The document explains decimals and how they allow us to track smaller quantities in the base-10 number system. It demonstrates decimals using a cash register with slots for $1s, $10s, $100s, etc. and introduces hypothetical smaller coins like "itties" and "bitties". Decimals are then represented by writing the coins in a cash register as a number with a decimal point, like 0.43 representing 4 $1s and 3 $10s. The document also provides steps for comparing decimal values by lining up the numbers and determining which has the largest digit in each place value slot from left to right.

1. Decimals
Back to Algebra–Ready Review Content.

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
Arithmetic of 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 2
100,000
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. 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

31. 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+
.

32. 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 .

33. 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. .

34. 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

35. 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–
.

36. 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
.

37. 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

38. 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.

39. 47
7x
9
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals
6

40. we start the multiplication by
multiplying the top with the
bottom right most digit.
47
7x
9For example,
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals
6

41. we start the multiplication by
multiplying the top with the
bottom right most digit.
47
7x
4x7=28
9For example,
6
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

42. we start the multiplication by
multiplying the top with the
bottom right most digit.
47
7x
8
record
the 8
carry
the 2
4x7=28
9For example,
6
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

43. we start the multiplication by
multiplying the top with the
bottom right most digit.
47
7x
8
record
the 8
carry
the 2
4x7=287x7=49,
9For example,
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals
6

44. we start the multiplication by
multiplying the top with the
bottom right most digit.
47
7x
8
record
the 8
carry
the 2
4x7=287x7=49,
49+2=51
9For example,
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals
6

45. we start the multiplication by
multiplying the top with the
bottom right most digit.
47
7x
8
record
the 8
carry
the 2
4x7=287x7=49,
1
record
the 1
49+2=51
9For example,
carry
the 5
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals
6

46. we start the multiplication by
multiplying the top with the
bottom right most digit.
47
7x
8
record
the 8
carry
the 2
4x7=287x7=49,
1
record
the 1
carry
the 5
49+2=51
9
9x7=63,
63+5= 68
For example,
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals
6

47. we start the multiplication by
multiplying the top with the
bottom right most digit.
47
7x
8
record
the 8
carry
the 2
4x7=287x7=49,
1
record
the 1
carry
the 5
49+2=51
9
9x7=63,
63+5= 68
8
record
the 8
carry
the 6
6
For example,
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals
6

48. we start the multiplication by
multiplying the top with the
bottom right most digit.
47
7x
8
record
the 8
carry
the 2
4x7=287x7=49,
1
record
the 1
carry
the 5
49+2=51
9
9x7=63,
63+5= 68
8
record
the 8
carry
the 6
6
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
6
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

49. we start the multiplication by
multiplying the top with the
bottom right most digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example, 47
7
8
record
the 8
1
record
the 1
9
8
record
the 8
carry
the 6
6
6x
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

50. we start the multiplication by
multiplying the top with the
bottom right most digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example, 47
7
8
record
the 8
4x6=24
1
record
the 1
9
8
record
the 8
carry
the 6
6
6x
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

51. we start the multiplication by
multiplying the top with the
bottom right most digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example, 47
7
8
record
the 8
4x6=24
1
record
the 1
←record
9
8
record
the 8
carry
the 6
6
6
carry
the 2
4
x
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

52. we start the multiplication by
multiplying the top with the
bottom right most digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example, 47
7
8
record
the 8
4x6=247x6=42,
1
record
the 1
←record
42+2=44
9
8
record
the 8
carry
the 6
6
6
carry
the 2
4
x
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

53. we start the multiplication by
multiplying the top with the
bottom right most digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example, 47
7
8
record
the 8
carry
the 4
4x6=247x6=42,
1
record
the 1
←record
42+2=44
9
8
record
the 8
carry
the 6
6
6
carry
the 2
44
x
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

54. we start the multiplication by
multiplying the top with the
bottom right most digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example, 47
7
8
record
the 8
carry
the 4
4x6=247x6=42,
1
record
the 1
←record
42+2=44
9
9x6=54
54+4= 58
8
record
the 8
carry
the 6
6
6
carry
the 2
44
x
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

55. we start the multiplication by
multiplying the top with the
bottom right most digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example, 47
7
8
record
the 8
carry
the 4
4x6=247x6=42,
1
record
the 1
←record
42+2=44
9
9x6=54
54+4= 58
8
record
the 8
carry
the 6
6
6
carry
the 2
4485
x
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

56. we start the multiplication by
multiplying the top with the
bottom right most digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
record
the 8
carry
the 4
4x6=247x6=42,
1
record
the 1
←record
42+2=44
9
9x6=54
54+4= 58
8
record
the 8
carry
the 6
6
6
carry
the 2
Finally, we obtain the answer
by adding the two columns.
4485+
x
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

57. we start the multiplication by
multiplying the top with the
bottom right most digit.
When this is completed, we
proceed with the multiplication to
the next digit of the bottom number.
For example,
Because we are in a
place value system, the
result of the multiplication
must be placed in the correct slots,
so it is shift one place to the left.
47
7
8
record
the 8
carry
the 4
4x6=247x6=42,
1
record
the 1
←record
42+2=44
9
9x6=54
54+4= 58
8
record
the 8
carry
the 6
6
6
carry
the 2
Finally, we obtain the answer
by adding the two columns.
4485
8526 5
+
x
Let's review the multiplication of two
multiple digit numbers. Such a problem
is treated as multiple problems of
multiplying with a single digit number.
Multiplication and Division of Decimals

58. To multiply two decimal numbers, do exactly the same–then insert
the decimal point in the product at the correct place for the final
answer.
Multiplication and Division of Decimals

59. To multiply two decimal numbers, do exactly the same–then insert
the decimal point in the product at the correct place for the final
answer.
Multiplication and Division of Decimals
Example E. Multiply 9.74 x 6.7

60. 47
7
81
9
86
6
4485
8526 5
x
To multiply two decimal numbers, do exactly the same–then insert
the decimal point in the product at the correct place for the final
answer.
Multiplication and Division of Decimals
Example E. Multiply 9.74 x 6.7
Ignore the decimal points and multiply
974 x 67 = 65258.

61. I. count the total number of places to the right of the decimal point in both
decimal numbers,
47
7
81
9
86
6
4485
8526 5
x
To multiply two decimal numbers, do exactly the same–then insert
the decimal point in the product at the correct place for the final
answer. To locate the position of the decimal point:
Multiplication and Division of Decimals
Example E. Multiply 9.74 x 6.7
Ignore the decimal points and multiply
974 x 67 = 65258. Put back the decimal
points to count the number of places after
them, which is 3.

62. I. count the total number of places to the right of the decimal point in both
decimal numbers,
47
7
81
9
86
6
4485
8526 5
x
To multiply two decimal numbers, do exactly the same–then insert
the decimal point in the product at the correct place for the final
answer. To locate the position of the decimal point:
Multiplication and Division of Decimals
Example E. Multiply 9.74 x 6.7
Ignore the decimal points and multiply
974 x 67 = 65258. Put back the decimal
points to count the number of places after
them, which is 3.
.
.
There are 3
places after the
decimal point

63. I. count the total number of places to the right of the decimal point in both
decimal numbers,
II. take the decimal point at the right end of their product, count to the left
the same total–number of places, to place the decimal point.
47
7
81
9
86
6
4485
8526 5
x
To multiply two decimal numbers, do exactly the same–then insert
the decimal point in the product at the correct place for the final
answer. To locate the position of the decimal point:
Multiplication and Division of Decimals
Example E. Multiply 9.74 x 6.7
.
.
There are 3
places after the
decimal point
Ignore the decimal points and multiply
974 x 67 = 65258. Put back the decimal
points to count the number of places after
them, which is 3.

64. I. count the total number of places to the right of the decimal point in both
decimal numbers,
II. take the decimal point at the right end of their product, count to the left
the same total–number of places, to place the decimal point.
47
7
81
9
86
6
4485
8526 5
x
To multiply two decimal numbers, do exactly the same–then insert
the decimal point in the product at the correct place for the final
answer. To locate the position of the decimal point:
Multiplication and Division of Decimals
Example E. Multiply 9.74 x 6.7
.
.
There are 3
places after the
decimal point
Move the decimal point of the product
3 places to the left for the answer.
Ignore the decimal points and multiply
974 x 67 = 65258. Put back the decimal
points to count the number of places after
them, which is 3.

65. I. count the total number of places to the right of the decimal point in both
decimal numbers,
II. take the decimal point at the right end of their product, count to the left
the same total–number of places, to place the decimal point.
47
7
81
9
86
6
4485
526 5
x
To multiply two decimal numbers, do exactly the same–then insert
the decimal point in the product at the correct place for the final
answer. To locate the position of the decimal point:
Multiplication and Division of Decimals
Example E. Multiply 9.74 x 6.7
.
.
There are 3
places after the
decimal point
Move the decimal point of the product
3 places to the left for the answer.
So move the decimal point
3 places left.
.. 8
Ignore the decimal points and multiply
974 x 67 = 65258. Put back the decimal
points to count the number of places after
them, which is 3.

66. I. count the total number of places to the right of the decimal point in both
decimal numbers,
II. take the decimal point at the right end of their product, count to the left
the same total–number of places, to place the decimal point.
47
7
81
9
86
6
4485
526 5
x
To multiply two decimal numbers, do exactly the same–then insert
the decimal point in the product at the correct place for the final
answer. To locate the position of the decimal point:
Multiplication and Division of Decimals
Example E. Multiply 9.74 x 6.7
.
.
There are 3
places after the
decimal point
Move the decimal point of the product
3 places to the left for the answer.
So move the decimal point
3 places left.
.. 8
Hence 9.74 x 6.7 = 65.258
Ignore the decimal points and multiply
974 x 67 = 65258. Put back the decimal
points to count the number of places after
them, which is 3.

67. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
Remove the trailing 0’s to the right for the multiplication decimal numbers.
b. Multiply 0.00012 x 0.00700.

68. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
Remove the trailing 0’s to the right for the multiplication decimal numbers.
We can drop the extra trailing 0’s in 1.200 x 0.700
b. Multiply 0.00012 x 0.00700.

69. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
Remove the trailing 0’s to the right for the multiplication decimal numbers.
We can drop the extra trailing 0’s in 1.200 x 0.700 = 1.2 x 0.7.
b. Multiply 0.00012 x 0.00700.

70. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
There are two places
after the decimal points.
Remove the trailing 0’s to the right for the multiplication decimal numbers.
We can drop the extra trailing 0’s in 1.200 x 0.700 = 1.2 x 0.7.
b. Multiply 0.00012 x 0.00700.

71. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
There are two places
after the decimal points.
Remove the trailing 0’s to the right for the multiplication decimal numbers.
We can drop the extra trailing 0’s in 1.200 x 0.700 = 1.2 x 0.7.
Multiply 12 x 7 = 84.
b. Multiply 0.00012 x 0.00700.

72. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
There are two places
after the decimal points.
Remove the trailing 0’s to the right for the multiplication decimal numbers.
We can drop the extra trailing 0’s in 1.200 x 0.700 = 1.2 x 0.7.
Multiply 12 x 7 = 84.
So move the decimal point
two places left to place the
decimal point.
0.8 4.
b. Multiply 0.00012 x 0.00700.

73. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
So 1.200 x 0.700 = 0.84
There are two places
after the decimal points.
Remove the trailing 0’s to the right for the multiplication decimal numbers.
We can drop the extra trailing 0’s in 1.200 x 0.700 = 1.2 x 0.7.
Multiply 12 x 7 = 84.
So move the decimal point
two places left to place the
decimal point.
0.8 4.
b. Multiply 0.00012 x 0.00700.

74. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
So 1.200 x 0.700 = 0.84
b. Multiply 0.00012 x 0.00700.
There are two places
after the decimal points.
Remove the trailing 0’s to the right for the multiplication decimal numbers.
We can drop the extra trailing 0’s in 1.200 x 0.700 = 1.2 x 0.7.
Multiply 12 x 7 = 84.
So move the decimal point
two places left to place the
decimal point.
0.
0.00012 x 0.00700 = 0.00012 x 0.007 and 12 x 7 = 84.
8 4.

75. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
So 1.200 x 0.700 = 0.84
b. Multiply 0.00012 x 0.00700.
There are two places
after the decimal points.
Remove the trailing 0’s to the right for the multiplication decimal numbers.
We can drop the extra trailing 0’s in 1.200 x 0.700 = 1.2 x 0.7.
Multiply 12 x 7 = 84.
So move the decimal point
two places left to place the
decimal point.
8 4. = 12 x 7
0.
0.00012 x 0.00700 = 0.00012 x 0.007 and 12 x 7 = 84.
There are eight places after the decimal points so move the point eight
place left and fill in 0’s as we move:
8 4.

76. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
So 1.200 x 0.700 = 0.84
b. Multiply 0.00012 x 0.00700.
There are two places
after the decimal points.
Remove the trailing 0’s to the right for the multiplication decimal numbers.
We can drop the extra trailing 0’s in 1.200 x 0.700 = 1.2 x 0.7.
Multiply 12 x 7 = 84.
So move the decimal point
two places left to place the
decimal point.
8 4. = 12 x 7
0.
0.00012 x 0.00700 = 0.00012 x 0.007 and 12 x 7 = 84.
There are eight places after the decimal points so move the point eight
place left and fill in 0’s as we move:
8 4.
0.0 0 0 0 0 0
8 places

77. Multiplication and Division of Decimals
Example F. a. Multiply 1.200 x 0.700
So 1.200 x 0.700 = 0.84
b. Multiply 0.00012 x 0.00700.
There are two places
after the decimal points.
Remove the trailing 0’s to the right for the multiplication decimal numbers.
We can drop the extra trailing 0’s in 1.200 x 0.700 = 1.2 x 0.7.
Multiply 12 x 7 = 84.
So move the decimal point
two places left to place the
decimal point.
8 4. = 12 x 7
0.
0.00012 x 0.00700 = 0.00012 x 0.007 and 12 x 7 = 84.
There are eight places after the decimal points so move the point eight
place left and fill in 0’s as we move:
8 4.
0.0 0 0 0 0 0
8 placesHence 0.00012 x 0.00700 = 0.00000084.

78. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual
65
1.3

79. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual
65
1.3
= 1.3 ÷ 65
65
1.3Calculate
by long division.

80. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual
65
1.3
)6 5 1 . 3
= 1.3 ÷ 65
65
1.3Calculate
by long division.

81. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual and
leave the decimal point in the same position for the quotient .
65
1.3
)6 5 1 . 3
.
= 1.3 ÷ 65
65
1.3Calculate
by long division.
the decimal point place

82. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual and
leave the decimal point in the same position for the quotient .
65
1.3
)6 5 1 . 3
0 . 0
= 1.3 ÷ 65
65
1.3Calculate
by long division.
the decimal point place

83. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual and
leave the decimal point in the same position for the quotient .
65
1.3
)6 5 1 . 3 0
0 . 0
= 1.3 ÷ 65
65
1.3Calculate
by long division.
the decimal point place
Pack trailing 0’s
so it’s enough to
enter a quotient

84. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual and
leave the decimal point in the same position for the quotient .
65
1.3
)6 5 1 . 3 0
1 3 0
2.
0
= 1.3 ÷ 65
65
1.3Calculate
by long division.
the decimal point place
00 Pack trailing 0’s
so it’s enough to
enter a quotient

85. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual and
leave the decimal point in the same position for the quotient .
65
1.3
)6 5 1 . 3 0
1 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65
65
1.3Calculate
by long division.
the decimal point place
00 Pack trailing 0’s
so it’s enough to
enter a quotient

86. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual and
leave the decimal point in the same position for the quotient .
Example H. a. Compute 0.0013 ÷ 0.00065
We change a problem of dividing two decimal numbers to a problem that is
a decimal number divided by an integer.
65
1.3
)6 5 1 . 3 0
1 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65
65
1.3Calculate
by long division.
00
the decimal point place
Pack trailing 0’s
so it’s enough to
enter a quotient

87. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual and
leave the decimal point in the same position for the quotient .
Example H. a. Compute 0.0013 ÷ 0.00065
We change a problem of dividing two decimal numbers to a problem that is
a decimal number divided by an integer. Write the problem as a fraction then
move the decimal points in tandem until the numerator is an integer.
65
1.3
)6 5 1 . 3 0
1 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65
65
1.3Calculate
by long division.
0.00065
0.0013
Write 0.0013 ÷ 0.00065 as
00
the decimal point place
Pack trailing 0’s
so it’s enough to
enter a quotient

88. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual and
leave the decimal point in the same position for the quotient .
Example H. a. Compute 0.0013 ÷ 0.00065
.
move 5 places so the numerator is an integer.
We change a problem of dividing two decimal numbers to a problem that is
a decimal number divided by an integer. Write the problem as a fraction then
move the decimal points in tandem until the numerator is an integer.
65
1.3
)6 5 1 . 3 0
1 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65
65
1.3Calculate
by long division.
0.00065
0.0013
Write 0.0013 ÷ 0.00065 as
. =
.
65
13
.
0 0
00
the decimal point place
Pack trailing 0’s
so it’s enough to
enter a quotient

89. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual and
leave the decimal point in the same position for the quotient .
Example H. a. Compute 0.0013 ÷ 0.00065
.
move 5 places so the numerator is an integer.
We change a problem of dividing two decimal numbers to a problem that is
a decimal number divided by an integer. Write the problem as a fraction then
move the decimal points in tandem until the numerator is an integer.
65
1.3
)6 5 1 . 3 0
1 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65
65
1.3Calculate
by long division.
0.00065
0.0013
Write 0.0013 ÷ 0.00065 as =
.
65
13
.
0 0
= 2
Hence 0.0013 ÷ 0.00065 = 2
00
the decimal point place
Pack trailing 0’s
so it’s enough to
enter a quotient
.

90. Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual and
leave the decimal point in the same position for the quotient .
Example H. a. Compute 0.0013 ÷ 0.00065
.
move 5 places so the numerator is an integer.
We change a problem of dividing two decimal numbers to a problem that is
a decimal number divided by an integer. Write the problem as a fraction then
move the decimal points in tandem until the numerator is an integer.
65
1.3
)6 5 1 . 3 0
1 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65
65
1.3Calculate
by long division.
0.00065
0.0013
Write 0.0013 ÷ 0.00065 as =
.
65
13
.
0 0
= 2
Hence 0.0013 ÷ 0.00065 = 2
00
the decimal point place
Pack trailing 0’s
so it’s enough to
enter a quotient
.

91. Multiplication and Division of Decimals
Example H. b. Compute 0.00013 ÷ 0.65

92. Multiplication and Division of Decimals
Example H. b. Compute 0.00013 ÷ 0.65
0.65
0.00 013
Write 0.00013 ÷ 0.65 as

93. Multiplication and Division of Decimals
Example H. b. Compute 0.00013 ÷ 0.65
.
move 2 places
0.65
0.00 013
Write 0.00013 ÷ 0.65 as .
=
.65
0 013.

94. Multiplication and Division of Decimals
Example H. b. Compute 0.00013 ÷ 0.65
.
move 2 places
)65 0 .1 3
0.65
0.00 013
Write 0.00013 ÷ 0.65 as .
=
.65
0 013.
Calculate this by long division:

95. Multiplication and Division of Decimals
Example H. b. Compute 0.00013 ÷ 0.65
.
move 2 places
)65 0 .1 3 0
1 3 0
0 20 .0
0
0.65
0.00 013
Write 0.00013 ÷ 0.65 as .
=
.65
0 013.
Hence 0.0013 ÷ 0. 65 = 0.002.
Calculate this by long division: