Section 7-1 – Introduction to Decimals
Decimals are an alternative was of representing fractions using our base-10 system.
Connecting Decimals and Integers: Decimals and integers are both base-10 place value systems.
The value of each digit depends on its place, and the value of each place (to the right) is one-tenth
the value of the previous place.
thousands
103
1,000
hundreds
102
100
tens
101
10
ones
100
1
tenths
10–1
0.1
hundredths
10–2
0.01
thousandths
10–3
0.001
Examples: Using the Base-10 blocks, if we let the Flat = 1, then how would you represent:
a) 2.3
b) 1.25
Examples:
c) 3.01
Use words to describe:
a) 5.45
b) 4.06
c) 0.302
Examples: Write the following decimals in expanded form with exponents:
a) 2.41
b) 25.0457
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Connecting Decimals and Fractions
By using the concept of place value, decimals can be translated into fractions.
Examples: Convert the following decimals to fractions
a) 0.35
b) 2.25
c) 4.501
Examples: Convert the following fractions to decimals
a)
63
100
b)
315
10, 000
c)
7
20
d)
27
250
e)
1
125
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When fractions are converted to decimals, some will terminate and some will not.
What causes a fraction to terminate?
a
in simplest form can be written as a terminating decimal if and
b
only if the prime factorization of the denominator contains no primes other than ______ or ______.
Theorem: A rational number
Examples: Which of the following fractions can be written as terminating decimals?
a)
1
250
b)
7
34
c)
1
384
d) 21/28
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METHODS
FOR
ORDERING DECIMALS:
Lining up the Decimals
1. Line up the numbers by place value.
2. Start at the left and find the first place where the place values are different
3. Compare these digits
Example: Put in order from smallest to largest:
0.587, 0.059, 0.524
__________, __________, __________
Using equivalent fractions
Example: Put in order from smallest to largest:
0.023, 0.0223, 0.233
__________, __________, __________
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Section 7-2 – Operations on Decimals
Addition/Subtraction
“Line up the decimals” or “Convert to Fractions”
Examples: Add or Subtract as indicated using the standard algorithm and using Base-10 blocks.
a) 2.35 + 0.74
b) 0.73 – 0.25
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Multiplication
Example: Multiply 3.16 × 2.4 using the standard algorithm taught in school
Using Fractions: 3.16 × 2.4 =
316 24
100 10
316 24
10 2 101
316 24
103
7584
1000
This is the reason that there are 3 places behind the decimal place!!!!
Note: Use your estimation skills to make sure the location of the decimal place seems reasonable!
When multiplying decimals, if the denominators are 10m and 10n , then the resulting product
will contain 10m n , therefore there will be m+n digits to the right of the decimal point.
Using the Lattice Method:
3.16 × 2.4 = _______________
3
.
1
6
2
.
4
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Division
Examples: Divide using long division and the standard method of “moving the decimal place
of the divisor if necessary”
a)
2.5 53.08
b)
1.25 4.4
Why do we move the decimal place on the divisor??
4.4 ÷ 1.25 =
4.4
1.25
4.4 102
1.25 102
440
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SCIENTIFIC NOTATION
A number is in scientific notation if it is in the form a × 10b where 1
a
10 and b is an integer.
Examples: Write each of the following in scientific notation
a)
4,520,000,000
b)
0.000006234
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Examples: Write each of the following as standard numerals
a)
9.97 × 105
b)
2.224 × 10–4
Examples: Perform the indicated operation
a)
(9 × 107 ) × (4 × 10 5)
b)
(24 × 1013 ) ÷ (8 × 103)
ROUNDING DECIMALS
Example:
Round 152.0398 to the nearest:
a) Hundred
152.0398
b) Ten
152.0398
c) One
152.0398
d) Tenth
152.0398
e) Hundredth
152.0398
f) Thousandth
152.0398
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Section 7-3 – Nonterminating Decimals
Repeating Decimals
Examples: Convert the following to decimals using long division
a)
1
= _______________________
7
7 1.000000000
b)
2
= __________________________
13
13 2.000000000
When dividing by 7, the only non-zero remainders were 1, 2, 3, 4, 5, 6.
As soon as a remainder re-occurs, you will see the repeating pattern.
The repeating block of digits is called the repetend.
Therefore, when dividing by 7, the repetend cannot contain more than 6 digits.
When dividing by 13, the repetend cannot contain more than 12 digits.
Example: Write
1
2 3
7
as a decimal. Then write , , and
as decimals.
9
9 9
9
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Converting a Repeating Decimal into a Rational Number
Examples:
a) Write 0.3 as a fraction
b) Write 0.36 as a fraction
c) Write 0.245 as a fraction
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Ordering Repeating Decimals
Example: Which is larger 0.23456 or 0.2345 ?
0.23456 = 0.23456456…
0.2345 = 0.2345345…
Section 7-4 – Percents
Percent means “per-hundred”
n%
n
100
Examples: Write each of the following as percents
a) 0.45 = 0.45 100
100
(0.45)100
=
100
b) 1.26 =
c) 0.005 =
d) 1 =
e) 0.3 =
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Fractions can be converted to percents by using the following proportion:
a
b
n
100
Examples: Write each of the following as percents
a)
2
=
5
b)
9
=
40
Examples: Convert the following percents to decimals:
a) 5.8% =
b) ¼ % =
c) 150%
Note: Estimation skills should tell you if your answer seems reasonable!
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Applications Involving Percents:
Example: What is 15% of 300?
Example: Ricky took a 35-question test and he got 20 questions right.
What percent of the problems did he get right?
Example: A candy bar contains 12 grams of fat. If this is 19% of the maximum recommended
daily intake of fat, what is the maximum recommended daily intake of fat?
Example: A realtor earned $9975 from a home sale that involved a 7% commission.
What was the selling price of the home?
Example: Kirk made $45,000 last year and received a 2% raise. How much does he make now?
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Example: A used car originally cost $1,700. One year later, it was worth $1,400.
What is the percent of decrease in value?
Example: Mark bought his house in 2000 for $59,000. It was recently appraised at $95,000.
What is the percent of increase in value?
Example: Amy bought a dress for $144 on a “20% off” clearance rack. What was the original price
of the dress?
Mental Math with Percents
It is relatively easy to find 50%, 25%, 10%, and 5% mentally
Examples: Compute the following percents mentally:
a) What is 50% of 250?
b) What is 25% of 800?
c) What is 10% of $35.00
d) What is 15% of $35.00
e) What is 1% of 160?
f) What is 150% of 20?
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