decimals chapter five 5.1 introductions to decimalsintroductions to decimals 5.2 adding &...

DecimalsDecimals

Chapter FiveChapter Five

5.1 Introductions to Decimals

5.2 Adding & Subtracting Decimals

5.3 Multiplying Decimals & Circumference of a Circle

5.4 Dividing Decimals

5.5 Fractions, Decimals, & Order of Operations

5.6 Equations Containing Decimals

Introduction to DecimalsIntroduction to Decimals

Section 5.1

Whole- number part

Decimal point

Decimal part

16.743

Like fractional notation, decimal notation Like fractional notation, decimal notation is used to denote a part of a whole. is used to denote a part of a whole. Numbers written in decimal notation are Numbers written in decimal notation are called called decimal numbersdecimal numbers, or simply , or simply decimalsdecimals. The decimal 16.734 has three . The decimal 16.734 has three parts.parts.

3

The position The position of each digit of each digit in a number in a number determines its determines its place value.place value.

1 6 7 3 4 o

nes

tho

usa

nd

ths

hu

nd

red

s

ten

s

ten

ths

hu

nd

red

ths

ten

-th

ou

san

dth

s

hu

nd

red

-th

ou

san

dth

s

Place Value

decimal point

100 10 11

1001

10,000

110

1100,000

11000

4

Notice that the value of each place is Notice that the value of each place is

of the value of the place to its left.of the value of the place to its left.

110

5Martin-Gay, Prealgebra, 5ed

16.73416.734

The digit 3 is in the hundredths place, soThe digit 3 is in the hundredths place, so

its value is 3 hundredths or .its value is 3 hundredths or .3

100


Writing (or Reading) a Decimal Writing (or Reading) a Decimal in Wordsin Words

Step 1. Write the whole-number part in words.Step 1. Write the whole-number part in words.Step 2. Write “and” for the decimal point.Step 2. Write “and” for the decimal point.Step 3. Write the decimal part in words as Step 3. Write the decimal part in words as

though it were a whole number, though it were a whole number, followed by the place value of the last followed by the place value of the last digit.digit.


Writing a Decimal in WordsWriting a Decimal in Words

Write the decimal 143.056 in words.Write the decimal 143.056 in words.

143.056

one hundred forty-three and fifty-six thousandths

whole-number part decimal part


A decimal written in words can be written A decimal written in words can be written in standard form by reversing the in standard form by reversing the procedure.procedure.

Writing Decimals in Writing Decimals in Standard FormStandard Form

Write one hundred six and five hundredths in Write one hundred six and five hundredths in standard form.standard form.

one hundred six and five hundredths

106 . 05

decimal partwhole-number part decimal

5 must be in thehundredths place

5 must be in thehundredths place

9

Helpful Hint

When writing a decimal from words to When writing a decimal from words to decimal notation, make sure the last decimal notation, make sure the last digit is in the correct place by inserting digit is in the correct place by inserting 0s after the decimal point if necessary.0s after the decimal point if necessary.

For example, For example,

three and fifty-four thousandths is 3.054 three and fifty-four thousandths is 3.054

thousandths place


Once you master writing and reading Once you master writing and reading decimals correctly, then you write a decimals correctly, then you write a decimal as a fraction using the fractions decimal as a fraction using the fractions associated with the words you use when associated with the words you use when you read it.you read it.

0.90.9

is read “nine tenths” and written as a is read “nine tenths” and written as a fraction asfraction as

9

10

Writing Decimals as FractionsWriting Decimals as Fractions


twenty-one hundredthstwenty-one hundredths

and written as a fraction asand written as a fraction as

2121

100100

0.011 is read as 0.011 is read as

eleven thousandthseleven thousandths

and written as a fraction asand written as a fraction as

1111

10001000

0.21 is read as0.21 is read as


Notice that the number of decimal places Notice that the number of decimal places in a decimal number is the same as the in a decimal number is the same as the number of zeros in the denominator of the number of zeros in the denominator of the equivalent fraction. We can use this fact equivalent fraction. We can use this fact to write decimals as fractions.to write decimals as fractions.

0 3737

100.

2 decimal places

2 zeros

0 02929

1000.

3 decimal places

3 zeros

13

Comparing DecimalsComparing Decimals

One way to compare decimals is to compare their One way to compare decimals is to compare their graphs on a number line. Recall that for any two graphs on a number line. Recall that for any two numbers on a number line, the number to the left numbers on a number line, the number to the left is smaller and the number to the right is larger. To is smaller and the number to the right is larger. To compare 0.3 and 0.7 look at their graphs.compare 0.3 and 0.7 look at their graphs.

0 10.3

3

10

7

10

0.70.3 < 0.7 or

0.7 > 0.314

Martin-Gay, Prealgebra, 5ed

Comparing decimals by comparing their Comparing decimals by comparing their graphs on a number line can be time graphs on a number line can be time consuming, so we compare the size of consuming, so we compare the size of decimals by comparing digits in decimals by comparing digits in corresponding places.corresponding places.


Comparing Two Positive Comparing Two Positive DecimalsDecimals

Compare digits in the same places from left to Compare digits in the same places from left to right. When two digits are not equal, the number right. When two digits are not equal, the number with the larger digit is the larger decimal. If with the larger digit is the larger decimal. If necessary, insert 0s after the last digit to the necessary, insert 0s after the last digit to the right of the decimal point to continue comparing.right of the decimal point to continue comparing.

Compare hundredths place digits Compare hundredths place digits

3 5<35.638 35.657<

35.638 35.657


Helpful Hint

For any decimal, writing 0s after the last For any decimal, writing 0s after the last digit to the right of the decimal point does digit to the right of the decimal point does not change the value of the number. not change the value of the number.

8.5 8.5 == 8.50 8.50 == 8.500, and so on 8.500, and so on

When a whole number is written as a When a whole number is written as a decimal, the decimal point is placed to the decimal, the decimal point is placed to the right of the ones digit.right of the ones digit.

15 15 == 15.0 15.0 == 15.00, and so on 15.00, and so on 17Martin-Gay, Prealgebra, 5ed

We We round the decimal partround the decimal part of a decimal of a decimal number in nearly the same way as we number in nearly the same way as we round whole numbers. The only difference round whole numbers. The only difference is that we drop digits to the right of the is that we drop digits to the right of the rounding place, instead of replacing these rounding place, instead of replacing these digits by 0s. For example,digits by 0s. For example,

Rounding DecimalsRounding Decimals

63.782 rounded to the nearest hundredth is63.782 rounded to the nearest hundredth is

63.7863.7818


Rounding Decimals To a Place Value to Rounding Decimals To a Place Value to the Right of the Decimal Pointthe Right of the Decimal Point

Step 1. Locate the digit to the right of the Step 1. Locate the digit to the right of the given place value.given place value.

Step 2. If this digit is 5 or greater, add 1 to Step 2. If this digit is 5 or greater, add 1 to the digit in the given place value and the digit in the given place value and drop all digits to the right. If this drop all digits to the right. If this digit is less than 5, drop all digits to digit is less than 5, drop all digits to the right of the given place.the right of the given place.


Rounding Decimals to a Place ValueRounding Decimals to a Place Value

Round 326.4386 to the nearest tenth.Round 326.4386 to the nearest tenth.Locate the digit to the right of the tenths place.

326.4386

tenths place

digit to the right

Since the digit to the right is less than 5, drop it and all digits to its right.

326.4386 rounded to the nearest tenths is 326.420Martin-Gay, Prealgebra, 5ed

Adding and Subtracting Adding and Subtracting DecimalsDecimals

Section 5.2

Adding or Subtracting Adding or Subtracting DecimalsDecimals

Step 1. Write the decimals so that the decimal Step 1. Write the decimals so that the decimal points line up vertically.points line up vertically.

Step 2. Add or subtract as for whole numbers.Step 2. Add or subtract as for whole numbers.

Step 3. Place the decimal point in the sum or Step 3. Place the decimal point in the sum or difference so that it lines up vertically difference so that it lines up vertically with the decimal points in the problem.with the decimal points in the problem.


Helpful Hint

Recall that 0s may be inserted to the right of Recall that 0s may be inserted to the right of the decimal point after the last digit without the decimal point after the last digit without changing the value of the decimal. This may changing the value of the decimal. This may be used to help line up place values when be used to help line up place values when adding or subtracting decimals.adding or subtracting decimals.

85 13.26 becomesbecomes

85.00 13.26

71.74

two 0s insertedtwo 0s inserted

23

Helpful Hint

Don’t forget that the decimal point in a Don’t forget that the decimal point in a whole number is after the last digit.whole number is after the last digit.


Estimating sums, differences, products, Estimating sums, differences, products, and quotients of decimal numbers is an and quotients of decimal numbers is an important skill whether you use a important skill whether you use a calculator or perform decimal operations calculator or perform decimal operations by hand.by hand.

Estimating Operations on Estimating Operations on DecimalsDecimals


Add 23.8 + 32.1.Add 23.8 + 32.1.

Estimating When Adding Estimating When Adding DecimalsDecimals

ExactExact EstimateEstimate23.8

+32.155.9

rounds to 24rounds to 32

56

This is a reasonable answer.


Helpful Hint

When rounding to check a calculation, When rounding to check a calculation, you may want to round the numbers to a you may want to round the numbers to a place value of your choosing so that place value of your choosing so that your estimates are easy to compute your estimates are easy to compute mentally.mentally.


Evaluate x + y for x = 5.5 and y = 2.8.

Evaluating with DecimalsEvaluating with Decimals

x + y = ( ) + ( )

Replace x with 5.5 and y with 2.8 in x + y.

5.5 2.8

= 8.3


Multiplying Decimals and Multiplying Decimals and Circumference of a CircleCircumference of a Circle

Section 5.3

Multiplying decimals is similar to Multiplying decimals is similar to multiplying whole numbers. The multiplying whole numbers. The difference is that we place a decimal difference is that we place a decimal point in the product.point in the product.

0.7 x 0.03 = 7

10

3

100x

1 decimal place

2 decimal places

21

1000

= 0.021

=

3 decimal places

Multiplying DecimalsMultiplying Decimals


Multiplying DecimalsMultiplying Decimals

Step 1. Multiply the decimals as though Step 1. Multiply the decimals as though they were whole numbers.they were whole numbers.

Step 2. The decimal point in the product Step 2. The decimal point in the product is placed so the number of is placed so the number of decimal places in the product is decimal places in the product is equal to the equal to the sumsum of the number of of the number of decimal places in the factors.decimal places in the factors.


Multiply 32.3 x 1.9.Multiply 32.3 x 1.9.

Estimating when Multiplying Estimating when Multiplying DecimalsDecimals

ExactExact EstimateEstimate

32.3

1.9

290.7323.061.37

rounds to 32rounds to 2

64



Multiplying Decimals by Multiplying Decimals by Powers of 10Powers of 10

There are some patterns that occur There are some patterns that occur when we multiply a number by a power when we multiply a number by a power of ten, such as 10, 100, 1000, 10,000, of ten, such as 10, 100, 1000, 10,000, and so on.and so on.


76.543 76.543 xx 10 10 == 765.43 765.43

76.543 76.543 xx 100 100 == 7654.3 7654.3

76.543 76.543 xx 100,000 100,000 == 7,654,300 7,654,300

Decimal point moved 1 place to the right.

Decimal point moved 2 places to the right.

Decimal point moved 5 places to the right.

2 zeros

5 zeros

1 zero

The decimal point is moved the same number of The decimal point is moved the same number of places as there are zeros in the power of 10.places as there are zeros in the power of 10.

34

Multiplying Decimals by Multiplying Decimals by Powers of 10Powers of 10


Multiplying by Powers of 10 such Multiplying by Powers of 10 such as 10, 100, 1000 or 10,000, . . .as 10, 100, 1000 or 10,000, . . .

Move the decimal point to the Move the decimal point to the rightright the the same number of places as there are same number of places as there are zeroszeros in the power of 10. in the power of 10.

Multiply: 3.4305 x 100

Since there are two zeros in 100, move the decimal place two places to the right.

3.4305 x 100 = 343.053.4305 =


Move the decimal point to the Move the decimal point to the left left the same the same number of places as there are number of places as there are decimal decimal placesplaces in the power of 10. in the power of 10.

Multiplying by Powers of 10 such Multiplying by Powers of 10 such as 0.1, 0.01, 0.001, 0.0001, . . .as 0.1, 0.01, 0.001, 0.0001, . . .

Multiply: 8.57 x 0.01

Since there are two decimal places in 0.01, move the decimal place two places to the left.

8.57 x 0.01 = 0.0857

Notice that zeros had to be inserted..

008.57 =

36

The distance around a polygon is called its The distance around a polygon is called its perimeterperimeter..

The distance around a circle is called the The distance around a circle is called the circumferencecircumference..

This distance depends on the This distance depends on the radiusradius or the or the diameterdiameter of the circle. of the circle.

Finding the Circumference Finding the Circumference of a Circleof a Circle


Circumference of a CircleCircumference of a Circle

r

d

CCircumference ircumference == 2· 2···rradiusadius

or or

CCircumference ircumference == ··ddiameteriameter

CC == 2 2 rr or or CC == dd


The symbol The symbol is the Greek letter pi, is the Greek letter pi, pronounced “pie.” It is a constant pronounced “pie.” It is a constant between 3 and 4. A decimal between 3 and 4. A decimal approximation for approximation for is 3.14. is 3.14.

A fraction approximation forA fraction approximation foris . is .

22

7


Find the circumference of a circle Find the circumference of a circle whosewhose r radius is adius is 44 inches. inches.

4 inches

CC == 2 2rr == 2 2··44 == 8 8inchesinches

88inches is the inches is the exact circumference of circumference of this circle.this circle.

If we replace If we replace with the approximation 3.14, with the approximation 3.14, CC == 8 8 8(3.14) 8(3.14) == 25.12 inches. 25.12 inches.

25.12 inches is the 25.12 inches is the approximate circumference of the circle.circumference of the circle. 40

Dividing DecimalsDividing Decimals

Section 5.4

The only difference is the placement of a decimal The only difference is the placement of a decimal point in the point in the quotientquotient. .

If the divisor is a whole number, divide as for If the divisor is a whole number, divide as for whole numbers; then place the decimal point in whole numbers; then place the decimal point in the quotient directly above the decimal point in the quotient directly above the decimal point in the dividend. the dividend. 8

- 5 0 42 52-2 52

0

divisor

quotient

dividend0 4

Division of decimal numbers is similar to Division of decimal numbers is similar to division of whole numbers. division of whole numbers.

63 52.92

42

863 529.2

- 504252-252

0

4

If the divisor is If the divisor is notnot a whole number, we need to move a whole number, we need to move the decimal point to the right until the divisor is a the decimal point to the right until the divisor is a whole number before we divide. whole number before we divide.

divisor dividend6 3 52 92. .

63 529 2. .


Dividing by a DecimalDividing by a Decimal

Step 1. Move the decimal point in the divisor Step 1. Move the decimal point in the divisor to the right until the divisor is a whole to the right until the divisor is a whole number.number.

Step 2. Move the decimal point in the Step 2. Move the decimal point in the dividend to the right the dividend to the right the same same numbernumber ofof placesplaces as the decimal as the decimal point was moved in Step 1.point was moved in Step 1.

Step 3. Divide. Place the decimal point in the Step 3. Divide. Place the decimal point in the quotient directly over the moved quotient directly over the moved decimal point in the dividend.decimal point in the dividend.


Divide 258.3 ÷ 2.8Divide 258.3 ÷ 2.8

Estimating When Dividing Estimating When Dividing DecimalsDecimals

ExactExact EstimateEstimate

28. 2583. - 252 63 - 56 70 - 56 140 -140 0

rounds to 3 300100


92.25


There are patterns that occur when dividing by There are patterns that occur when dividing by powers of 10, such as 10, 100, 1000, and so on.powers of 10, such as 10, 100, 1000, and so on.

The decimal point moved The decimal point moved 1 place to the left.1 place to the left.

1 zero

3 zeros

The decimal point moved The decimal point moved 3 places to the left.3 places to the left.

The pattern suggests the following rule.The pattern suggests the following rule.

.45 6210

456.2

1 0000 4562

,.456.2


Move the decimal point of the dividend to the leftleft the same number of places as there are zeroszeros in the power of 10.

Dividing Decimals by Powers of 10 Dividing Decimals by Powers of 10 such as 10, 100, or 1000, . . .such as 10, 100, or 1000, . . .

Notice that this is the same pattern as multiplying by powers of 10 such as 0.1, 0.01, or 0.001. Because dividing by a power of 10 such as 100 is the same as multiplying by its reciprocal , or 0.01. 1

100463 7100

463 71

100463 7 0 01 4 637

.. . . .

To divide by a number is the same as multiplying by its reciprocal. 47

Section 5.5

Fractions, Decimals, and Fractions, Decimals, and Order of OperationsOrder of Operations

To write a fraction as a decimal, divide To write a fraction as a decimal, divide the numerator by the denominator.the numerator by the denominator.

Writing Fractions as DecimalsWriting Fractions as Decimals

49

3 = 3 4 = 0.754

2 = 2 5 = 0.405


Comparing Fractions and Comparing Fractions and DecimalsDecimals

To compare decimals and fractions, write To compare decimals and fractions, write the fraction as an equivalent decimal.the fraction as an equivalent decimal.

50

Compare 0.125 and Compare 0.125 and .14

1 = 0.254

Therefore, 0.125 < 0.25Therefore, 0.125 < 0.25


1.1. Do all operations within grouping Do all operations within grouping symbols such as parentheses or symbols such as parentheses or brackets.brackets.

2.2. Evaluate any expressions with Evaluate any expressions with exponents.exponents.

3.3. Multiply or divide in order from left to Multiply or divide in order from left to right.right.

4. 4. Add or subtract in order from left to Add or subtract in order from left to right.right.

Order of OperationsOrder of Operations


Using the Order of Operations

Simplify ( – 2.3)2 + 4.1(2.2 + 3.1)

Simplify inside parentheses.

( – 2.3)2 + 4.1(2.2 + 3.1)

= ( – 2.3)2 + 4.1(5.3)

= 5.29 + 4.1(5.3) Write ( – 2.3)2 as 5.29.

= 5.29 + 21.73 Multiply.

= 27.02 Add.Martin-Gay, Prealgebra, 5ed

Finding the Area of a TriangleFinding the Area of a Triangle

base

height

AA basebase • • heightheight==11

22

AA bhbh==1

2


Equations Containing Equations Containing DecimalsDecimals

Section 5.6

Steps for Solving an Equation Steps for Solving an Equation in in xx

Step 1. If fractions are present, multiply Step 1. If fractions are present, multiply both sides of the equation by the both sides of the equation by the LCD of the fractions.LCD of the fractions.

Step 2. If parentheses are present, use the Step 2. If parentheses are present, use the distributive property.distributive property.

Step 3. Combine any like terms on each Step 3. Combine any like terms on each side of the equation.side of the equation.


Steps for Solving an Equation . . .Steps for Solving an Equation . . .

Step 4. Use the addition property of equality Step 4. Use the addition property of equality to rewrite the equation so that variable to rewrite the equation so that variable terms are on one side of the equation terms are on one side of the equation and constant terms are on the other and constant terms are on the other side.side.

Step 5. Divide both sides by the numerical Step 5. Divide both sides by the numerical coefficient of coefficient of xx to solve. to solve.

Step 6. Check the answer in the Step 6. Check the answer in the original original equation.equation.


– 0.01(5a + 4) = 0.04 – 0.01(a + 4)

Solving Equations with DecimalsSolving Equations with Decimals

Multiply both sides by 100. – 1(5a + 4) = 4 – 1(a + 4)

Apply the distributive property.

– 5a – 4 = 4 – a – 4

Add a to both sides. – 4a – 4 = 4 – 4

Add 4 to both sides and simplify.

– 4a = 4

Divide both sides by 4. a = – 1

decimals chapter five 5.1 introductions to decimalsintroductions to decimals 5.2 adding &...

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