EM3SRB_G6_24A-24H.qxd
11/3/08
3:54 PM
Page 59
Whole Numbers
U.S. Traditional Long Division: Single-Digit Divisors
U.S. traditional long division is another method you can use to divide.
Share $957 among 5 people.
Step 1: Share the
$100
Step 2: Trade 4
s.
$100
s for 40 $10 s.
That makes 45 $10 s in all.
1
5
7
59
5
4
Ò Each person gets 1 $100 .
Ò 1 $100 each for 5 people
Ò 4 $100 s are left.
Step 3: Share the
$10
1
5
7
59
5
45
Ò 45 $10 s are to be shared.
Step 4: Share the
s.
19 Ò Each person gets 9 $10 s.
5
7
59
5
45
45 Ò 9 $10 s each for 5 people
0 Ò 0 $10 s are left.
191
5
7
59
5
45
45
07
5
2
$1
s.
Ò Each person gets 1
$1
.
Ò 7 s are to be shared.
Ò1
$1
Ò2
$1
each for 5 people
s are left.
$957 / 5 ∑ $191 R$2
Each person gets $191; $2 are left over.
Divide.
1. 840 / 7 ?
2. 69
8
4
3. 45
3
9
4. 5,280 / 6 ?
Check your answers on page 424.
24E
EM3SRB_G6_24A-24H.qxd
11/12/08
12:02 PM
Page 60
Whole Numbers
U.S. traditional long division is not limited to
dividing money.
Note
The “leading” 0 in the quotient
is shown in the problem to help
you understand the long division
method. It should not be
included in the answer.
3,628 / 5 ?
Think about the problem as dividing 3,628 into 5 equal shares.
Step 1: Start with the thousands.
Step 2: So trade 3 thousands for 30 hundreds.
Share the hundreds.
Ò There are not enough thousands
0
to share 5 ways.
53
6
2
8
07
6
2
8
53
35
1
Step 3: Trade 1 hundred for 10 tens.
Step 4: Trade 2 tens for 20 ones.
Share the tens.
072
53
6
2
8
35
12
10
2
Ò
Ò
Ò
Ò
Each share gets 7 hundreds.
36 hundreds
7 hundreds 5 shares
1 hundred is left.
Share the ones.
Ò Each share gets 2 tens.
Ò 10 tens 2 tens
Ò 2 tens 5 shares
Ò 2 tens are left.
0725
53
6
2
8
35
12
10
28
25
3
Ò Each share gets 5 ones.
Ò 20 ones 8 ones
Ò 5 ones 5 shares
Ò 3 ones are left.
3,628 / 5 ∑ 725 R3
1. 5,376 / 6 = ?
2. 68
,5
8
6
3. 46
,9
2
3
Check your answers on page 424.
24F
4. 8,029 / 3 = ?
EM3SRB_G6_24A-24H.qxd
11/5/08
11:05 AM
Page 61
Whole Numbers
U.S. Traditional Long Division: Multidigit Divisors
You can use U.S. traditional long division to divide by larger numbers.
Share $681 among 21 people.
Make a table of easy multiples of the divisor.
This can help you decide how many to share at each step.
1
* 21
2 * 21
3 * 21
4 * 21
5 * 21
6 * 21
8 * 21
10 * 21
21
42
Double 21.
63
Add 2 21 and 1 21.
84
Double 2 21.
105
Halve 10 21.
126
Double 3 21.
168
Double 4 21.
210
Move decimal point one place to the right.
Step 1: There are not enough [$100]s to
Step 2: Trade the 5 [$10]s for 50 [$1]s.
share 21 ways, so trade 6 [$100]s
for 60 [$10]s.
Share the 51 [$1]s.
Share the 68 [$10]s.
3
216
8
1
63
5
Ò Each person gets 3 [$10]s.
Ò There are 68 [$10]s to share.
Ò 3 [$10]s 21
Ò 5 [$10]s are left.
32
216
8
1
63
51
42
9
Ò Each person gets 2 [$1]s.
Ò 50 [$1]s 1 [$1]
Ò 2 [$1]s 21
Ò 9 [$1]s are left.
$681 / 21 ∑ $32 R$9
24G
EM3SRB_G6_24A-24H.qxd
11/5/08
11:27 AM
Page 62
Whole Numbers
7720 / 25 ?
Make a table of easy multiples of the divisor.
1
* 25
2 * 25
3 * 25
4 * 25
5 * 25
6 * 25
8 * 25
10 * 25
25
50
Double 25.
75
Add 2 25 and 1 25.
100
Double 2 25.
125
Halve 10 25.
150
Double 3 25.
200
Double 4 25.
250
Move decimal point one place to the right.
Step 1: There are not enough thousands to
share 25 ways, so trade the thousands
for hundreds. Share the hundreds.
3
257
7
2
0
75
2
Ò
Ò
Ò
Ò
Each share gets 3 hundreds.
77 hundreds
3 hundreds 25 shares
2 hundreds are left.
Step 2: Trade the hundreds for tens.
Share the tens.
30 Ò There are not enough tens to share.
7
2
0
257
75
22 Ò 20 tens 2 tens
Step 3: Trade the tens for ones.
Share the ones.
308 Ò Each share gets 8 ones.
7
2
0
257
75
220 Ò 22 tens 0 ones
200 Ò 8 ones 25 shares
20 Ò 20 ones are left.
Beginning in the late 1920s and early 1930s, the
U.S. Treasury issued a small number of large
bills, including $500, $1,000, $5,000, $10,000,
and $100,000 bills. By the mid-1940s, the
Treasury stopped making these bills, and in 1969
President Nixon removed them from circulation
because they were rarely used and were
attractive to counterfeiters.
7720 / 25 ∑ 308 R20
Divide
1. 650 / 25 ?
2. 7,720 / 25 ?
3. 135
,8
1
9
Check your answers on page 424.
24H
4. 485
,2
8
6
EM3SRB_G6_60A-60I.qxd
11/6/08
12:22 PM
Page 103
Decimals and Percents
U.S. Traditional Long Division: Decimal Dividends
You can use U.S. traditional long division to divide money in
dollars-and-cents notation.
Share $5.29 among 3 people.
Step 1: Share the dollars.
1
5
.2
9
3$
3
2
Ò Each person gets 1 dollar.
Ò 1 dollar each for 3 people
Ò 2 dollars are left.
Step 2: Trade the dollars for dimes. Share the dimes.
1.7
3$
5
.2
9
3
22
2 1
1
Ò Each person gets 7 dimes. Write a decimal point
to show amounts less than a dollar.
Ò 20 dimes 2 dimes
Ò 7 dimes each for 3 people
Ò 1 dime is left.
Step 3: Trade the dime for pennies. Share the pennies.
1.76
3$
5
.2
9
3
22
2 1
19
18
1
Ò Each person gets 6 pennies.
Ò 10 pennies 9 pennies
Ò 6 pennies each for 3 people
Ò 1 penny is left.
Each person gets $1.76. There is 1¢ left.
$5.29 / 3 ∑ $1.76 R1¢
Divide.
1. $7.26 / 6 = ?
2. 7$
8
.6
1
3. 7$
5
.6
2
4. $8.04 / 3 = ?
Check your answers on page 424A.
60E
EM3SRB_G6_60A-60I.qxd
11/5/08
11:31 AM
Page 104
Decimals and Percents
You can use U.S. long division to divide decimals that do not represent money.
3.97 / 5 ?
Step 1: Trade the ones for tenths and share the tenths.
.7
53
.9
7
35
4
Ò
Ò
Ò
Ò
Each share gets 7 tenths. Write a decimal point in the quotient.
3 ones 9 tenths 39 tenths
7 tenths 5 35 tenths
4 tenths are left.
Step 2: Trade the remaining tenths for hundredths. Share the hundredths.
.79
53
.9
7
35
47
45
2
Ò Each share gets 9 hundredths.
Ò 4 tenths 7 hundredths 47 hundredths
Ò 9 hundredths 5 45 hundredths
Ò 2 hundredths are left.
At this point, you can either round 0.79 to 0.8 and write 3.97 / 5 ≈ 0.8,
or you can continue dividing into the thousandths.
Step 3: Continue dividing into the thousandths. Add a 0 to the end of 3.97.
(Adding 0s or “padding” a decimal with 0s doesn’t change its value.)
.794
53
.9
7
0
35
47
45
20
20
0
Ò Each share gets 4 thousandths.
Ò 3.97 3.970
Ò 2 hundredths 0 thousandths 20 thousandths
Ò 4 thousandths 5 20 thousandths
Ò No thousandths are left.
3.97 / 5 0.794
Divide.
1. 8.28 / 4 ?
2. 49
.6
4
3. 68
.6
7
Check your answers on page 424A.
60F
4. 38.65 / 5 = ?
EM3SRB_G6_60A-60I.qxd
11/26/08
1:58 PM
Page 105
Decimals and Percents
U.S. Traditional Long Division: Decimal Divisors
To use U.S. traditional long division to divide by a decimal
number, such as 0.6 or 3.5, you can find an equivalent
problem that has no decimal in the divisor. The answer to
the equivalent problem is the same as the answer to your
original problem.
Step 1: Think of the division problem as a fraction.
Step 2: Use the multiplication rule to find an equivalent
fraction that has no decimal in the denominator.
Step 3: Think of the equivalent fraction as a division problem.
Step 4: Solve the division problem. The answer to the
equivalent problem is the same as the answer to the
original problem.
194 / 0.4 ?
Step 1: Think of the division problem as a fraction.
194 / 0.4 194
0.4
Step 2: Find an equivalent fraction with no decimal in the denominator.
194 10
0.4 10
1940
4
Step 3: Think of the equivalent fraction as a division problem.
1940
4
1940 / 4
Step 4: Solve the equivalent division problem.
485
9
4
0
41
16
34
32
20
20
0
1940
194
Because 4 and 0.4 are equivalent fractions, the division problems 1940 / 4 and 194 / 0.4
are equivalent. So the answer to 1940 / 4 is the same as the answer to 194 / 0.4.
194 / 0.4 485
60G
EM3SRB_G6_60A-60I.qxd
11/12/08
12:05 PM
Page 106
Decimals and Percents
U.S. Traditional Long Division:
Decimal Divisors and Dividends
Sometimes both the divisor (the number you are dividing by) and the dividend
(the number being divided) are decimal numbers. To use U.S. traditional long
division in such cases, you can first find an equivalent problem that has no decimal
in the divisor. (Having a decimal part in the dividend is okay.) The answer to the
equivalent problem is the same as the answer to your original problem.
3.78 / 0.7 ?
Step 1: Think of the division problem as a fraction.
3.78 / 0.7 3.78
0.7
Step 2: Find an equivalent fraction with no decimal in the denominator.
3.78 10
0.7 10
37.8
7
Step 3: Think of the equivalent fraction as a division problem.
37.8
7
37.8 / 7
Step 4: Solve the division problem.
5.4
7
.8
73
35
28
28
0
37.8
3.78
Because 7 and 0.7 are equivalent fractions, the division problems
37.8 / 7 and 3.78 / 0.7 are equivalent. So the answer to 37.8 / 7 is the
same as the answer to 3.78 / 0.7.
3.78 / 0.7 5.4
Divide.
1. 784 / 0.7 ?
2. 36.9 / 1.5 ?
3. 4.68 / 0.03 ?
Check your answers on page 424A.
60H
4. 3.05 / 0.005 ?
EM3SRB_G6_60A-60I.qxd
11/5/08
12:24 PM
Page 107
Decimals and Percents
U.S. Traditional Long Division:
Renaming Fractions as Decimals
U.S. traditional long division can be used to rename fractions as decimals.
.375
.0
0
0
83
24
60
56
40
40
0
Use U.S. traditional long division
3
to rename 8 as a decimal.
Step 1: Write 38 as a division problem. Write 3 with several 0s
after the decimal point: 3.000. (You can always add more 0s
if you need them.)
83
.0
0
0
Step 2: Solve the division problem. Stop when the remainder is 0,
or when you have enough precision for your purposes,
or when you notice a repeating pattern.
This division problem divided evenly in three decimal places.
3
8
0.375
Use U.S. traditional long division
9
to rename 11 as a decimal.
.818181
119
.0
0
0
0
0
0
88
20
11
90
88
20
11
90
88
20
11
9
Step 1: Write 191 as a division problem. Write 9 with several 0s
after the decimal point: 9.000. (You can always add more 0s
if you need them.)
119
.0
0
0
Step 2: Solve the division problem. Stop when the remainder is 0,
or when you have enough precision for your purposes,
or when you notice a repeating pattern.
The digits 8 and 1 in the quotient appear to repeat forever.
9
11
0.818181… 0.81
Use long division to rename these fractions as decimals.
1.
2
3
2.
3
11
3.
8
9
4.
5
6
Check your answers on page 424A.
60I