0.42) = 5.41 x 200 = 2.55 0.42126 1,082

Dividing with Decimals: There are 2 types of division problems involving decimal points: No decimal in the divisor Decimal in the divisor

Division with Decimals: NO decimal point in the divisor… Step 1: Write the problem in the traditional long division format Step 2: Move the decimal point in the dividend straight up into the quotient Step 3: Divide as usual Remember to divide out one more place than you are rounding to…

Division with Decimals: Yes…Decimal point in the divisor… Step 1: Write the problem in the traditional long division format Step 2: Move the decimal point in the divisor to the far right of the divisor Step 3: Move the decimal point the SAME number of places in the dividend Step 4: Move the decimal point in the dividend straight up into the quotient Step 5: Divide as usual Remember to divide out one more place than you are rounding to…

Division Practice: Practice Problems: Find the quotient for each. 3.753  3 = 8.7  100 = 245.9 ÷ 1000 = 0.65 ÷ 5 = 1.251 0.087 0.2459 0.13

Division Practice: Practice Problems: Find the quotient for each. 428.6 ÷ 2 = 2.436 ÷ 0.12 = 4.563 ÷ 0.003 = 21.35 ÷ 0.7 = 214.3 20.3 1,521 30.5

Division Practice: Practice Problems: Find the quotient for each. 97.31 ÷ 5 = 0.8542 ÷ 0.2 = 67.337 ÷ 0.02 = 1500.4 ÷ 1000 = 19.462 4.271 3,369.5 1.5004

Problem Solving with Decimals: Follow the correct Order of Operations only remember to apply the rules that go with decimals. P.E.M.D.A.S. P – Parenthesis E – Exponents M- Multiplication D – Division A – Addition S – Subtraction Do whichever one comes first working from left to right

Order of Operations Practice: Practice Problems: Solve each by following the correct order of operations. 2.3 x 4  2 + 4 = 3.5  7 + 2.15 x 0.13 = 2(7 – 2.49) + 0.3 = 14  0.2 + (3.1 – 2.56) x 2 = 8.6 0.7795 9.32 71.08

Order of Operations Practice: Practice Problems: Solve each by following the correct order of operations. 5 + (7.8 – 5.5)2 – 14.3 = (40 ÷ 0.5 • 7) + 5 – 14 = -8 • 0.75 + 15.23 – 4 = -4.01 551 5.23

Percents: Understanding Percent: A percent is one way to represent a part of a whole. “Percent” means per 100 Sometimes a percent can have a decimal. A percent can be more than 100. A percent can be less than 1. When you write a fraction as a percent: Change the fraction to a decimal value then change it to a percent.

Percents, Decimals, and Fractions: To change between formats… Fractions Decimals Percents Divide the numerator by the denominator Move the decimal point to the right 2 places and add a % sign

Percents, Decimals, and Fractions: To go the other direction… Fractions Decimals Percents Put the # (to the right of the decimal) on top. The # on the bottom will represent the appropriate place value. Reduce to lowest terms Move the decimal point to the left 2 places and add drop the % sign

Practice Problems: .8 80% .166 16.6% 52% 325% .32 .06 3 Fractions Decimals Percents 4 5 1 6 .52 3.25 32% 6% .8 80% .166 16.6% 52% 325% .32 .06 13 25 1 4 8 3 50 3

Proportions: A proportion shows that two ratios are equal. 2 = 4 3 6 5 = 17.5 7 24.5 3 = 27 2 18

Ratio Equivalency: To check the equivalency of two ratios, you CROSS MULTIPLY. (If your products are equal, your ratios are equal). 3 = 12 5 20 (3)(20) = (12)(5) 60 = 60 EQUAL

Ratio Equivalency: To check the equivalency of two ratios, you CROSS MULTIPLY. (If your products are equal, your ratios are equal). 2.4 = 13 3 15 (2.4)(15) = (13)(3) 36 = 39 NOT EQUAL

Proportion Practice: Check to see if the proportions are equal or not. 1 2 3 = 9 2 5 1 7 21 14 6 8 Equal Not Equal Equal

Proportion Practice: Check to see if the proportions are equal or not. 3 = 4 2.5 6.5 5¾ 11½ 8 9 5 13 20 Not Equal Equal Not Equal

Solving Proportions: When you know three of the four parts of a proportion, you can CROSS MULTIPLY then DIVIDE to find the missing value.

(divide by the number with the variable) Solving Proportions: Cross Multiply 4 = x 5 20 (4)(20) = (x)(5) 80 = 5x 5 5 16 = x 9 = 3 x 8 (9)(8) = (3)(x) 72 = 3x 72 = 3x 3 3 24 = x Show what you are multiplying in your first line…in your second line show your products Divide (divide by the number with the variable)

Solving Proportions Practice: Solve for the missing value. 3 = X 2 5 6 12 7 24 6 = X 17.5 16

Solving Proportions Practice: Solve for the missing value. 2.5 = X 10 5 4 18 11 33 9 = X 5.5 13.2

Solving Proportions Practice Problems: Practice: Solve each. One person can move 120 barrels in one hour. How many barrels can that person move in 2.5 hours? One person could move 300 barrels in 2.5 hours

Solving Proportions Practice Problems: Practice: Solve each. A baseball player hits 55 times in 165 at bats. At this rate, how many at bats will he need to have to reach 70 hits? The player would need 210 at bats to reach 70 hits

Solving Proportions Practice Problems: Practice: Solve each. In her garden, Maggie plans to plant 8 blue petunias for every 12 red geraniums. If she buys a total of 70 plants, how many plants are petunias? 28 plants are petunias

Solving Proportions Practice Problems: Practice: Solve each. The sun is shining on two buildings (short and tall) creating 30 ft and 45 ft shadows. The tall building is 60 ft tall. What is the height of the shorter building? The shorter building was 40 feet tall

Solving Percent Problems: A proportion setup can be used to solve percent problems. Set the problem up as a proportion and solve for the missing information. When solving percent problems, think of the proportion set-up as: Partial % = “is” 100 % “of”

Solving Percent Problems using a Proportion Setup: Step 1: Put your numbers in the correct places Step 2: Solve the proportion by cross- multiplying then dividing

Solving Percent Problems Practice: 23 is 20% of what? Find 80% of 40 24 is what % of 72? 40 is 50% of what? Find 6½ % of 24 5 is 5.5% of what? 115 32 33.3% 80 1.56 90.90

Solving Percent Problems Practice: Find 8% of 150 108 is 72% of what? 3.75 is what % of 50 12 150 7.5%

Applications Using Percents: TAX Tax = (Purchase Price) x (Percent of Tax) OR % = Amount of Tax 100 Purchase Price TOTAL COST = Purchase Price + Tax

Tax Application Example: You buy a television set for $289. The local tax rate is 7.5%. Find 1) the amount of tax and 2) the total cost of your purchase. (orig amt) (tax) (total cost) $289 x 0.075 1445 +20230 21.675 $289.00 + 21.68 $310.68 (Tax) $21.675 becomes $21.68…must round because it is money

Applications Using Percents: DISCOUNT Discount = (Original Cost) (Percent of Discount) OR % = Amount of Discount 100 Original Cost Original Cost - Amount of Discount DISCOUNTED PRICE

Discount Application Example: You buy a microwave oven for $135. You can save 25% if you shop at today’s sale. Find 1) the amount of discount and 2) the discounted price of your purchase. (orig amt) (discount) (discounted price) $135.00 - 27.00 $108.00 $135 x 0.20 $27.00 (discount)

Applications Using Percents: MARK-UPS Mark-ups = (Original Cost) (Percent of Mark-up) OR % = Amount of Mark-up 100 Original Cost Original Cost + Amount of Mark-up MARK-UP

Mark-Up Application Example: I buy t-shirts for $3.00. I turn around and mark them up 75% and sell them. Find 1) the amount of mark-up and 2) the mark-up price. (orig amt) (mark-up) (mark-up price) $3.00 + 2.25 $5.25 $3.00 x 0.75 1500 + 21000 $2.2500 (mark-up)

Applications Using Percents: COMMISSION Commission = (Total Sales) (Percent of Commission) OR % = Commission 100 Total Sales Salary + Commission TOTAL PAY

Commission Example: $135,000 $135,000 + 6,750 (commission) x 0.05 Tony has a base salary of $22,000 a year. He makes 5% commission on all of his sales. Over the course of a year, he has a total sales amount of $135,000. Find 1) the amount of his commission and 2) his total pay for the year. $135,000 + 6,750 $141,750 $135,000 x 0.05 $6,750 (base salary) (commission) (total pay) (commission)

Applications Using Percents: In order to find Percent of Increase or Percent of Decrease you must first find the Amount of Increase or Amount of Decrease. To find the amount of increase or the amount of decrease, find the difference between the original amount and the second amount.

Applications Using Percents: PERCENT OF INCREASE Percent of Increase = Amount of Increase Original Amount OR % = Amount of Increase 100 Original Amount

Percent of Increase Example: I buy a box of pencils for $4.00 and sell it for $5.00. what is my percent of increase? $5.00 - $4.00 $4.00 Find the difference between the two amounts… divide by the original amount $1.00 $4.00 = .25 = 25% increase Convert to a percent

Applications Using Percents: PERCENT OF DECREASE Percent of Decrease = Amount of Decrease Original Amount OR % = Amount of Decrease 100 Original Amount

Percent of Decrease Example: I buy a box of books for $10.00 and sell it for $8.00. What is my percent of decrease? $10.00 - $8.00 $10.00 Find the difference between the two amounts… divide by the original amount $2.00 $10.00 = .20 = 20% decrease Convert to a percent

Applications Using Percents: SIMPLE INTEREST I = P  R  T I = Interest P = Principal R = Percentage Rate T = Time (in years) Total Amount = Principal + Interest

Simple Interest Application Example: I had to borrow $15,000 to buy a new car. My interest rate was 5%. My loan was for 5 years. Find 1) how much interest will I pay for borrowing $15,000 and 2) the total amount of my loan. I = P  R  T I = ($15,000) (0.05) (5) I = $3,750 $15,000 + 3,750 $18,500 - Principal - Interest - Total amt of loan

Applications Using Percents: MONTHLY PAYMENT OF A LOAN principal + interest Monthly payment = Total # of payments

Monthly Payment of a Loan Example: If my total loan for the purchase of a new car is $18,750 and I’m going to pay it over the course of 5 years, what is my monthly payment? $18,750 60 mo (Loan amount) (Number of payments) = $312.50/mo Monthly payment

Review the things that you need to review. Study the things that you need to spend more time on. Ask questions about things you don’t understand. PRACTICE…PRACTICE…PRACTICE