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Decimals
Back to Algebra–Ready Review Content.
DecimalsIn order to track smaller and smaller quantities we include base-10 fractions to the whole number-system.
$100’s $1’s$10’s
DecimalsIn 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.
*
$100’s $1’s$10’s
DecimalsIn 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, $100’s, ...etc. For a moment let’s assume that US Treasury not only makes
*
(dime),101$ (penny),100
1$
, a “itty”, and10001$ , a “bitty”, etc...10000
1$
but also makes smaller value coins
of
$100’s $1’s$10’s
DecimalsIn 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
101
$
pennies itties
1001
$ 10001
$
(dime),101$ (penny),100
1$
, a “itty”, and10001$ , a “bitty”, etc...10000
1$
but also makes smaller value coins
100001
$
of
bitties
*
$100’s $1’s$10’s
DecimalsIn 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
* 101
$ 1001
$ 10001
$
(dime),101$ (penny),100
1$
, a “itty”, and10001$ , a “bitty”, etc...10000
1$
but also makes smaller value coins
100001
$
of
Note that 10 itties = 1 penny and 10 bitties = 1 itty, etc...
*
dimes pennies itties bitties
$100’s $1’s$10’s
DecimalsIn 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
* 101
$ 1001
$ 10001
$
(dime),101$ (penny),100
1$
, a “itty”, and10001$ , a “bitty”, etc...10000
1$
but also makes smaller value coins
Let’s further assume each slot only hold up to 9 bills or coinsso we may record the money stored in the register
100001
$
of
Note that 10 itties = 1 penny and 10 bitties = 1 itty, etc...
*
dimes pennies itties bitties
$100’s $1’s$10’s
DecimalsIn 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
* 101
$ 1001
$ 10001
$
(dime),101$ (penny),100
1$
, a “itty”, and10001$ , a “bitty”, etc...10000
1$
# # # # ##
but also makes smaller value coins
Let’s further assume each slot only hold up to 9 bills or coinsso we may record the money stored in the register
100001
$
of
Note that 10 itties = 1 penny and 10 bitties = 1 itty, etc...
#
*
dimes pennies itties bitties
$100’s $1’s$10’s
Arithmetic of DecimalsIn 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
* 101
$ 1001
$ 10001
$
(dime),101$ (penny),100
1$
, a “itty”, and10001$ , 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 coinsso we may record the money stored in the register
100001
$
of
Note that 10 itties = 1 penny and 10 bitties = 1 itty, etc...
#
*
dimes pennies itties bitties
.
$100’s* $1’s$10’s* 101
$ 1001
$ 10001
$
4 5 63
For example,
.
Decimals
dimes pennies itties bitties
$100’s* $1’s$10’s*
dimes
101
$
pennies itties
1001
$ 10001
$
4 5 63
For example,
. 43 5 6is written as
.
Decimals
bitties
$100’s* $1’s$10’s*
dimes
101
$
pennies itties
1001
$ 10001
$
4 5 63
For example,
. 43 5 6is written as
.
4 $1’s3 $10’s
Decimals
bitties
$100’s* $1’s$10’s*
dimes
101
$
pennies itties
1001
$ 10001
$
4 5 63
For example,
. 43 5 6is written as
.
4 $1’s3 $10’s
(5 dimes)(6 pennies)
105 100
6$$
Decimals
bitties
$100’s* $1’s$10’s*
dimes
101
$
pennies itties
1001
$ 10001
$
4 5 63
For example,
. 43 5 6is written as
.
4 $1’s3 $10’s
(5 dimes)(6 pennies)
105 100
6$$
$100’s $1’s$10’s* 101
$ 1001
$ 10001
$
4 5 0 7
Decimals
8
100001$
bitties
.
$100’s* $1’s$10’s*
dimes
101
$
pennies itties
1001
$ 10001
$
4 5 63
For example,
. 43 5 6is written as
.
4 $1’s3 $10’s
(5 dimes)(6 pennies)
105 100
6$$
$100’s $1’s$10’s* 101
$ 1001
$ 10001
$
4 5 0 7
4 $1’s
4is written as .
Decimals
8
100001$
bitties
.
$100’s* $1’s$10’s*
dimes
101
$
pennies itties
1001
$ 10001
$
4 5 63
For example,
. 43 5 6is written as
.
4 $1’s3 $10’s
(5 dimes)(6 pennies)
105 100
6$$
$100’s $1’s$10’s*
dimes
101
$
pennies itties
1001
$ 10001
$
4 5 0 7
4 $1’s (no penny)1000$
(5 dimes)105$
10007$
4 75 0is written as .
Decimals
8
100001$
(8 bitties)100008
$
bitties
.
8
(7 itties)
Comparing Decimal NumbersDecimals
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.)
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.)
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
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
0.00980.0100.00199
1. line up by the decimal points
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
0.00980.0100.00199
1. line up by the decimal points
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
0.00980.0100.00199
1. line up by the decimal points
2. scan the digits in each slot from left to right
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
0.00980.0100.00199
1. line up by the decimal points
2. scan the digits in each slot from left to right
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
0.00980.0100.00199
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
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
0.00980.0100.00199
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
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
0.00980.0100.00199
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.
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 10,000 100,0001 1 1 1 1
1,000,0001
Decimals
10’s
ones tenths hundredths thousandthsten–thousandths
Decimal point
hundred–thousandths millionths.tens
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 thousandthsten–thousandths
100,000
Decimal point
hundred–thousandths millionths.
1 1 1 1 11,000,000
1
Hence
2 . 3 4 5 6 7is 2 + 10 100 1,000 10,000 100,000
3 4 5 6 7+ + + +
Threetenths
Fourhundredths
Fivethousandths
Six ten-thousandths
Seven hundred-thousandth
Decimals
10’s
tens
Two
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 2 100,00034,567
Decimals
Hence
2 . 3 4 5 6 7is 2 + 10 100 1,000 10,000 100,000
3 4 5 6 7+ + + +
Threetenths
Fourhundredths
Fivethousandths
Six ten-thousandths
Seven hundred-thousandth
1’s 10 100 1,000 10,000
ones tenths hundredths thousandthsten–thousandths
100,000
Decimal point
hundred–thousandths millionths.
1 1 1 1 11,000,000
110’s
tens
.
Two
DecimalsTo 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
DecimalsTo 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 80 . 6 5 7 +
.
DecimalsTo 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 80 . 6 5 7 +
1
53
1
6
1
9 .
DecimalsTo 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 80 . 6 5 7 +
1
53
1
6
1
9So the sum is 9.635. .
DecimalsTo 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 80 . 6 5 7 +
1
53
1
6
1
9So the sum is 9.635. .
b. Subtract 0.078 – 0.0293
DecimalsTo 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 80 . 6 5 7 +
1
53
1
6
1
9So the sum is 9.635. .
b. Subtract 0.078 – 0.0293
0 . 0 7 80 . 0 2 9 3 –
.
DecimalsTo 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 80 . 6 5 7 +
1
53
1
6
1
9So the sum is 9.635. .
b. Subtract 0.078 – 0.0293
0 . 0 7 80 . 0 2 9 3 –
0
Add 0’s at the end of the decimal expansion,then subtract
.
DecimalsTo 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 80 . 6 5 7 +
1
53
1
6
1
9So the sum is 9.635. .
b. Subtract 0.078 – 0.0293
0 . 0 7 80 . 0 2 9 3 –
8400 . 7
0
Add 0’s at the end of the decimal expansion,then subtract
DecimalsTo 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 80 . 6 5 7 +
1
53
1
6
1
9So the sum is 9.635. .
b. Subtract 0.078 – 0.0293
0 . 0 7 80 . 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.
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
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
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
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
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 7x7=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
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 7x7=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
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 7x7=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
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 7x7=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
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 7x7=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
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 7x7=49,
1
record the 1
carry the 5
49+2=51
9
9x7=63, 63+5= 68
8
record the 8
carry the 6
6When 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
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
78
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
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
78
record the 8
4x6=24
1
record the 1
9
8
record the 8
carry the 6
6
6
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
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
78
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
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
78
record the 8
4x6=24 7x6=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
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
78
record the 8
carry the 4
4x6=24 7x6=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
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
78
record the 8
carry the 4
4x6=24 7x6=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
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
78
record the 8
carry the 4
4x6=24 7x6=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
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
78
record the 8
carry the 4
4x6=24 7x6=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
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
78
record the 8
carry the 4
4x6=24 7x6=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.
44
85
85 2 6 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
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
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
47781
9
866
4485
85 2 6 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 6.7Ignore the decimal points and multiply 974 x 67 = 65258.
I. count the total number of places to the right of the decimal point in both decimal numbers,
47781
9
866
4485
85 2 6 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 6.7Ignore 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.
I. count the total number of places to the right of the decimal point in both decimal numbers,
47781
9
866
4485
85 2 6 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 6.7Ignore 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
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.
47781
9
866
4485
85 2 6 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.
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.
47781
9
866
4485
85 2 6 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 product3 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.
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.
47781
9
866
4485
5 2 6 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 product3 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.
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.
47781
9
866
4485
5 2 6 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 product3 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual
651.3
Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual
651.3
= 1.3 ÷ 65651.3Calculate
by long division.
Multiplication and Division of Decimals
Example G. Compute by long division.
To divide a decimal number by an integer, do long division as usual
651.3
)6 5 1 . 3 = 1.3 ÷ 65651.3Calculate
by long division.
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 .
651.3
)6 5 1 . 3.
= 1.3 ÷ 65651.3Calculate
by long division. the decimal point place
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 .
651.3
)6 5 1 . 30 . 0
= 1.3 ÷ 65651.3Calculate
by long division. the decimal point place
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 .
651.3
)6 5 1 . 3 00 . 0
= 1.3 ÷ 65651.3Calculate
by long division. the decimal point place
Pack trailing 0’s so it’s enough to enter a quotient
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 .
651.3
)6 5 1 . 3 01 3 0
2.
0
= 1.3 ÷ 65651.3Calculate
by long division. the decimal point place
00 Pack trailing 0’s so it’s enough to enter a quotient
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 .
651.3
)6 5 1 . 3 01 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65651.3Calculate
by long division. the decimal point place
00 Pack trailing 0’s so it’s enough to enter a quotient
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.
651.3
)6 5 1 . 3 01 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65651.3Calculate
by long division.
00
the decimal point place
Pack trailing 0’s so it’s enough to enter a quotient
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.
651.3
)6 5 1 . 3 01 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65651.3Calculate
by long division.
0.000650.0013
Write 0.0013 ÷ 0.00065 as
00
the decimal point place
Pack trailing 0’s so it’s enough to enter a quotient
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.
651.3
)6 5 1 . 3 01 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65651.3Calculate
by long division.
0.000650.0013
Write 0.0013 ÷ 0.00065 as .
= .
6513
. 0 0
00
the decimal point place
Pack trailing 0’s so it’s enough to enter a quotient
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.
651.3
)6 5 1 . 3 01 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65651.3Calculate
by long division.
0.000650.0013
Write 0.0013 ÷ 0.00065 as
= .
6513
. 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
.
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.
651.3
)6 5 1 . 3 01 3 0
2.
0Hence 1.3 ÷ 65 = 0.02
= 1.3 ÷ 65651.3Calculate
by long division.
0.000650.0013
Write 0.0013 ÷ 0.00065 as
= .
6513
. 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
.
Multiplication and Division of DecimalsExample H. b. Compute 0.00013 ÷ 0.65
Multiplication and Division of DecimalsExample H. b. Compute 0.00013 ÷ 0.65
0.650.00 013
Write 0.00013 ÷ 0.65 as
Multiplication and Division of DecimalsExample H. b. Compute 0.00013 ÷ 0.65
. move 2 places
0.650.00 013
Write 0.00013 ÷ 0.65 as .
= . 650 013.
Multiplication and Division of DecimalsExample H. b. Compute 0.00013 ÷ 0.65
. move 2 places
)65 0 .1 3
0.650.00 013
Write 0.00013 ÷ 0.65 as .
= . 650 013.
Calculate this by long division:
Multiplication and Division of DecimalsExample H. b. Compute 0.00013 ÷ 0.65
. move 2 places
)65 0 .1 3 01 3 0
0 20 . 0
0
0.650.00 013
Write 0.00013 ÷ 0.65 as .
= . 650 013.
Hence 0.0013 ÷ 0. 65 = 0.002.
Calculate this by long division:
