Decimals
1
• Decimals are absolutely amazing
– We have only 10 symbols, yet can represent any number,
large or small
– We use zero (0) as a place holder to allow us to do this
2
Some Older Number Systems
3
• Can you imagine doing arithmetic with Roman Numerals?
29 plus 28 becomes:
XXIX+ XXVII = ?
How do you put together the X, V, and I?
4
5
We Use Place Value
11211 vs. 12111: What does the position of the 2 say about the
value of the number?
For numbers greater than one we have:
ones
tens
hundreds…
For numbers less than one we have:
tenths
hundredths …
We separate those less than one by using a decimal point;
one is the focus point, not zero!
6
For Numbers Less Than One
One tenth is 1/10 is 0.1
One thousandth is 1/1000 is 0.001
What is 0.321?
7
We can think in terms of adding
345.26 is
3 hundreds
300
4 tens
40
5 ones
5
2 tenths
2/10
6 hundredths
6/100
300 + 40 + 5 + 2/10 + 6/100 = 345.26
8
Examples
9
Decimals to Fractions
• We can write decimals as fractions (and/or mixed numbers)
0.51 = 51/100
0.125 = 126/1000
We could also write 0.125 as 1/10 + 2/100 + 6/1000, find a
common denominator, and add the fractions!
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Mixed Numbers
5.7 = 5 7/10
11.345 = 11 345/1000
11
Fractions as Decimals
6/10 = 0.6
45/100 = 0.45
85/10 = 8.5
306/100 = 3.06
How do I know that there is something to the left of the decimal
point? If the numerator is larger than the denominator
12
Examples/Review
• Three and nine tenths in decimals =
• Eight hundred four and three hundred ninety-nine thousandths
in decimals =
• 47.65 in words =
• 3601/1000 in decimals =
13
Solution
• Three and nine tenths in decimals = 3.9
• Eight hundred four and three hundred ninety-nine
thousandths in decimals = 804.099
• 47.65 in words = forty seven and 65 hundredths
• 3601/1000 in decimals = 3.601
14
Place Value
• Our base-ten place-value numeration system is adequate for
expressing all whole numbers and decimal numbers.
• The value of each digit in a numeral is determined by the
position of the digit in the numeral.
• Digits in different places have different values.
• Finally, the number 1, not the decimal point, serves as the
focal point of decimal numbers.
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Order and Rounding
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Which is larger?
• 23.456 or 23.556?
• 3.045 or 3.405?
• 25.1 or 25.13?
• 0.61 or 0.076?
• -3.451 or -3.441?
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Solution
• 23.456 or 23.556?
• 3.045 or 3.405?
• 25.1 or 25.13?
• 0.61 or 0.076?
• -3.451 or -3.441? It is less negative, so larger
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Note:
76 = 76.0 = 76.000, etc.
Similarly
0.35 = .35, etc. and 02 (which makes no sense) is still 2
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Rounding
• Remember
– If we round 56 to the nearest ten, we have 60
– If we round 3465 to the nearest thousand we have 3000
• Rule:
– if the number in the place to the right of where we are
rounding to is <5, drop it:
343 rounded to the nearest 10 is 340
– If the number in the place to the left of where we are
rounding to is ≥ 5, go up:
345 rounded to the nearest 10 is 350
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Rounding
• Similarly, 2.954 rounded to the nearest hundredth is 2.95
• What is 2.954 rounded to the nearest tenth???
Nearest one?
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Solution
• What is 2.954 rounded to the nearest tenth???
3.0
Nearest one? 3 (or 3.0, they are the same)
22
Examples
Suzie has an annual income of $41,568.72. She is paying her
taxes, and the tax table use amounts to the nearest dollar.
What income value should she use in determining her tax?
Jane has an income of $154,321.25. For her income bracket, the
tax tables use amounts to the nearest hundred dollars. What
income value should she use in determining her tax?
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Solution
• $41,568.72 to the nearest dollar is $41,569
• $154,321.25 to the nearest hundred dollars is $154,300
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More Examples
• Which number(s) round to 0.06?
0.612
0.066
0.586
0.506
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Solution
• 0.0612 and 0.0586
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Adding and Subtracting Decimals
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How?
• Can think of in two ways:
–
need a common denominator, or
–
line up the decimal points
23.85 + 1.604 = 2385/100 + 1604/1000;
common denominator is 1000,
(23850 + 1604)/1000 = 25454/1000 = 25.454
23.850
1.604
25.454
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Estimating
• 27.6 + 519.25 = ?
We know that this has to be close to 30 + 500 or 530.
This helps us figure out where the decimal point goes
If the number seems wrong, check where your decimal
point is.
The exact answer is 546.85, which is near 530
This is especially useful for division!
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Examples: Estimate and Solve
2.5 + 4.1 =
47.14 + 409.567 =
100.009 + 6.08 + 9.034 =
What is wrong with:
7.03
2.008
19.16
3.1415
3.6042
30
Solutions
2.5 + 4.1 = 6.6; estimate = 6
47.14 + 409.567 = 456.707; estimate = 50 + 400 = 450
100.009 + 6.08 + 9.034 = 115.123; estimate = 100 + 6 + 9 = 115
What is wrong with: The decimal points were not kept straight!
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Subtraction
• Same as addition, but may need to borrow:
– Get a common denominator or
– Line up the decimal points
• 82.75 – 1.59 =
change the 82 to 81 and make the one into 10/10
82.75
-15.9
66.85
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Examples: Estimate and Solve
7.6 – 2.1 =
28. – 3.3 =
6.4 – 3.04 =
33
Solutions
7.6 – 2.1 = 5.5; estimate 7.5 – 2 = 5.5
28 – 3.3 = 24.7; estimate 28 – 3 = 25
6.4 – 3.04 = 3.36; estimate 6.4 – 3 = 3.4
34
Example
Find the total cost of owning this car:
monthly car payment
$256.63
monthly car insurance
$47.52
average gas per month
$95.33
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Solution
To find the total monthly cost, we need to add up all the
monthly expenses
256.63 + 47.52 + 95.33 = ?
estimate: 250 + 50 + 100 = 400
actual $399.48
36
Example
• The average temperature throughout the US, for many years
is 52.85 °F. In 1998, the warmest year, the average was
55.08 °F. How much warmer than usual was it in 1998?
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Solution
• 52.85 – 55.08 = 2.23°F
38
Example
• To send a 2 lb package by parcel post cost $4.90. To send it
express mail costs $16.30. How much more does it cost to
send the package by express mail than by regular mail?
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Solution
• To send a 2 lb package by parcel post cost $4.90. To send it
express mail costs $16.30. How much more does it cost to
send the package by express mail than by regular mail?
• 16.30 – 4.90 = $11.40
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Example; Mean Chocolate Consumption
• Which country has the most chocolate consumption per person?
• The least?
• How much more chocolate do Austrians eat than Germans?
25
23.36
Pounds of Chocolate/yr
20.13
20
19.47
18.04
17.93
Germany
Norway
15
10
5
0
Austria
Switzerland
Ireland
Country
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Solution
• Which country has the most chocolate consumption per
person? Switzerland
• The least? Norway
• How much more chocolate do Austrians eat than Germans?
2.09 lb
42
Example
The screen of the 2007 iPhone was 4.5” by 2.4”. What was
its perimeter
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Solution
The screen of the 2007 iPhone was 4.5” by 2.4”. What was
its perimeter
Perimeter = 2 L + 2 W = 2 (4.5) + 2( 2.4) =
4.5+4.5+2.4+2.4 = 13.8”
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Multiplying and Dividing Decimals
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It is like multiplying fractions:
• 0.6 x 0.03 = 6/10 x 3/100 = 18/1000 = 0.018
• Or a rule: Count the number of digits to the right of the
decimal point in the problem, and you need that many
in the answer; just multiply the numbers as if they are whole
• 0.6 x 0.03: 1 place plus 2 places = 3 places; 6 x 3 = 18, but
need three decimal places, so 0.018
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Estimation
• Figuring out how big the number should be is a helpful check
• 0.112 x 0.6 =
– You have 1/10 time 6/10, so you know it is near 6/100
• 112 x 6 = 672, but it has to be near 6/100, so 0.0672
• Or, 112/1000 x 6/100
• Or, there are four decimals in the problem, four in the answer
47
Multiplying by Power of Ten
• 23.6951 x 10 = 236.951
• 23.6891 x 0.1 = 2.36951
• Just move the decimal point to the right or left, depending on
how many decimal places there are in the power of ten
48
Examples
• 23.7 x 10 =
• 203.004 x 100 =
• 7.62 x 0.1 =
• 1.9 x 0.01 =
• 7682 x 0.001 =
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Solutions
• 23.7 x 10 = 237
• 203.004 x 100 = 20,300.4
• 7.62 x 0.1 = 0.762
• 1.9 x 0.01 = 0.019
• 7682 x 0.001 = 7.682
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Examples
• 0.7 x 0.9 =
• 6.8 x 0.3 =
• 2.0005 x 5.5 =
• 31.006 x 3.71 =
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Solutions
• 0.7 x 0.9 = 0.63
• 6.8 x 0.3 = 2.04 (estimate 7 x 0.3)
• 2.0005 x 5.5 = 11.00275 (estimate 2 x 5.5)
• 31.006 x 3.71 = 115.03226 (estimate 31 x 4)
52
Example
The circumference of a circle is 2π times the radius,
where π = 3.141549…
• Approximating π as 3.14, find the circumference of a circle
that has a diameter of 10 cm.
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Solution
• If the diameter is 10, the radius is 5
2 π r = 2 x π x 5 = 10 π = 31.4
54
Example
In 2010 the average US airplane passenger paid $0.1433 per
mile to fly. If it is 905 mils from Atlanta to Minneapolis, find the
fare. (Round to the nearest cent)
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Solution
In 2010 the average US airplane passenger paid $0.1433 per
mile to fly. If it is 905 mils from Atlanta to Minneapolis, find the
fare. (Round to the nearest cent)
0.1433 x 905 = $129.69
Estimate: 0.14 x 1000
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Dividing Decimals and Order of
Operations
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Dividing by a Whole Number
• The key is to fix the decimal point! Estimation can help
270.2 / 7
• Estimate 270/7 ~ 40; you can check your result against this
• Write the problem, but fix the decimal point above where it
should be!
xxx.xxxxx
7)270.6
• Do the long division. 7 doesn’t divide 2, 7 divides 27, 3 times,
7 divides 60, 8 times, THEN PUT THE DECIMAL POINT
270.2 / 7 = 38.6
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Examples
• 8.32 / 32 =
• 34.08 / 48 =
59
Solutions
• 8.32 / 32 = 0.26; estimate 8/32 ~ 4
• 34.08 / 48 = 0.71; estimate 34/48 ~ 3/4 = 0.75
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Dividing by a Decimal
• DON’T!!!!
• Multiply the numerator and denominator by the same power of
10 so that the denominator is no longer a decimal
• 10.764 / 2.3 = ?
Multiply top and bottom (numerator and denominator) by 10
107.64 / 23
• Then divide a decimal by a whole number
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Examples
• 17.5 / 0.48 =
1750 / 48 =
• 23.4 / 0.57=
• 272.356 / 28.4 =
62
Solution
• 17.5 / 0.48 =
1750 / 48 = 36.46; estimate 1750/50 = 175/5 = 35
• 23.4 / 0.57= 41.05; rewrite 2340 / 57; estimate 2400/60 = 240/6
• 272.356 / 28.4 = 9.59; rewrite 2723.56 / 284;
estimate 3000/300 = 30/3 = 10
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Order of Operations
• PEMDAS: Please Excuse My Dear Aunt Sally
• P: Parentheses; simplify quantities in parentheses, inside to
outside, and remove the parentheses
• E: Exponents; evaluate any expressions with exponents
• M and D: Multiplication and Division; perform any
multiplication and division from left to right
• A and S: Addition and Subtraction; perform any addition and
subtraction from left to right
• M and D are done at the same time, as are A and S;
My and Dear are adjectives, Aunt and Sally are nouns;
they stay together.
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Examples
723.6 / 1000 x 10 =
No P, no E, M & D, working from left to right
Step 1: 723.6/1000 = 0.7236
Step 2: 0.7236 x 10 = 7.236
What happens if we mix the order?
if we multiply 1000 x 10 = 10,000
723.6 / 10,000 = 0.07236: WRONG!!!
65
Example
• 0.5 ( 8.6 – 1.2) =
P first: 8.6 – 1.2 = 7.4
No E
M and D: 0.5 x 7.4 = 3.7
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Example
• [5.68 + (0.9)2 / 100 ] / 0.2 =
67
Solution
• [5.68 + (0.9)2 / 100 ] / 0.2 =
P: can’t simplify [ ] until we do some work
Inside the [ ] first
E: (0.9) 2 = 0.81
M & D: 0.81 / 100 = 0.0081
A & S: 5.68 + 0.0081 = 5.6881
Now we have [ ] evaluated
[ ] / 0.2 = 5.6881 / 0.2 = 56.881 / 2 = 28.4405
68
Example
• 897.8 / 100 x 10 =
• 8.69 ( 3.2 – 1.8) =
• [ 20.06 – (1.2)2 / 10 ] / 0.02 =
69
Solution
• 897.8 / 100 x 10 = 89.78
• 8.69 ( 3.2 – 1.8) = 12.166
• [ 20.06 – (1.2)2 / 10 ] / 0.02 = 995.8
70
Fractions as Decimals
• Writing decimals as fractions is easy: denominator is a power
of 10
• Writing fractions as decimals is a bit harder – may not have a
finite number of digits.
• To make a fraction a decimal, just divide:
1/4 = 1 ÷ 4 = 0.25
4/5 = 4 ÷ 5 = 0.8
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When it Doesn’t Come out Even
2/3 = ?
0.6666….: The … means keep writing 6 for a long as you want;
it will never end
_
You can also write 0.6 = 0.66…
72
Example
Write as a decimal, round to nearest thousandth:
• 28/13 =
• 2 3/16 =
• 3 5/16 =
• 7/9 =
73
Solution
• 28/13 = 2.154
• 2 3/16 = 2.188
• 3 5/16 = 3.313
• 7/9 = 0.77777… = 0.8
74
Which is Larger?
• 1/2 or 0.54?
• 0.44… or 4/9?
• 5/7 or 0.72?
75
Solution
• 1/2 or 0.54?
• 0.44… or 4/9? Same
• 5/7 or 0.72?
76
Summary
Decimal Arithmetic:
• Estimate the result
• Multiplication:
– You can make decimals fractions and multiply, or
– Count the number of decimal places (riskier)
• Division:
– When dividing by a hole number, mark the decimal point
– When dividing by decimals, GET RID OF THE DECIMALS
IN THE DENOMINATOR!
77
Summary Cont.
Order of operations
• Clear parentheses, inside to out
• Clear exponents
• Multiplication and Division, from left to right
• Addition and Subtraction, from left to right
PEMDAS: Please Excuse My Dear Aunt Sally
M and D go together (adjectives)
Aunt and Sally go together (nouns)
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Solution
• 8.32 / 32
• 34.08 / 48
79