decimals, addition and subtraction of decimalsecimals, addition and subtraction of decimals

1. Decimals

2. Decimals
In order to track smaller and smaller quantities we include base-
10 fractions to the whole number-system.

3. $100’s $1’s$10’s
Decimals
In order to track smaller and smaller quantities we include base-
10 fractions to the whole number-system. Let’s demonstrate
this with a cash register that holds $1’s, $10’s, $100’s, ...etc.
*

4. $100’s $1’s$10’s
Decimals
In order to track smaller and smaller quantities we include base-
10 fractions to the whole number-system. Let’s demonstrate
this with a cash register that holds $1’s, $10’s, $100’s, ...etc.
For a moment let’s assume that US Treasury not only makes
*
(dime),
10
1$ (penny),
100
1$
, a “itty”, and
1000
1
$ , a “bitty”, etc...
10000
1$
but also makes smaller value coins
of

5. $100’s $1’s$10’s
Decimals
In order to track smaller and smaller quantities we include base-
10 fractions to the whole number-system. Let’s demonstrate
this with a cash register that holds $1’s, $10’s, $100’s, ...etc.
For a moment let’s assume that US Treasury not only makes
*
dimes
10
1
$
pennies itties
100
1
$ 1000
1
$
(dime),
10
1$ (penny),
100
1$
, a “itty”, and
1000
1
$ , a “bitty”, etc...
10000
1$
but also makes smaller value coins
10000
1
$
of
bitties
*

6. $100’s $1’s$10’s
Decimals
In order to track smaller and smaller quantities we include base-
10 fractions to the whole number-system. Let’s demonstrate
this with a cash register that holds $1’s, $10’s, $100’s, ...etc.
For a moment let’s assume that US Treasury not only makes
* 10
1
$ 100
1
$ 1000
1
$
(dime),
10
1$ (penny),
100
1$
, a “itty”, and
1000
1
$ , a “bitty”, etc...
10000
1$
but also makes smaller value coins
10000
1
$
of
Note that 10 itties = 1 penny and 10 bitties = 1 itty, etc...
*
dimes pennies itties bitties

7. $100’s $1’s$10’s
Decimals
In order to track smaller and smaller quantities we include base-
10 fractions to the whole number-system. Let’s demonstrate
this with a cash register that holds $1’s, $10’s, $100’s, ...etc.
For a moment let’s assume that US Treasury not only makes
* 10
1
$ 100
1
$ 1000
1
$
(dime),
10
1$ (penny),
100
1$
, a “itty”, and
1000
1
$ , a “bitty”, etc...
10000
1$
but also makes smaller value coins
Let’s further assume each slot only hold up to 9 bills or coins
so we may record the money stored in the register
10000
1
$
of
Note that 10 itties = 1 penny and 10 bitties = 1 itty, etc...
*
dimes pennies itties bitties

8. $100’s $1’s$10’s
Decimals
In order to track smaller and smaller quantities we include base-
10 fractions to the whole number-system. Let’s demonstrate
this with a cash register that holds $1’s, $10’s, $100’s, ...etc.
For a moment let’s assume that US Treasury not only makes
* 10
1
$ 100
1
$ 1000
1
$
(dime),
10
1$ (penny),
100
1$
, a “itty”, and
1000
1
$ , a “bitty”, etc...
10000
1$
# # # # ##
but also makes smaller value coins
Let’s further assume each slot only hold up to 9 bills or coins
so we may record the money stored in the register
10000
1
$
of
Note that 10 itties = 1 penny and 10 bitties = 1 itty, etc...
#
*
dimes pennies itties bitties

9. $100’s $1’s$10’s
Decimals
In order to track smaller and smaller quantities we include base-
10 fractions to the whole number-system. Let’s demonstrate
this with a cash register that holds $1’s, $10’s, $100’s, ...etc.
For a moment let’s assume that US Treasury not only makes
* 10
1
$ 100
1
$ 1000
1
$
(dime),
10
1$ (penny),
100
1$
, a “itty”, and
1000
1
$ , a “bitty”, etc...
10000
1$
# # # # ##
simply as . # # # # where the #’s = 0,1,.., or 9.# # #
The decimal point (the divider)
but also makes smaller value coins
Let’s further assume each slot only hold up to 9 bills or coins
so we may record the money stored in the register
10000
1
$
of
Note that 10 itties = 1 penny and 10 bitties = 1 itty, etc...
#
*
dimes pennies itties bitties
.

10. $100’s* $1’s$10’s* 10
1
$ 100
1
$ 1000
1
$
4 5 63
For example,
.
Decimals
dimes pennies itties bitties

11. $100’s* $1’s$10’s*
dimes
10
1
$
pennies itties
100
1
$ 1000
1
$
4 5 63
For example,
.43 5 6is written as
.
Decimals
bitties

12. $100’s* $1’s$10’s*
dimes
10
1
$
pennies itties
100
1
$ 1000
1
$
4 5 63
For example,
.43 5 6is written as
.
4 $1’s
3 $10’s
Decimals
bitties

13. $100’s* $1’s$10’s*
dimes
10
1
$
pennies itties
100
1
$ 1000
1
$
4 5 63
For example,
.43 5 6is written as
.
4 $1’s
3 $10’s
(5 dimes)
(6 pennies)
10
5
100
6$
$
Decimals
bitties

14. $100’s* $1’s$10’s*
dimes
10
1
$
pennies itties
100
1
$ 1000
1
$
4 5 63
For example,
.43 5 6is written as
.
4 $1’s
3 $10’s
(5 dimes)
(6 pennies)
10
5
100
6$
$
$100’s $1’s$10’s* 10
1
$ 100
1
$ 1000
1
$
4 5 0 7
Decimals
8
10000
1$
bitties
.

15. $100’s* $1’s$10’s*
dimes
10
1
$
pennies itties
100
1
$ 1000
1
$
4 5 63
For example,
.43 5 6is written as
.
4 $1’s
3 $10’s
(5 dimes)
(6 pennies)
10
5
100
6$
$
$100’s $1’s$10’s* 10
1
$ 100
1
$ 1000
1
$
4 5 0 7
4 $1’s
4is written as .
Decimals
8
10000
1$
bitties
.

16. $100’s* $1’s$10’s*
dimes
10
1
$
pennies itties
100
1
$ 1000
1
$
4 5 63
For example,
.43 5 6is written as
.
4 $1’s
3 $10’s
(5 dimes)
(6 pennies)
10
5
100
6$
$
$100’s $1’s$10’s*
dimes
10
1
$
pennies itties
100
1
$ 1000
1
$
4 5 0 7
4 $1’s
(no penny)
100
0$
(5 dimes)10
5$
1000
7$
4 75 0is written as .
Decimals
8
10000
1$
(8 bitties)
10000
8
$
bitties
.
8
(7 itties)

17. Comparing Decimal Numbers
Decimals
Because decimal numbers may be viewed as coins stored in a
base-10 cash registers, therefore to determine which decimal
numbers is the largest is similar to finding which cash register
contains more money (in coins.)

18. Comparing Decimal Numbers
Example A.
List 0.0098, 0.010, 0.00199
from the largest to the smallest.
Decimals
Because decimal numbers may be viewed as coins stored in a
base-10 cash registers, therefore to determine which decimal
numbers is the largest is similar to finding which cash register
contains more money (in coins.)

19. Comparing Decimal Numbers
1. line up the numbers by their decimal points,
Example A.
List 0.0098, 0.010, 0.00199
from the largest to the smallest.
Decimals
Because decimal numbers may be viewed as coins stored in a
base-10 cash registers, therefore to determine which decimal
numbers is the largest is similar to finding which cash register
contains more money (in coins.) Specifically, to compare
multiple decimal numbers to see which is largest, we

20. Comparing Decimal Numbers
1. line up the numbers by their decimal points,
Example A.
List 0.0098, 0.010, 0.00199
from the largest to the smallest.
Decimals
Because decimal numbers may be viewed as coins stored in a
base-10 cash registers, therefore to determine which decimal
numbers is the largest is similar to finding which cash register
contains more money (in coins.) Specifically, to compare
multiple decimal numbers to see which is largest, we
0.0098
0.010
0.00199
1. line up by the decimal points

21. Comparing Decimal Numbers
1. line up the numbers by their decimal points,
2. scan the digits, i.e. the number of coins, in the slot from
left to right,
Example A.
List 0.0098, 0.010, 0.00199
from the largest to the smallest.
Decimals
Because decimal numbers may be viewed as coins stored in a
base-10 cash registers, therefore to determine which decimal
numbers is the largest is similar to finding which cash register
contains more money (in coins.) Specifically, to compare
multiple decimal numbers to see which is largest, we
0.0098
0.010
0.00199
1. line up by the decimal points

22. Comparing Decimal Numbers
1. line up the numbers by their decimal points,
2. scan the digits, i.e. the number of coins, in the slot from
left to right,
Example A.
List 0.0098, 0.010, 0.00199
from the largest to the smallest.
Decimals
Because decimal numbers may be viewed as coins stored in a
base-10 cash registers, therefore to determine which decimal
numbers is the largest is similar to finding which cash register
contains more money (in coins.) Specifically, to compare
multiple decimal numbers to see which is largest, we
0.0098
0.010
0.00199
1. line up by the decimal points
2. scan the digits in each slot from
left to right

23. Comparing Decimal Numbers
1. line up the numbers by their decimal points,
2. scan the digits, i.e. the number of coins, in the slot from
left to right,
3. the one with the 1st largest digit is the largest quantity.
Example A.
List 0.0098, 0.010, 0.00199
from the largest to the smallest.
Decimals
Because decimal numbers may be viewed as coins stored in a
base-10 cash registers, therefore to determine which decimal
numbers is the largest is similar to finding which cash register
contains more money (in coins.) Specifically, to compare
multiple decimal numbers to see which is largest, we
0.0098
0.010
0.00199
1. line up by the decimal points
2. scan the digits in each slot from
left to right

24. Comparing Decimal Numbers
1. line up the numbers by their decimal points,
2. scan the digits, i.e. the number of coins, in the slot from
left to right,
3. the one with the 1st largest digit is the largest quantity.
Example A.
List 0.0098, 0.010, 0.00199
from the largest to the smallest.
Decimals
Because decimal numbers may be viewed as coins stored in a
base-10 cash registers, therefore to determine which decimal
numbers is the largest is similar to finding which cash register
contains more money (in coins.) Specifically, to compare
multiple decimal numbers to see which is largest, we
0.0098
0.010
0.00199
1. line up by the decimal points
2. scan the digits in each slot from
left to right
1st largest digit, so it’s
the largest number

25. Comparing Decimal Numbers
1. line up the numbers by their decimal points,
2. scan the digits, i.e. the number of coins, in the slot from
left to right,
3. the one with the 1st largest digit is the largest quantity.
Example A.
List 0.0098, 0.010, 0.00199
from the largest to the smallest.
Decimals
Because decimal numbers may be viewed as coins stored in a
base-10 cash registers, therefore to determine which decimal
numbers is the largest is similar to finding which cash register
contains more money (in coins.) Specifically, to compare
multiple decimal numbers to see which is largest, we
0.0098
0.010
0.00199
1. line up by the decimal points
2. scan the digits in each slot from
left to right
1st largest digit, so it’s
the largest number
2nd largest digit, so it’s
the 2nd largest number

26. Comparing Decimal Numbers
1. line up the numbers by their decimal points,
2. scan the digits, i.e. the number of coins, in the slot from
left to right,
3. the one with the 1st largest digit is the largest quantity.
Example A.
List 0.0098, 0.010, 0.00199
from the largest to the smallest.
Decimals
Because decimal numbers may be viewed as coins stored in a
base-10 cash registers, therefore to determine which decimal
numbers is the largest is similar to finding which cash register
contains more money (in coins.) Specifically, to compare
multiple decimal numbers to see which is largest, we
0.0098
0.010
0.00199
1. line up by the decimal points
2. scan the digits in each slot from
left to right
1st largest digit, so it’s
the largest number
2nd largest digit, so it’s
the 2nd largest number
So listing them from the largest
to the smallest, we have:
0.010, 0.0098, 0.00199.

27. Here are the official names of some of the base-10-denominator
fractions. Note the suffix “ ’th ” at the end their names.
1’s 10 100 1,000 10,000 100,000
1 1 1 1 1
1,000,000
1
Decimals
10’s
ones tenths hundredths thousandths
ten–
thousandths
Decimal point
hundred–
thousandths millionths
.tens

28. Here are the official names of some of the base-10-denominator
fractions. Note the suffix “ ’th ” at the end their names.
1’s 10 100 1,000 10,000
ones tenths hundredths thousandths
ten–
thousandths
100,000
Decimal point
hundred–
thousandths millionths
.
1 1 1 1 1
1,000,000
1
Hence
2 . 3 4 5 6 7
is 2 +
10 100 1,000 10,000 100,000
3 4 5 6 7
+ + + +
Three
tenths
Four
hundredths
Five
thousandths
Six
ten-
thousandths
Seven
hundred-
thousandth
Decimals
10’s
tens
Two

29. Here are the official names of some of the base-10-denominator
fractions. Note the suffix “ ’th ” at the end their names.
In fraction it’s 2100,000
34,567
Decimals
Hence
2 . 3 4 5 6 7
is 2 +
10 100 1,000 10,000 100,000
3 4 5 6 7
+ + + +
Three
tenths
Four
hundredths
Five
thousandths
Six
ten-
thousandths
Seven
hundred-
thousandth
1’s 10 100 1,000 10,000
ones tenths hundredths thousandths
ten–
thousandths
100,000
Decimal point
hundred–
thousandths millionths
.
1 1 1 1 1
1,000,000
110’s
tens
.
Two

30. Fact About Shifting the Decimal Points in a Fraction
Decimals

31. Fact About Shifting the Decimal Points in a Fraction
Given a fraction, if the decimal points of the numerator and the
denominator are shifted in the same direction with the same
number of spaces, the resulting fraction is an equivalent
fraction, i.e. it’s the same.
Decimals

32. Fact About Shifting the Decimal Points in a Fraction
Example A. a. Convert the following fractions to decimals.
100
3
Decimals
Given a fraction, if the decimal points of the numerator and the
denominator are shifted in the same direction with the same
number of spaces, the resulting fraction is an equivalent
fraction, i.e. it’s the same.

33. Fact About Shifting the Decimal Points in a Fraction
Example A. a. Convert the following fractions to decimals.
100
3
To change from base-10-denominator fractions to decimals,
1. line up the top and bottom decimal points,
Decimals
Given a fraction, if the decimal points of the numerator and the
denominator are shifted in the same direction with the same
number of spaces, the resulting fraction is an equivalent
fraction, i.e. it’s the same.

34. Fact About Shifting the Decimal Points in a Fraction
Example A. a. Convert the following fractions to decimals.
100
3
= 100
3
1. Line up the
decimal points.
To change from base-10-denominator fractions to decimals,
1. line up the top and bottom decimal points,
.
.
Decimals
Given a fraction, if the decimal points of the numerator and the
denominator are shifted in the same direction with the same
number of spaces, the resulting fraction is an equivalent
fraction, i.e. it’s the same.

35. Fact About Shifting the Decimal Points in a Fraction
Example A. a. Convert the following fractions to decimals.
100
3
= 100
3
1. Line up the
decimal points.
To change from base-10-denominator fractions to decimals,
1. line up the top and bottom decimal points,
2. slide the pair of points in tandem left to behind the 1 in the
denominator, and pack 0’s in the skipped slots in the numerator.
.
.
Decimals
Given a fraction, if the decimal points of the numerator and the
denominator are shifted in the same direction with the same
number of spaces, the resulting fraction is an equivalent
fraction, i.e. it’s the same.

36. Fact About Shifting the Decimal Points in a Fraction
Example A. a. Convert the following fractions to decimals.
100
3
= 100
3
=
1. Line up the
decimal points.
To change from base-10-denominator fractions to decimals,
1. line up the top and bottom decimal points,
2. slide the pair of points in tandem left to behind the 1 in the
denominator, and pack 0’s in the skipped slots in the numerator.
1.00
0.03
2. Move the pair of points in tandem until the
denominator is 1 and pack 0’s in the skipped
slots in the numerator.
.
.
.
.
Decimals
Given a fraction, if the decimal points of the numerator and the
denominator are shifted in the same direction with the same
number of spaces, the resulting fraction is an equivalent
fraction, i.e. it’s the same.

37. Fact About Shifting the Decimal Points in a Fraction
Example A. a. Convert the following fractions to decimals.
100
3
= 100
3
=
1. Line up the
decimal points.
To change from base-10-denominator fractions to decimals,
1. line up the top and bottom decimal points,
2. slide the pair of points in tandem left to behind the 1 in the
denominator, and pack 0’s in the skipped slots in the numerator.
1.00
0.03
2. Move the pair of points in tandem until the
denominator is 1 and pack 0’s in the skipped
slots in the numerator.
.
.
.
.
The new numerator is the decimal form of the fraction.
Decimals
Given a fraction, if the decimal points of the numerator and the
denominator are shifted in the same direction with the same
number of spaces, the resulting fraction is an equivalent
fraction, i.e. it’s the same.

38. 1
0.03
Fact About Shifting the Decimal Points in a Fraction
Example A. a. Convert the following fractions to decimals.
100
3
= 100
3
=
1. Line up the
decimal points.
To change from base-10-denominator fractions to decimals,
1. line up the top and bottom decimal points,
2. slide the pair of points in tandem left to behind the 1 in the
denominator, and pack 0’s in the skipped slots in the numerator.
1.00
0.03 =
2. Move the pair of points in tandem until the
denominator is 1 and pack 0’s in the skipped
slots in the numerator.
.
.
.
. = 0.03
The new numerator is the decimal form of the fraction.
The decimal form
of the fraction
Decimals
Given a fraction, if the decimal points of the numerator and the
denominator are shifted in the same direction with the same
number of spaces, the resulting fraction is an equivalent
fraction, i.e. it’s the same.

39. b. Convert the following fractions to decimals.
10000
16.35
Decimals

40. b. Convert the following fractions to decimals.
10000
16.35
=
10000
16.35
.
1. Line up the
decimal points.
Decimals

41. b. Convert the following fractions to decimals.
10000
16.35
= =
10000
16.35
. 1 0000
0.0016 35
.
.
.
1. Line up the
decimal points. 2. Move the pair of points in tandem until
the denominator is 1 and pack 0’s in the
skipped slots in the numerator.
Decimals
.

42. b. Convert the following fractions to decimals.
10000
16.35
= =
10000
16.35
. 1 0000
0.0016 35
.
.
. = 0.001635
1. Line up the
decimal points. 2. Move the pair of points in tandem until
the denominator is 1 and pack 0’s in the
skipped slots in the numerator.
Decimals
.
The decimal form
of the fraction

43. b. Convert the following fractions to decimals.
10000
16.35
= =
10000
16.35
. 1 0000
0.0016 35
.
.
. = 0.001635
1. Line up the
decimal points. 2. Move the pair of points in tandem until
the denominator is 1 and pack 0’s in the
skipped slots in the numerator.
Recall that x =
x
1.
Decimals
.
The decimal form
of the fraction

44. To change a decimal number of the form
b. Convert the following fractions to decimals.
10000
16.35
= =
10000
16.35
. 1 0000
0.0016 35
.
.
. = 0.001635
1. Line up the
decimal points. 2. Move the pair of points in tandem until
the denominator is 1 and pack 0’s in the
skipped slots in the numerator.
0 . # # # # to a fraction:
Recall that x =
x
1.
Decimals
.
1. Put “1.” in the denominator and line up the decimal points.
0 . # # # #
1 . =
The decimal form
of the fraction

45. To change a decimal number of the form
b. Convert the following fractions to decimals.
10000
16.35
= =
10000
16.35
. 1 0000
0.0016 35
.
.
. = 0.001635
1. Line up the
decimal points. 2. Move the pair of points in tandem until
the denominator is 1 and pack 0’s in the
skipped slots in the numerator.
0 . # # # # to a fraction:
Recall that x =
x
1.
Decimals
.
1. Put “1.” in the denominator and line up the decimal points.
2. Slide the decimal point of the
numerator to end
of the number. 0 . # # # #
1 . =
The decimal form
of the fraction

46. To change a decimal number of the form
b. Convert the following fractions to decimals.
10000
16.35
= =
10000
16.35
. 1 0000
0.0016 35
.
.
. = 0.001635
1. Line up the
decimal points. 2. Move the pair of points in tandem until
the denominator is 1 and pack 0’s in the
skipped slots in the numerator.
0 . # # # # to a fraction:
Recall that x =
x
1.
Decimals
.
1. Put “1.” in the denominator and line up the decimal points.
2. Slide the decimal point of the
numerator to end
of the number. 0 . # # # #
1 .
0 . # # # #
1 .
.=
Drag the decimal point
to the end of the number
The decimal form
of the fraction

47. To change a decimal number of the form
b. Convert the following fractions to decimals.
10000
16.35
= =
10000
16.35
. 1 0000
0.0016 35
.
.
. = 0.001635
1. Line up the
decimal points. 2. Move the pair of points in tandem until
the denominator is 1 and pack 0’s in the
skipped slots in the numerator.
0 . # # # # to a fraction:
Recall that x =
x
1.
Decimals
.
1. Put “1.” in the denominator and line up the decimal points.
2. Slide the decimal point of the
numerator to end
of the number.
3. Pack a “0” for
each move to the right.
0 . # # # #
1 .
0 . # # # #
1 .
.
.0000
=
Drag the decimal point
to the end of the number
then fill in a “0” for each move.
The decimal form
of the fraction

48. Example B. Convert the following decimals to fractions.
a. 0.023
Decimals

49. Example B. Convert the following decimals to fractions.
a. 0.023
1. Insert “1.” in the denominator
and line up the decimal points.
Decimals

50. Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .
1. Insert “1.” in the denominator
and line up the decimal points.
Decimals

51. Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .
1. Insert “1.” in the denominator
and line up the decimal points.
2. Slide the pair of points in
tandem right, to the back of the
last non-zero digit in the
numerator, and pack 0’s in the
skipped slots in the denominator.
Decimals

52. 1. Insert “1.” in the denominator
and line up the decimal points.
2. Slide the pair of points in
tandem right, to the back of the
last non-zero digit in the
numerator, and pack 0’s in the
skipped slots in the denominator.
Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .0 0 0
0 . 0 2 3
1 .
=
.
.
Decimals

53. 1. Insert “1.” in the denominator
and line up the decimal points.
2. Slide the pair of points in
tandem right, to the back of the
last non-zero digit in the
numerator, and pack 0’s in the
skipped slots in the denominator.
Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .0 0 0 1000
23=
0 . 0 2 3
1 .
=
.
.
Decimals

54. 1. Insert “1.” in the denominator
and line up the decimal points.
2. Slide the pair of points in
tandem right, to the back of the
last non-zero digit in the
numerator, and pack 0’s in the
skipped slots in the denominator.
Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .0 0 0 1000
23=
0 . 0 2 3
1 .
=
.
.
b. 37. 25
Decimals

55. 1. Insert “1.” in the denominator
and line up the decimal points.
2. Slide the pair of points in
tandem right, to the back of the
last non-zero digit in the
numerator, and pack 0’s in the
skipped slots in the denominator.
Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .0 0 0 1000
23=
0 . 0 2 3
1 .
=
.
.
b. 37. 25
we only need to convert the decimal 0.25 to a fraction.
Since 37.25 = 37 + 0.25
Decimals

56. 1. Insert “1.” in the denominator
and line up the decimal points.
2. Slide the pair of points in
tandem right, to the back of the
last non-zero digit in the
numerator, and pack 0’s in the
skipped slots in the denominator.
Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .0 0 0 1000
23=
0 . 0 2 3
1 .
=
.
.
b. 37. 25
0 . 2 5
1 .
we only need to convert the decimal 0.25 to a fraction.
Since 37.25 = 37 + 0.25
0. 2 5 =
Decimals

57. 1. Insert “1.” in the denominator
and line up the decimal points.
2. Slide the pair of points in
tandem right, to the back of the
last non-zero digit in the
numerator, and pack 0’s in the
skipped slots in the denominator.
Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .0 0 0 1000
23=
0 . 0 2 3
1 .
=
.
.
b. 37. 25
0 . 2 5
1 .
we only need to convert the decimal 0.25 to a fraction.
Since 37.25 = 37 + 0.25
0. 2 5 =
0 . 2 5
1 . 0 0
=
.
.
Decimals

58. 1. Insert “1.” in the denominator
and line up the decimal points.
2. Slide the pair of points in
tandem right, to the back of the
last non-zero digit in the
numerator, and pack 0’s in the
skipped slots in the denominator.
Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .0 0 0 1000
23=
0 . 0 2 3
1 .
=
.
.
b. 37. 25
0 . 2 5
1 .
we only need to convert the decimal 0.25 to a fraction.
Since 37.25 = 37 + 0.25
0. 2 5 =
0 . 2 5
1 . 0 0
=
.
. 100
25
=
Decimals

59. 1. Insert “1.” in the denominator
and line up the decimal points.
2. Slide the pair of points in
tandem right, to the back of the
last non-zero digit in the
numerator, and pack 0’s in the
skipped slots in the denominator.
Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .0 0 0 1000
23=
0 . 0 2 3
1 .
=
.
.
b. 37. 25
0 . 2 5
1 .
we only need to convert the decimal 0.25 to a fraction.
Since 37.25 = 37 + 0.25
0. 2 5 =
0 . 2 5
1 . 0 0
=
.
. 100
25
=
4
1
=
Decimals

60. 1. Insert “1.” in the denominator
and line up the decimal points.
2. Slide the pair of points in
tandem right, to the back of the
last non-zero digit in the
numerator, and pack 0’s in the
skipped slots in the denominator.
Example B. Convert the following decimals to fractions.
a. 0.023
0 . 0 2 3
1 .0 0 0 1000
23=
0 . 0 2 3
1 .
=
.
.
b. 37. 25
0 . 2 5
1 .
we only need to convert the decimal 0.25 to a fraction.
Since 37.25 = 37 + 0.25
0. 2 5 =
0 . 2 5
1 . 0 0
=
.
. 100
25
=
4
1
=
Therefore 37.25 = 37
4
1
Decimals

61. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Decimals

62. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.

63. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1

64. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1
)4 1

65. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1
)4 1
1. Place a decimal point above
the decimal point of the
denominator. This is the
decimal point of the quotient.

66. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1
)4 1.
1. Place a decimal point above
the decimal point of the
denominator. This is the
decimal point of the quotient.
.

67. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1
)4 1.
1. Place a decimal point above
the decimal point of the
denominator. This is the
decimal point of the quotient.
2. Add 0’s to the right of the
dividend to perform the division
then perform the long division.
.

68. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1
)4 1.
1. Place a decimal point above
the decimal point of the
denominator. This is the
decimal point of the quotient.
2. Add 0’s to the right of the
dividend to perform the division
then perform the long division.
.
0 0

69. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1
)4 1.
1. Place a decimal point above
the decimal point of the
denominator. This is the
decimal point of the quotient.
2. Add 0’s to the right of the
dividend to perform the division
then perform the long division.
.
0 0
2
8

70. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1
)4 1.
1. Place a decimal point above
the decimal point of the
denominator. This is the
decimal point of the quotient.
2. Add 0’s to the right of the
dividend to perform the division
then perform the long division.
.
0 0
2
8
2 0

71. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
0
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1
)4 1.
1. Place a decimal point above
the decimal point of the
denominator. This is the
decimal point of the quotient.
2. Add 0’s to the right of the
dividend to perform the division
then perform the long division.
.
0 0
2
8
2 0
5
2 0

72. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
=Therefore
0
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1
)4 1.
1. Place a decimal point above
the decimal point of the
denominator. This is the
decimal point of the quotient.
2. Add 0’s to the right of the
dividend to perform the division
then perform the long division.
.
0 0
2
8
2 0
5
2 0
4
1
2 5.0

73. Adding 0’s to the right end of a decimal number does not
change the decimal number. Hence 1.2 = 1.20 = 1.200 etc..
since the extra 0’s do not carry any value.
Example C. Convert the fractions into decimals.
=Therefore
0
Decimals
We add 0’s to carry out long division to convert a fraction to a
decimal.
4
1
)4 1.
1. Place a decimal point above
the decimal point of the
denominator. This is the
decimal point of the quotient.
2. Add 0’s to the right of the
dividend to perform the division
then perform the long division.
.
0 0
2
8
2 0
5
2 0
4
1
2 5.0
Using similar division method, we list some of the common
fractions and their decimals expansions on the next slide.

74. =
Decimals
2
1
0.50 =
4
1
0.25 =
5
1
0.20 =
10
1 0.10
=
20
1
0.05 =
25
1
0.04 =
50
1
0.02 =
100
1
0.01
Here is a list of common fractions and their decimal expansions.

75. =
Decimals
2
1
0.50 =
4
1
0.25 =
5
1
0.20 =
10
1 0.10
=
20
1
0.05 =
25
1
0.04 =
50
1
0.02 =
100
1
0.01
A helpful way to remember some of these conversion is to
relate them to money.
Here is a list of common fractions and their decimal expansions.

76. =
Decimals
2
1
Here is a list of common fractions and their decimal expansions.
0.50 =
4
1
0.25 =
5
1
0.20 =
10
1 0.10
=
20
1
0.05 =
25
1
0.04 =
50
1
0.02 =
100
1
0.01
A helpful way to remember some of these conversion is to
relate them to money.
=
2
1 0.50
=
4
1
0.25
=
5
1
0.20
=
10
1 0.10
A half-dollar is 50 cents.
A quarter is 25 cents.
A fifth of a dollar is 20 cents.
A tenth of a dollar is 10 cents (dime.)
=100
1
0.01 One hundredth of a dollar is 1 cents (penny.)
=
20
1
0.05 One twenty of a dollar is 5 cents (nickel).

77. Decimals
To add decimal numbers, we line up the decimal point the
set the its position then add as usual.
Example D.
a. Add 8.978 + 0.657

78. Decimals
To add decimal numbers, we line up the decimal point the
set the its position then add as usual.
Example D.
a. Add 8.978 + 0.657
8 . 9 7 8
0 . 6 5 7+
.

79. Decimals
To add decimal numbers, we line up the decimal point the
set the its position then add as usual.
Do the same for subtracting decimals.
Example D.
a. Add 8.978 + 0.657
8 . 9 7 8
0 . 6 5 7+
1
53
1
6
1
9 .

80. Decimals
To add decimal numbers, we line up the decimal point the
set the its position then add as usual.
Do the same for subtracting decimals.
Example D.
a. Add 8.978 + 0.657
8 . 9 7 8
0 . 6 5 7+
1
53
1
6
1
9So the sum is 9.635. .

81. Decimals
To add decimal numbers, we line up the decimal point the
set the its position then add as usual.
Do the same for subtracting decimals.
Example D.
a. Add 8.978 + 0.657
8 . 9 7 8
0 . 6 5 7+
1
53
1
6
1
9So the sum is 9.635. .
b. Subtract 0.078 – 0.0293

82. Decimals
To add decimal numbers, we line up the decimal point the
set the its position then add as usual.
Do the same for subtracting decimals.
Example D.
a. Add 8.978 + 0.657
8 . 9 7 8
0 . 6 5 7+
1
53
1
6
1
9So the sum is 9.635. .
b. Subtract 0.078 – 0.0293
0 . 0 7 8
0 . 0 2 9 3–
.

83. Decimals
To add decimal numbers, we line up the decimal point the
set the its position then add as usual.
Do the same for subtracting decimals.
Example D.
a. Add 8.978 + 0.657
8 . 9 7 8
0 . 6 5 7+
1
53
1
6
1
9So the sum is 9.635. .
b. Subtract 0.078 – 0.0293
0 . 0 7 8
0 . 0 2 9 3–
0
Add 0’s at the end
of the decimal
expansion,
then subtract
.

84. Decimals
To add decimal numbers, we line up the decimal point the
set the its position then add as usual.
Do the same for subtracting decimals.
Example D.
a. Add 8.978 + 0.657
8 . 9 7 8
0 . 6 5 7+
1
53
1
6
1
9So the sum is 9.635. .
b. Subtract 0.078 – 0.0293
0 . 0 7 8
0 . 0 2 9 3–
8400 . 7
0
Add 0’s at the end
of the decimal
expansion,
then subtract

85. Decimals
To add decimal numbers, we line up the decimal point the
set the its position then add as usual.
Do the same for subtracting decimals.
Example D.
a. Add 8.978 + 0.657
8 . 9 7 8
0 . 6 5 7+
1
53
1
6
1
9So the sum is 9.635. .
b. Subtract 0.078 – 0.0293
0 . 0 7 8
0 . 0 2 9 3–
8400 . 7
0
Hence 0.078 – 0.0293 = 0.0487.
Add 0’s at the end
of the decimal
expansion,
then subtract.