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Lesson 31 ! page 1
Lesson 31
Decimal Fractions
Expressing parts of a whole is a mathematical problem with several solutions. One way to do this is the ordinary fractions
(classically called the “common” or “vulgar” fractions) we have been using in this book so far. Another way to express parts
of a whole is with decimals.
U.S. currency uses a decimal system, dollars and cents. Calculators default to decimals when displaying results. Decimals
form the basis of the metric system. Many people feel more comfortable with decimal representations than with fractions.
Decimals have many advantages – and some disadvantages we will also explore.
Place Value Revisited
The number system we use for whole numbers is called a Place Value system, because the placement of the digit in the
number tells its value. It is a Base Ten system, because each place is worth ten times the next place to the right.1
For example, in the number 5,432,106 each digit has a place that defines its value. The rightmost digit is in the ones
place, so we have 6 ones. The next digit, 0, holds the tens place – it tells us that there are no tens. We need the zero,
because otherwise the 1 in the hundreds place would mean 1 ten rather than 1 hundred. Each place value is a power of 10,
because it comes from multiplying the place to the right by 10.
108
107
millions
106
105
104
thousands
103
102
101
1
hundreds
tens
ones
hundreds
tens
ones
hundreds
tens
ones
5,
4
3
2,
1
0
6
The big idea for decimal fractions is to expand the place value system to the right, using the same principle that each place
value is ten times the one to the right. Ten times the place to the right means the same as 1/10 the place to the left, so after
the ones place we have the fractional places 1/10, 1/100, 1/1000, etc. We write a decimal point after the ones place to
indicate that the digits to the right represent fractional parts of a whole.
103
102
101
1
thousands
hundreds
tens
ones
2,
1
0
6.
tenths
hundredths
thousandths
0
1
2
1/101
1/102
1/103
The ones place is highlighted above because it is the center of the number names. To the left of the ones place is the tens,
to the right is the tenths. The number names proceed in the same order to the right as they did to the left, but the name is
followed by the suffix “th” to indicate a fraction.
Say It, Don’t Spray It
Careful with those “th”s.
When you’re writing a decimal number with no whole number part, you properly include a
zero in the ones place.
0.012
In practice, however, many people leave the leading zero off:
0.012
Use your judgment about the requirements of a particular situation.
The names also tell us that other systems are possible! Think about Roman numerals (not a place value system), or about the binary
numbers used in computers (not a base ten system).
1
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 31 ! page 2
Reading and Writing Decimals
To read a decimal number properly, say “and” for the decimal point, the regular name for the number after the decimal point,
then the place value of the last digit on the right. The number,
2,106.012
is formally read “two thousand, one hundred six and twelve thousandths”
The formal name is rarely used, (most people would say “point oh one two” rather than “and twelve thousandths”) but it is
helpful because it reveals that the decimal is really a fraction.
Let’s break down the decimal 0.012 into its parts to see how it is equal to twelve thousandths, that is 12/1000.
103
102
101
1
thousands
hundreds
tens
ones
2,
1
0
6.
tenths
hundredths
thousandths
0
1
2
1/101
1/102
1/103
Just as the 2 in the thousands place means two groups of a thousand, the 2 in the thousandths place means there are two
parts of size 1/1000. In expanded form, this number is equal to
! 1$
! 1 $
! 1 $
2 1000 + 1 100 + 0 10 + 6 1 + 0 # & + 1#
+
2
#" 1000 &%
" 10 %
" 100 &%
(
) (
) ( ) ()
The fraction part (after the decimal point) is
! 1$
! 1 $
! 1 $
0 # & + 1#
+ 2#
&
" 10 %
" 100 %
" 1000 &%
1
2
1 10
2
+
=
•
+
100 1000 100 10 1000
10
2
12
=
+
=
1000 1000 1000
= 0+
Example: Fill in the missing columns in the chart.
Decimal
Formal Name
0.97
Fraction
“ninety-seven hundredths”
97
100
25.017
0.3
“twenty-five and seventeen thousandths”
“three tenths”
25
17
1000
3
10
© 2010 Cheryl Wilcox
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Lesson 31 ! page 3
From Fraction to Decimal
If a fraction has a denominator that is a power of ten, you can write it as a decimal just by filling in the numerator in the place
value chart with the rightmost digit in the place value of the denominator.
For example, the fraction 7/100 has a denominator of 100. Find the hundredths place on the chart and write 7 there:
1
ones
tenths
hundredths
1/101
1/102
0.
thousandths
tenthousandths
hundredthousandths
1/103
1/104
1/105
7
Then flll in any zeros that come between the decimal point and the numerator.
1
ones
tenths
0.
hundredths
0
7
1/101
1/102
thousandths
tenthousandths
hundredthousandths
1/103
1/104
1/105
7
= 0.07
100
Zeros to the right of the last non-zero digit do not change the value of the number. The number 0.07 is different from 0.7, but
0.07 and 0.0700 are equivalent.
Example: Write the fractions as decimals. Use the place value chart for reference.
1
ones
tenths
hundredths
thousandths
tenthousandths
hundredthousandths
1/10
1/100
1/1000
1/10.000
1/100,000
.
19
= 0.19
100
19
= 0.019
1000
199
= 0.199
1000
199
= 0.00199
100,000
Suppose you have just any random fraction, without a special power of ten denominator. To change any fraction to a
decimal, we read the fraction bar as a division symbol, and do long division or divide on a calculator.
1
= 1÷ 4
4
0.25
4 1.00
0.8
0.20
0.20
0.00
or
Notice that if you change 1/4 to an equivalent fraction with denominator 100, you get the same result.
1 1 25 25
= •
=
= 0.25
4 4 25 100
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 31 ! page 4
Example: Write the fractions as decimals.
3
= 3 ÷ 5 = 0.6
5
21
= 21÷ 25 = 0.84
25
7
= 7 ÷ 20 = 0.35
20
27
= 27 ÷ 50 = 0.54
50
Sometimes the division does not work out evenly, resulting in what we call a repeating decimal. For example,
1
= 1÷ 3
3
0.333
3 1.000
0.9
0.100
0.090
0.010
You can see from the long division that it can never terminate.
The calculator displays as many 3s as it can fit onto the screen, but it
cannot show infinitely many 3s as it should.
To write 3s on to infinity is impossible, so we have three choices:
We can round the decimal to some predetermined place value. For example, your calculator rounds repeating
decimals to the number of digits in its display.
We can write a few 3s, followed by three dots (ellipses) that mean “and so forth.” 0.3333… This notation is
considered casual, and somewhat unmathematical. It can be confusing in some circumstances because it is not
specified exactly what the repeating part is.
The correct mathematical notation is to put a bar over the repeating part of the decimal. This is difficult to type but
easy to write by hand.
0.3 = 0.33333333333333333333333333333333...
When you change a fraction to a decimal, it will either terminate, as 1/4 did, or repeat, as 1/3 did. However, sometimes the
repeating part has one or more leading digits before it begins, and sometimes the repeating pattern has more than one digit.
Example: Write the fractions as decimals.
2
= 2 ÷ 3 = 0.6
3
1
= 1÷ 6 = 0.16
6
5
= 5 ÷ 11= 0.45
11
41
= 41÷ 90 = 0.45
90
Did you notice that your
calculator rounded up
because the last digit is
greater than 5?
The one in the tenths place
does not repeat.The
repeating part of the decimal
begins at the 6. Notice how
your calculator rounds the
last digit to 7.
Here the two digit pattern 45
repeats forever.
Compare this to the previous
example. Here the four in
the tenths place does not
repeat. The repeating
pattern of 5s begins in the
hundredths place.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 31 ! page 5
Summary (and More)
Decimals are an alternate way to write fractions of a whole amount. Decimal notation is equivalent to writing a fraction with a
denominator that is a power of ten.
To convert a decimal to a fraction, write the number after the decimal point as the numerator and use the place value of
the rightmost digit as the denominator.
0.0047 =
0047
47
=
10000 10,000
A quick way to calculate the denominator is to write the numeral 1 followed by as many zeros as there are places after the
decimal point. You can see in this example the decimal point is followed by four digits and the denominator of the fraction
has four zeros.
To convert a fraction to a decimal, interpret the fraction bar as a division symbol. If the denominator of the fraction has
any factors other than 2 and 5 (the factors of 10), the division will not be even, and the decimal will have a repeating part.
5
5
=
22 2 • 11
5
= 5 ÷ 22 = 0.227
22
If the fraction can be converted to an equivalent fraction with denominator equal to a power of ten, then the decimal will
terminate. (Convert by creating 2•5 combinations.)
3
3
5 • 5 • 5 375
=
•
=
8 2 • 2 • 2 5 • 5 • 5 1000
3
= 3 ÷ 8 = 0.375
8
If you are converting a mixed number, concentrate on the fraction part. You can append the whole number part before the
decimal point once you’ve converted.
302
(
)
43
= 302 + 43 ÷ 99 = 302 + 0.43 = 302.43
99
The disadvantage of decimal representations is the loss of precision. While the infinitely repeating form of a decimal fraction
is precise, for practical purposes we must round repeating decimals before we can use them.
The advantage of decimal representations lies in the ease of arithmetic operations, as we’ll see in the next section. Because
we are working in our familiar place value system, the arithmetic operations are similar to those with whole numbers, rather
than the special rules for fractions.
!
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 31 ! page 6
Lesson 31: Decimal Fractions
Worksheet
Name _________________________________
1
ones
tenths
hundredths
thousandths
tenthousandths
hundredthousandths
1/10
1/100
1/1000
1/10.000
1/100,000
.
Fill in the missing columns in the chart. Use the place value chart for reference.
Decimal
Formal Name
Fraction
5.07
“three hundred eighty-one ten-thousandths”
13
Write the decimals as fractions. Simplify the fractions to lowest terms.
5.2
5.02
5.20
0.502
Write the fractions as decimals. Use a bar to indicate repeating decimals.
33
100
9
10
9
100
1
1000
8
25
39
40
49
90
6
11
1
9
2
9
1
3
7
9
© 2010 Cheryl Wilcox
33
100
Free Pre-Algebra
Lesson 31 ! page 7
This ruler has been enlarged so you can see the marks clearly. The true size is shown at the bottom of the page.
The upper part of the ruler, marked in inches, is divided into eighths.
The lower part of the ruler, marked in centimeters, is divided into tenths.
Write both the fraction and the decimal name for each point.
The convention for centimeters is to use tenths rather than simplifying to lower terms (write 5/10 instead of 1/2 for cm).
True Size
© 2010 Cheryl Wilcox
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Lesson 31 ! page 8
Lesson 31: Decimal Fractions
Homework 31A
1. a. Round 76,939 to the nearest
thousand.
b. Round 4,377,299 to the nearest
million.
23a 2
207a
Name __________________________________
2. Write four fractions equivalent to
3. Find the prime factorization of 207.
9
25
" 7
19 %
x! '
5. Simplify 60 $
20 &
# 60
6. Evaluate
7. Find equivalent fractions with a
common denominator.
8. Add, and write the answer as a
mixed number.
9. Subtract.
3
8
3 5
+
8 6
7
4. Simplify
a
8
when a = ! .
8
9
3
5
!2
8
6
5
6
10. Convert 50 inches to feet. (12 inches = 1 foot)
11. How many cups is 18 gallons?
(1 gallon = 4 quarts, 1 quart = 4 cups)
12. The height (in feet, after t seconds) of a rock thrown
down a deep well is given by the equation
h = !16t 2 ! 24t .
The bottom of the well is at –500 feet.
13. Convert feet to inches, then use the distance-rate-time
formula.
Has the rock already hit the bottom when t = 5 seconds?
© 2010 Cheryl Wilcox
A model train ran at the rate of 3 inches per second. How
long does it take to travel a 6 foot track?
Free Pre-Algebra
Lesson 31 ! page 9
Solve the equations.
14.
4
x = 12
9
16.
3
5 9
x! =
7
7 7
15. !7 y + 4 = 11
17. x +
3
=8
4
18. Fill in the blanks.
Decimal
Formal Name
Fraction
0.19
“two and four hundredths”
3
19. Write the decimals as fractions or mixed numbers. Simplify to lowest terms.
0.005
3.11
7.125
20. Write the fractions as decimals. Use a bar for repeating decimals.
1
2
1
3
2
3
1
4
1
5
1
6
1
7
1
8
1
9
© 2010 Cheryl Wilcox
4
10
Free Pre-Algebra
Lesson 31 ! page 10
Lesson 31: Decimal Fractions
Homework 31A Answers
1. a. Round 76,939 to the nearest
thousand.
77,000
b. Round 4,377,299 to the nearest
million.
2. Write four fractions equivalent to
3. Find the prime factorization of 207.
207 = 3 • 3 • 23
9
18 27 36
45
=
=
=
=
25 50 75 100 125
4,000,000
" 7
23a 2
4. Simplify
207a
=
19 %
x! '
5. Simplify 60 $
20 &
# 60
23 • a • a
=
3 • 3 • 23 • a
a
9
60 •
7
60
3
x ! 60 •
19
20
6. Evaluate
= 7x ! 57
8
9 = !8 ÷8= ! 8 • 1 = !1
8
9
9 8
9
!
7. Find equivalent fractions with a
common denominator.
8. Add, and write the answer as a mixed
number.
9. Subtract.
3
3
3 9
=
• =
8 2 • 2 • 2 3 24
3 5
9 20 29
5
+ =
+
=
=1
8 6 24 24 24
24
7
3
5
9
20
!2 =7
!2
8
6
24
24
=6
5
5 2 • 2 20
=
•
=
6 2 • 3 2 • 2 24
10. Convert 50 inches to feet. (12 inches = 1 foot)
50 in
1 ft
50
2
1
•
=
ft = 4
ft = 4 ft
1
12
6
12 in 12
12. The height (in feet, after t seconds) of a rock thrown
down a deep well is given by the equation
h = !16t 2 ! 24t .
The bottom of the well is at –500 feet.
Has the rock already hit the bottom when t = 5 seconds?
()
2
()
h = !16 5 ! 24 5
= !16 • 25 ! 24 • 5 = !429
No, it is still above –500 feet.
© 2010 Cheryl Wilcox
a
8
when a = ! .
8
9
33
20
13
!2
=4
24
24
24
11. How many cups is 18 gallons?
(1 gallon = 4 quarts, 1 quart = 4 cups)
18 gal
1
•
4 qts
1 gal
•
4 cups
1 qt
= 288 cups
13. Convert feet to inches, then use the distance-rate-time
formula.
A model train ran at the rate of 3 inches per second. How
long does it take to travel a 6 foot track?
6 ft 12 in
•
= 72 in
1
1 ft
72 = 3t
d = rt
72 = 3t
3t / 3 = 72 / 3
t = 24 seconds
Free Pre-Algebra
Lesson 31 ! page 11
Solve the equations.
14.
15. !7 y + 4 = 11
4
x = 12
9
4
x = 12
9
!7 y + 4 = 11
!7 y = 7
3
9 4
9
• x = 12 •
4 9
4
! 7 y + 4 ! 4 = 11! 4
! 7 y / !7 = 7 / !7
y = !1
x = 27
16.
3
5 9
x! =
7
7 7
17. x +
3x ! 5 = 9
3x = 14
x=
3x ! 5 + 5 = 9 + 5
3
=8
4
3
3 3
3
=8
x + ! = 8!
4
4 4
4
3
4 3
1
x = 8! =7 ! =7
4
4 4
4
x+
3x / 3 = 14 / 3
14
3
18. Fill in the blanks.
Decimal
Formal Name
0.19
Fraction
“nineteen hundredths”
19
100
2.04
3.4
“two and four hundredths”
2
“three and four tenths”
5
1
=
1000 200
3.11 = 3
11
100
7.125 = 7
20. Write the fractions as decimals. Use a bar for repeating decimals.
1
= 0.5
2
1
= 0.3
3
2
= 0.6
3
1
= 0.25
4
1
= 0.2
5
1
= 0.16
6
1
= 0.142857
7
1
= 0.125
8
1
= 0.1
9
© 2010 Cheryl Wilcox
100
3
19. Write the decimals as fractions or mixed numbers. Simplify to lowest terms.
0.005 =
4
125
1
=7
1000
8
4
10
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Lesson 31 ! page 12
Lesson 31: Decimal Fractions
Homework 31B
1. a. Round 509,887 to the nearest
ten thousand.
b. Round 84,087,361 to the nearest
million.
4. Simplify
38a 2b
418a
3
7. Find equivalent fractions with a
common denominator.
2
3
4
7
Name _________________________________________
2. Write four fractions equivalent to
3. Find the prime factorization of 418.
5
18
" 7
9%
y! '
5. Simplify 50 $
10 &
# 25
6. Evaluate
8. Subtract.
9. Add.
2 4
!
3 7
5
3
!1
when x =
.
x
5
2
4
+8
3
7
10. Convert 85 inches to feet. (12 inches = 1 foot)
11. How many minutes is 14 days?
(1 day = 24 hours, 1 hour = 60 minutes)
12. The height (in feet, after t seconds) of a rock thrown
down a deep well is given by the equation
h = !16t 2 ! 24t .
The bottom of the well is at –500 feet.
13. Convert 8 feet to inches, then use the distance-rate-time
formula.
Has the rock already hit the bottom when t = 6 seconds?
© 2010 Cheryl Wilcox
A model train ran at the rate of 3 inches per second. How
long does it take to travel an 8 foot track?
Free Pre-Algebra
Lesson 31 ! page 13
Solve the equations.
14. 8n + 14 = 12
16.
11
9
=x!
4
2
15.
2
y = 16
3
17.
7
3 10
w+
=
15
15 15
18. Fill in the blanks.
Decimal
Formal Name
Fraction
0.009
“three and seven tenths”
10
19. Write the decimals as fractions or mixed numbers. Simplify to lowest terms.
0.012
5.75
7.375
20. Write the fractions as decimals. Use a bar for repeating decimals.
1
11
10
11
1
12
11
12
1
40
39
40
7
9
7
10
77
100
© 2010 Cheryl Wilcox
23
1000