Decimals

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Decimals
Chapter 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

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Introduction to Decimals
Section 5.1
3
Like fractional notation, decimal notation is
used to denote a part of a whole. Numbers written
in decimal notation are called decimal numbers,
or simply decimals. The decimal 16.734 has three
parts.
16.743
Whole- number part
Decimal part
Decimal point
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The position of each digit in a number determines
its place value.
hundred-thousandths
thousandths
hundreds
tens
tenths
hundredths
ten-thousandths
ones
1 6 7 3 4
Place Value
100
10
1
decimal point
5
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16.734
The digit 3 is in the hundredths place, so its
value is 3 hundredths or .
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Writing (or Reading) a Decimal in Words

• Step 1. Write the whole-number part in words.
• Step 2. Write and for the decimal point.
• Step 3. Write the decimal part in words as
  though it were a whole number, followed by the
  place value of the last digit.

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Writing a Decimal in Words

• Write the decimal 143.056 in words.

143.056
one hundred forty-three
and
fifty-six thousandths
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A decimal written in words can be written in
standard form by reversing the procedure.
Writing Decimals in Standard Form
Write one hundred six and five hundredths in
standard form.
one hundred six and five hundredths
5 must be in the hundredths place
106
.
05
10
Helpful Hint

• When writing a decimal from words to decimal
  notation, make sure the last digit is in the
  correct place by inserting 0s after the decimal
  point if necessary.
• For example,
• three and fifty-four thousandths is 3.054

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Writing Decimals as Fractions
Once you master writing and reading decimals
correctly, then you write a decimal as a fraction
using the fractions associated with the words you
use when you read it.
0.9 is read nine
tenths and written as a fraction as

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twenty-one hundredths and written as a fraction
as
0.21 is read as
0.011 is read as eleven thousandths and written
as a fraction as
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2 zeros
3 decimal places
3 zeros
2 decimal places
Notice that the number of decimal places in a
decimal number is the same as the number of zeros
in the denominator of the equivalent fraction. We
can use this fact to write decimals as fractions.
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Comparing Decimals
One way to compare decimals is to compare their
graphs on a number line. Recall that for any two
numbers on a number line, the number to the left
is smaller and the number to the right is larger.
To compare 0.3 and 0.7 look at their graphs.
0
1
0.3
0.7
0.3 lt 0.7 or 0.7 gt 0.3
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Comparing decimals by comparing their graphs on a
number line can be time consuming, so we compare
the size of decimals by comparing digits in
corresponding places.
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Comparing Two Positive Decimals
Compare digits in the same places from left to
right. When two digits are not equal, the number
with the larger digit is the larger decimal. If
necessary, insert 0s after the last digit to the
right of the decimal point to continue
comparing. Compare hundredths place digits

35.638
35.657
3
5
lt
35.638
35.657
lt
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Helpful Hint
For any decimal, writing 0s after the last digit
to the right of the decimal point does not change
the value of the number.
8.5 8.50 8.500, and so on When a whole
number is written as a decimal, the decimal point
is placed to the right of the ones digit. 15
15.0 15.00, and so on
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Rounding Decimals
We round the decimal part of a decimal number in
nearly the same way as we round whole numbers.
The only difference is that we drop digits to the
right of the rounding place, instead of replacing
these digits by 0s. For example,
63.782 rounded to the nearest hundredth is
63.78
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Rounding Decimals To a Place Value to the Right
of the Decimal Point

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

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Rounding Decimals to a Place Value

• Round 326.4386 to the nearest tenth.

Locate the digit to the right of the tenths place.
326.4386
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 326.4
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Adding and Subtracting Decimals
Section 5.2
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Adding or Subtracting Decimals

• Step 1. Write the decimals so that the decimal
  points line up vertically.
• Step 2. Add or subtract as for whole numbers.
• Step 3. Place the decimal point in the sum or
  difference so that it lines up vertically with
  the decimal points in the problem.

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Helpful Hint
Recall that 0s may be inserted to the right of
the decimal point after the last digit without
changing the value of the decimal. This may be
used to help line up place values when adding or
subtracting decimals.
85 - 13.26 becomes
two 0s inserted
71.74
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Helpful Hint
Dont forget that the decimal point in a whole
number is after the last digit.
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Estimating Operations on Decimals
Estimating sums, differences, products, and
quotients of decimal numbers is an important
skill whether you use a calculator or perform
decimal operations by hand.
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Estimating When Adding Decimals
Add 23.8 32.1.
Exact
Estimate
rounds to
rounds to
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Helpful Hint
When rounding to check a calculation, you may
want to round the numbers to a place value of
your choosing so that your estimates are easy to
compute mentally.
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Evaluating with Decimals
Evaluate x y for x 5.5 and y 2.8.
Replace x with 5.5 and y with 2.8 in x y.
x y ( ) ( )
5.5
2.8
8.3
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Multiplying Decimals and Circumference of a Circle
Section 5.3
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Multiplying Decimals
Multiplying decimals is similar to multiplying
whole numbers. The difference is that we place a
decimal point in the product.
0.7 x 0.03

0.021
1 decimal place
2 decimal places
3 decimal places
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Multiplying Decimals

• Step 1. Multiply the decimals as though they
  were whole numbers.
• Step 2. The decimal point in the product is
  placed so the number of decimal places in the
  product is equal to the sum of the number of
  decimal places in the factors.

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Estimating when Multiplying Decimals
Multiply 32.3 x 1.9.
Exact
Estimate
rounds to
rounds to
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Multiplying Decimals by Powers of 10
There are some patterns that occur when we
multiply a number by a power of ten, such as 10,
100, 1000, 10,000, and so on.
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Multiplying Decimals by Powers of 10

• 76.543 x 10 765.43
• 76.543 x 100 7654.3
• 76.543 x 100,000 7,654,300

Decimal point moved 1 place to the right.
1 zero
Decimal point moved 2 places to the right.
2 zeros
Decimal point moved 5 places to the right.
5 zeros
The decimal point is moved the same number of
places as there are zeros in the power of 10.
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Multiplying by Powers of 10 such as 10, 100, 1000
or 10,000, . . .
Move the decimal point to the right the same
number of places as there are zeros 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
343.05
3.4305
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Multiplying by Powers of 10 such as 0.1, 0.01,
0.001, 0.0001, . . .
Move the decimal point to the left the same
number of places as there are decimal places in
the power of 10.
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
008.57
Notice that zeros had to be inserted.
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Finding the Circumference of a Circle
The distance around a polygon is called its
perimeter. The distance around a circle is called
the circumference. This distance depends on the
radius or the diameter of the circle.
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Circumference of a Circle
r
d
Circumference 2p radius or Circumference p
diameter C 2 p r or C p d
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p
The symbol p is the Greek letter pi, pronounced
pie. It is a constant between 3 and 4. A
decimal approximation for p is 3.14. A fraction
approximation for p is .
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4 inches
Find the circumference of a circle whose radius
is 4 inches.
C 2pr 2p 4 8p inches 8p inches is the
exact circumference of this circle.
If we replace ? with the approximation 3.14, C
8? ? 8(3.14) 25.12 inches. 25.12 inches is the
approximate circumference of the circle.
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Dividing Decimals
Section 5.4
42
Division of decimal numbers is similar to
division of whole numbers.
The only difference is the placement of a decimal
point in the quotient. If the divisor is a whole
number, divide as for whole numbers then place
the decimal point in the quotient directly above
the decimal point in the dividend.
8
0
4
- 5 0 4
2 5
2
-2 52
0
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If the divisor is not a whole number, we need to
move the decimal point to the right until the
divisor is a whole number before we divide.
8
4
- 504
25
2
-252
0
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Dividing by a Decimal

• Step 1. Move the decimal point in the divisor to
  the right until the divisor is a whole number.
• Step 2. Move the decimal point in the dividend
  to the right the same number of places as the
  decimal point was moved in Step 1.
• Step 3. Divide. Place the decimal point in the
  quotient directly over the moved decimal point in
  the dividend.

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Estimating When Dividing Decimals
Divide 258.3 2.8
Exact
Estimate
rounds to
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There are patterns that occur when dividing by
powers of 10, such as 10, 100, 1000, and so on.
The decimal point moved 1 place to the left.
The decimal point moved 3 places to the left.
The pattern suggests the following rule.
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Dividing Decimals by Powers of 10 such as 10,
100, or 1000, . . .
Move the decimal point of the dividend to the
left the same number of places as there are zeros
in the power of 10.
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Section 5.5
Fractions, Decimals, and Order of Operations
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To write a fraction as a decimal, divide the
numerator by the denominator.
Writing Fractions as Decimals
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To compare decimals and fractions, write the
fraction as an equivalent decimal.
Comparing Fractions and Decimals
Therefore, 0.125 lt 0.25
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Order of Operations
1. Do all operations within grouping symbols such
as parentheses or brackets. 2. Evaluate any
expressions with exponents. 3. Multiply or divide
in order from left to right. 4. Add or subtract
in order from left to right.
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Using the Order of Operations
Simplify ( 2.3)2 4.1(2.2 3.1)
( 2.3)2 4.1(2.2 3.1) ( 2.3)2
4.1(5.3)
Simplify inside parentheses.
5.29 4.1(5.3)
Write ( 2.3)2 as 5.29.
5.29 21.73
Multiply.
27.02
Add.
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Finding the Area of a Triangle
height
base
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Equations Containing Decimals
Section 5.6
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Steps for Solving an Equation in x

• Step 1. If fractions are present, multiply both
  sides of the equation by the LCD of the
  fractions.
• Step 2. If parentheses are present, use the
  distributive property.
• Step 3. Combine any like terms on each side of
  the equation.

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Steps for Solving an Equation . . .

• Step 4. Use the addition property of equality to
  rewrite the equation so that variable terms are
  on one side of the equation and constant terms
  are on the other side.
• Step 5. Divide both sides by the numerical
  coefficient of x to solve.
• Step 6. Check the answer in the original equation.

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Solving Equations with Decimals
0.01(5a 4) 0.04 0.01(a 4)
1(5a 4) 4 1(a 4)
Multiply both sides by 100.
5a 4 4 a 4
Apply the distributive property.
4a 4 4 4
Add a to both sides.
4a 4
Add 4 to both sides and simplify.
Divide both sides by 4.
a 1