Basic Math and Pre-Algebra - Decimals - Quick Guides

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Study GuideBasic Math and Pre-Algebra–Decimals1. Using the Place Value GridTheplace value gridhelps you understand how decimals work.Just like whole numbers, each digit in a decimal has a specific place and value.The only difference?Decimal places go to therightof the decimal point and represent parts of a whole (tenths,hundredths, thousandths, and so on).1.1Expanded NotaƟon with DecimalsDecimals can be written inexpanded notation, just like whole numbers.You simply multiply each digit by its place value.Example 1:Write 0.365 in expanded notation.First, identify each place value:•3 is in thetenthsplace→0.3•6 is in thehundredthsplace→0.06•5 is in thethousandthsplace→0.005So:0.365 = 0.3 + 0.06 + 0.005Now show it with multiplication:= (3 × 0.1) + (6 × 0.01) + (5 × 0.001)Or using powers of ten:= (3 × 10⁻¹) + (6 × 10⁻²) + (5 × 10⁻³)Notice that decimal places usenegative exponentsbecause they represent parts of 1.

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Study GuideExample 2:Write 5.26 in expanded notation.Break it into place values:•5 is in theonesplace•2 is in thetenthsplace•6 is in thehundredthsplaceSo:5.26 = 5 + 0.2 + 0.06With multiplication:= (5 × 1) + (2 × 0.1) + (6 × 0.01)Or with exponents:= (5 × 10⁰) + (2 × 10⁻¹) + (6 × 10⁻²)Expanded notation helps you clearly see the value of every digit.1.2WriƟng Decimals in WordsTo read or write a decimal in words:1.Start reading from the left.2.Say the whole number part.3.Say“and”to show the decimal point.4.Read the digits to the right as a whole number.5.End with the place value of the last digit.Example 3:Read 0.750.75 = seventy-five hundredths

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Study GuideExample 4:Read 45.32145.321 = forty-five and three hundred twenty-one thousandthsExample 5:Write “two hundred and three tenths.”Two hundred = 200Three tenths = 0.3So the number is:200.31.3Comparing DecimalsTo compare decimals, make sure they go to thesame number of placesto the right.You can do this by adding zeros at the end.Adding zeros doesnotchange the value of a decimal.Example 6:Which is greater, 0.37 or 0.365?Rewrite 0.37 as 0.370 so both numbers have three decimal places.0.3700.365Now compare digit by digit.Since 370 thousandths is greater than 365 thousandths,0.37 is greater than 0.365.Example 7:Order 0.66, 0.6587, and 0.661 from largest to smallest.First, write them to the same place value (ten-thousandths):0.66000.65870.6610

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Study GuideNow compare:Largest→0.661Next→0.66Smallest→0.6587So the order is:0.661, 0.66, 0.6587Remember:The number of digits after the decimal point doesnotdetermine size.For example, 0.5 is greater than 0.33, even though it has fewer digits.1.4Rounding DecimalsRounding decimals is almost the same as rounding whole numbers.Follow these steps:1.Underline the place value you are rounding to.2.Look at the digit immediately to the right.3.If that digit is5 or more, round up.4.If that digit is4 or less, keep the number the same.5.Drop (remove) all digits to the right of your rounding place.6.(You do not need to replace dropped digits with zeros.)Example 8:Round 0.478 to the nearest hundredth.Underline the hundredths place (7).Look at the thousandths digit (8).Since 8 is 5 or more, round up.0.478 becomes0.48

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Study GuideExample 9:Round 5.3743 to the nearest thousandth.Underline the thousandths place (4).Look at the next digit (3).Since 3 is less than 5, round down.5.3743 becomes5.374Why This MaƩersUnderstanding decimals helps you:•Work with money•Measure length and weight•Compare data•Estimate answers•Solve real-world problemsSummary•The place value grid helps you understand decimals by showing that each digit has a specificvalue based on its position (tenths, hundredths, thousandths, etc.).•Decimals can be written in expanded notation by breaking the number into the value of eachdigit, using multiplication or powers of ten with negative exponents.•When reading decimals in words, say the whole number part, use “and” for the decimal point,then name the decimal part using the place value of the last digit.•To compare decimals, line up the decimal points and add zeros if needed so the numbershave the same number of decimal places before comparing digits from left to right.•The number of digits after the decimal point does not determine which number is larger (forexample, 0.5 is greater than 0.33).•Rounding decimals follows the same rules as whole numbers: look at the digit to the right ofthe rounding place, round up if it is 5 or more, and round down if it is 4 or less.

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Study Guide2. Quiz: Using the Place Value Grid1. QuestionHow is the value one-tenth expressed as a power of 10?Answer Choices•10⁻¹•10⁰•10¹Correct Answer࿨࿩࿪10⁻¹Why This Is CorrectOne-tenth equals0.1.10⁻¹ = 1 ÷ 10 = 0.12. QuestionHow is the value hundred-thousandths expressed as a power of 10?Answer Choices•10⁻⁶•10⁻⁵•10⁵Correct Answer࿨࿩࿪10⁻⁵

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Study GuideWhy This Is CorrectHundred-thousandths means1 divided by 100,000.That equals 0.00001, which is10⁻⁵.3. QuestionWhich of the following is NOT an accepted expanded form for 0.0234?Answer Choices•(2 × 0.01) + (3 × 0.001) + (4 × 0.0001)•(2 × 10⁻¹) + (3 × 10⁻²) + (4 × 10⁻³)•(2 × 10⁻²) + (3 × 10⁻³) + (4 × 10⁻⁴)Correct Answer࿨࿩࿪(2 × 10⁻¹) + (3 × 10⁻²) + (4 × 10⁻³)Why This Is Correct0.0234 means:2 hundredths→2 × 10⁻²3 thousandths→3 × 10⁻³4 ten-thousandths→4 × 10⁻⁴But 2 × 10⁻¹ equals 0.2, which is too large.4. QuestionWhich of the following is an accepted expanded form for 10.405?Answer Choices•(10 × 10⁰) + (4 × 10⁻¹) + (5 × 10⁻³)•(1 × 10¹) + (4 × 10⁻¹) + (5 × 10⁻²)•(1 × 10¹) + (4 × 10⁻¹) + (5 × 10⁻³)

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Study GuideCorrect Answer࿨࿩࿪(1 × 10¹) + (4 × 10⁻¹) + (5 × 10⁻³)Why This Is Correct10.405 =1 in the tens place→1 × 10¹4 in the tenths place→4 × 10⁻¹5 in the thousandths place→5 × 10⁻³That matches the third choice.5. QuestionWhich of the following represents(3 × 10²) + (4 × 10⁰) + (5 × 10⁻²) + (6 × 10⁻⁴)?Answer Choices•304.506•304.056•304.0506Correct Answer࿨࿩࿪304.0506Why This Is Correct3 × 10² = 3004 × 10⁰= 45 × 10⁻² = 0.056 × 10⁻⁴= 0.0006Add them:300 + 4 + 0.05 + 0.0006 =304.0506

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Study Guide6. QuestionHow do you read 0.023?Answer Choices•twenty-three thousandths•twenty-three hundredths•twenty-three tenthsCorrect Answer࿨࿩࿪twenty-three thousandthsWhy This Is Correct0.023 has three digits after the decimal point.That makes it thousandths.7. QuestionHow is the number three hundred and three hundredths written?Answer Choices•0.303•300.03•300.003Correct Answer࿨࿩࿪300.03Why This Is CorrectThree hundred = 300Three hundredths = 0.03Together: 300.03

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Study Guide8. QuestionWhich list is correctly ordered from least to greatest?Answer Choices•0.45; 0.5; 0.396•0.375; 0.4; 0.51•0.8; 0.73; 0.652Correct Answer࿨࿩࿪0.375; 0.4; 0.51Why This Is Correct0.375 < 0.4 < 0.51The other lists are not in increasing order.9. QuestionWhich has the least value?Answer Choicesthirteen hundredthsfour tenthsfifty-three thousandthsCorrect Answer࿨࿩࿪fifty-three thousandths

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