Arithmetic of Decimals, Positives and Negatives

Transcription

1 18 Decimals DEFINITIONS & BASICS 1) Like things In addition and subtraction we must only deal with like things. Example: If someone asks you sheep + 2 sheep = you would be able to tell them 7 sheep. What if they asked you sheep + 2 penguins = We really can t add them together, because they aren t like things. 2) We do not need like things for multiplication and division. 3) Negative The negative sign means opposite direction. Example:.3 is just.3 in the opposite direction Example : is just in the opposite direction. Example: 7 = 12, because they are both headed in that direction 4) Decimal Deci is a prefix meaning 10. Since every place value is either 10 times larger or smaller than the place next to it, we call each place a decimal place. ) Place Values Every place on the left or right of the decimal holds a certain value Arithmetic of Decimals, Positives and Negatives Addition of Decimals LAWS & PROCESSES 1. Line up decimals 2. Add in columns 3. Carry by 10 s

2 19 EXAMPLE Add Line up decimals 2. Add in columns 3. Carry by 10 s. Carry the 1 and leave the 3. Subtraction of Decimals 1. Biggest on top 2. Line up decimals; subtract in columns. 3. Borrow by 10 s 4. Strongest wins. EXAMPLE Subtract , Biggest on top Line up decimals; subtract in columns 3. Borrow by 10 s. Carry the 1 and leave the Strongest one wins.

3 20 Multiplication of Decimals Multiplication of Decimals 1. Multiply each place value 2. Carry by 10 s 3. Add 4. Right size. 1. Add up zeros or decimals 2. Negatives EXAMPLES 3. Add the pieces together. 29, , , ,871,000 16,001,196 Start: , ,936 Last: , ,871,000 Next: , ,260

4 21 3. Add the pieces together. 4. Right size. Total number of decimal places = 3. Answer is negative. Final example with decimals: Start: Last: Next: Division of Decimals The only thing left is to count the number of decimal places. We have one in the first number and two in the second. Final answer: Division of Decimals 1. Move decimals 2. Add zeros 1. Set up. 2. Divide into first. 3. Multiply. 4. Subtract.. Drop down. 6. Write answer. 1. Remainder 2. Decimal

5 EXAMPLES Step 1. No decimals to set up. Go to Step 2. Step 2.We know that 8 goes into 42 about times. Step 3. Multiply 8 Step 4.subtract. Step. Bring down the 9 to continue on. Repeat steps 2- Step 2: 8 goes into 29 about 3 times. Step 3: Multiply 3 8 Step 4: subtract. 8 doesn t go into (remainder) Which means that = 3 R or in other words = 3 8 Example: Step 2: 22 goes into 8 about 2 times. Step 3: Multiply 2 22 = 44 Step 4: Subtract. Step : Bring down the next column 22 goes into 147 about????times. Let s estimate. 2 goes into 14 about 7 times try that. Multiply 22 7 = 14 Oops, a little too big

6 An example resulting in a decimal: Since 7 was a little too big, try 6. Multiply 6 22 = 132 Subtract. Bring down the next column. 22 goes into 1 about?????times. Estimate. 2 goes into 1 about 7 times. Try 7 Multiply 22 7 = 14. It worked. Subtract. Remainder = 267 R 1 or Write 9 4 as a decimal: Repeating decimal Step 1: Set it up. Write a few zeros, just to be safe Step 2: Divide into first. 9 goes into about 4 times. Step 3. Multiply 4 9 = 36 Step 4. Subtract. Step. Bring down the next column. Repeat steps 2-4 Step 2: 9 goes into about 4 times. Step 3: Multiply 4 9 = 36 Step 4: Subtract. Step. Bring down the next column. Repeat steps 2-4 Step 2: 9 goes into about 4 times. Step 3: Multiply 4 9 = 36 Step 4: Subtract. This could go on forever! Thus 9 4 = which we simply write by.4 The bar signifies numbers or patterns that repeat.

7 24 Two final examples: (.00) Step 1. Set it up and move the decimals Step 2. Divide into first Step 3. Multiply down Step 4. Subtract Step. Bring down Repeat steps 2- as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract Step. Bring down Repeat steps 2- as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract Step. Bring down Repeat steps 2- as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract Step : Bring down

8 Repeat steps 2- as necessary Step 2: Divide into first Step 3: Multiply down Step 4: Subtract ,680 Step 6: Write answer One negative in the original problem gives a negative answer. The decimal obviously keeps going. Round after a couple of decimal places. Two negatives make a positive COMMON MISTAKES - True in Multiplication and Division Since a negative sign simply means opposite direction, when we switch direction twice, we are headed back the way we started. Example: -(-) = Example: -(-2)(-1)(-3)(-) = = -30 Example: -(- -8) = -(- -) = - - False in Addition and Subtraction With addition and subtraction negatives and positives work against each other in a sort of tug o war. Whichever one is stronger will win. Example: Debt is negative and income is positive. If there is more debt than income, then the net result is debt. If we are $77 in debt and get income of $66 then we have a net debt of $ = -11 On the other hand if we have $77 dollars of income and $66 of debt, then the net is a positive $ = 11

9 26 Example: Falling is negative and rising is positive. An airplane rises 307 feet and then falls 23 feet, then the result is a rise of 284 feet: = 284 If, however, the airplane falls 307 feet and then rises 23 feet, then the result is a fall of 284 feet: = -284 Other examples: Discount is negative and markup or sales tax is positive. Warmer is positive and colder is negative. Whichever is greater will give you the sign of the net result. PERCENT Percent can be broken up into two words: per and cent meaning per hundred, or in other words, hundredths Example: =.07 = 7% =.31 = 31% =.3 = 3% Notice the shortcut from decimal to percents: move the decimal right two places. LAWS & PROCESSES Converting Percents Percents 1. If fraction, solve for decimals. 2. Move decimal 2 places. 3. OF means times. 1. Right for decimal to % 2. Left for % to decimal EXAMPLES Convert.2 to a percent.2= 2% Move the decimal two places to the right because we are turning this into a percent.2=2%

10 27 What is as a percent? 32=.162 Turn the fraction into a decimal by dividing.162=1.62% Move the decimal two places to the right because we are turning this into a percent 32 =1.62% Convert 124% to decimals 124%=1.24 Move the decimal two places to the left because we are turning this into a decimal 124%=1.24 Solving Of with Percents The most important thing that you should know about percents is that they never stand alone. If I were to call out that I owned 3%, the immediate response is, 3% of what? Percents always are a percent of something. For example, sales tax is about 6% or 7% of your purchase. Since this is so common, we need to know how to calculate this. If you buy $2 worth of food and the sales tax is 7%, then the actual tax is 7% of $2..07 $2 = $1.7 In math terms the word of means multiply. EXAMPLES What is 2% of 64? 2%=.2 Turn the percent into a decimal.2 64=16 Multiply the two numbers together 2% of 64 is 16 13%=.13 What is 13% of $2? Turn the percent into a decimal.13 2=3.2 Multiply the two numbers together 13% of $2 is $3.2 What is 30% of 90 feet? 30%=.30 Turn the percent into a decimal.30 90=27 Multiply the two numbers together 30% of 90 feet is 27 feet