u.s. traditional long division: single-digit divisors · 2009-10-21 · u.s. traditional long...

U.S. Traditional Long Division: Single-Digit Divisors

U.S. traditional long division is another method you can use to divide.

Step 1: Share the s.$100 Step 2: Trade 4 s for 40 s.

That makes 45 s in all.$10

$10$100

1915�9�5�7��5

45

�45

07

�5

2

Whole Numbers

Divide.

1. 840 / 7 � ? 2. 6�9�8�4� 3. 4�5�3�9� 4. 5,280 / 6 � ?Check your answers on page 424.

24E

Share $957 among 5 people.

Ò Each person gets 1 .

Ò 1 each for 5 people

Ò 4 s are left.$100

$100

$10015�9�5�7��5

4

15�9�5�7��5

45 Ò 45 s are to be shared.$10

Step 3: Share the s. $10

Ò Each person gets 9 s.

Ò 9 s each for 5 people

Ò 0 s are left.$10

$10

$10

Ò 7 s are to be shared.

Ò 1 each for 5 people

Ò 2 s are left.$1

$1

195�9�5�7��5

45

�45

0

Step 4: Share the s. $1

$957 / 5 ∑ $191 R$2Each person gets $191; $2 are left over.

Ò Each person gets 1 .$1

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Note

The “leading” 0 in the quotientis shown in the problem to helpyou understand the long divisionmethod. It should not beincluded in the answer.

1. 5,376 / 6 = ? 2. 6�8�,5�8�6� 3. 4�6�,9�2�3� 4. 8,029 / 3 = ?Check your answers on page 424.

24F

U.S. traditional long division is not limited to dividing money.

Whole Numbers

Step 1: Start with the thousands. Step 2: So trade 3 thousands for 30 hundreds.Share the hundreds.

3,628 / 5 � ?Think about the problem as dividing 3,628 into 5 equal shares.

Ò There are not enough thousandsto share 5 ways.

Ò Each share gets 7 hundreds.Ò 36 hundredsÒ 7 hundreds � 5 sharesÒ 1 hundred is left.

Step 3: Trade 1 hundred for 10 tens. Share the tens.

Step 4: Trade 2 tens for 20 ones. Share the ones.

3,628 / 5 ∑ 725 R3

075�3�6�2�8��35

1

05�3�6�2�8�

Ò Each share gets 2 tens.

Ò 10 tens � 2 tensÒ 2 tens � 5 sharesÒ 2 tens are left.

0725�3�6�2�8��35

12

�10

2

Ò Each share gets 5 ones.

Ò 20 ones � 8 onesÒ 5 ones � 5 sharesÒ 3 ones are left.

07255�3�6�2�8��35

12

�10

28

�25

3

EM3SRB_G6_24A-24H.qxd 11/12/08 12:02 PM Page 60

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.

24G

Whole Numbers

Step 1: There are not enough [$100]s toshare 21 ways, so trade 6 [$100]sfor 60 [$10]s.

Share the 68 [$10]s.

Step 2: Trade the 5 [$10]s for 50 [$1]s.

Share the 51 [$1]s.

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.

3

21�6�8�1��63

5

Ò Each person gets 2 [$1]s.

Ò 50 [$1]s � 1 [$1]Ò 2 [$1]s � 21

Ò 9 [$1]s are left.

3221�6�8�1�

�63

51

�42

9

$681 / 21 ∑ $32 R$9

1 * 21 21

2 * 21 42

3 * 21 63

4 * 21 84

5 * 21 105

6 * 21 126

8 * 21 168

10 * 21 210

Double 21.

Add 2 � 21 and 1 � 21.

Double 2 � 21.

Halve 10 � 21.

Double 3 � 21.

Double 4 � 21.

Move decimal point one place to the right.

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Divide

1. 650 / 25 � ? 2. 7,720 / 25 � ? 3. 13�5�,8�1�9� 4. 48�5�,2�8�6�Check your answers on page 424.

24H

Whole Numbers

Beginning in the late 1920s and early 1930s, theU.S. Treasury issued a small number of largebills, including $500, $1,000, $5,000, $10,000,and $100,000 bills. By the mid-1940s, theTreasury stopped making these bills, and in 1969President Nixon removed them from circulationbecause they were rarely used and wereattractive to counterfeiters.

Step 1: There are not enough thousands toshare 25 ways, so trade the thousandsfor hundreds. Share the hundreds.

Step 2: Trade the hundreds for tens. Share the tens.

7720 / 25 � ?Make a table of easy multiples of the divisor.

Ò Each share gets 3 hundreds.Ò 77 hundredsÒ 3 hundreds � 25 sharesÒ 2 hundreds are left.

Step 3: Trade the tens for ones.Share the ones.

7720 / 25 ∑ 308 R20

3

25�7�7�2�0��75

2

Ò There are not enough tens to share.

Ò 20 tens � 2 tens

3025�7�7�2�0��75

22

Ò Each share gets 8 ones.

Ò 22 tens � 0 ones

Ò 8 ones � 25 shares

Ò 20 ones are left.

30825�7�7�2�0��75

220

�200

20

1 * 25 25

2 * 25 50

3 * 25 75

4 * 25 100

5 * 25 125

6 * 25 150

8 * 25 200

10 * 25 250

Double 25.

Add 2 � 25 and 1 � 25.

Double 2 � 25.

Halve 10 � 25.

Double 3 � 25.

Double 4 � 25.

Move decimal point one place to the right.

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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

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.

Step 2: Trade the dollars for dimes. Share the dimes.

Step 3: Trade the dime for pennies. Share the pennies.

Ò Each person gets 1 dollar.

Ò 1 dollar each for 3 people

Ò 2 dollars are left.

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.

Ò Each person gets 6 pennies.

Ò 10 pennies � 9 pennies

Ò 6 pennies each for 3 people

Ò 1 penny is left.

Decimals and Percents

13�$�5�.2�9�

�3

2

1.73�$�5�.2�9�

�3

2 2

�2 1

1

1.763�$�5�.2�9�

�3

2 2

�2 1

19

�18

1

Each person gets $1.76. There is 1¢ left.$5.29 / 3 ∑ $1.76 R1¢

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3.97 / 5 � ?

Step 1: Trade the ones for tenths and share the tenths.

Step 2: Trade the remaining tenths for hundredths. Share the hundredths.

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.)

3.97 / 5 � 0.794

Ò Each share gets 7 tenths. Write a decimal point in the quotient.Ò 3 ones � 9 tenths � 39 tenthsÒ 7 tenths � 5 � 35 tenthsÒ 4 tenths are left.

Ò Each share gets 9 hundredths.

Ò 4 tenths � 7 hundredths � 47 hundredthsÒ 9 hundredths � 5 � 45 hundredthsÒ 2 hundredths are left.

Ò Each share gets 4 thousandths.Ò 3.97 � 3.970

Ò 2 hundredths � 0 thousandths � 20 thousandthsÒ 4 thousandths � 5 � 20 thousandthsÒ No thousandths are left.

.7945�3�.9�7�0�

� 3 5

47

� 45

20

� 20

0

.795�3�.9�7�

� 3 5

47

� 45

2

.7

5�3�.9�7�� 3 5

4

Divide.

1. 8.28 / 4 � ? 2. 4�9�.6�4� 3. 6�8�.6�7� 4. 38.65 / 5 = ?Check your answers on page 424A.

You can use U.S. long division to divide decimals that do not represent money.

60F


EM3SRB_G6_60A-60I.qxd 11/5/08 11:31 AM Page 104

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 � ?


194 / 0.4 � �109.44

�

Step 2: Find an equivalent fraction with no decimal in the denominator.

�109.44

��

1100

� � �19

440�


�19

440� � 1940 / 4

Step 4: Solve the equivalent division problem.

Because �19

440� and �

109.44

� are equivalent fractions, the division problems 1940 / 4 and 194 / 0.4are equivalent. So the answer to 1940 / 4 is the same as the answer to 194 / 0.4.

194 / 0.4 � 485


4854�1�9�4�0�

� 16

34

� 32

20

� 20

0

60G


3.78 / 0.7 � ?


3.78 / 0.7 � �30.7.78

�

Step 2: Find an equivalent fraction with no decimal in the denominator.

�30.7.78

��1100

� � �37

7.8�


�37

7.8� � 37.8 / 7

Step 4: Solve the division problem.

Because �377.8� and �30

.7.78

� are equivalent fractions, the division problems 37.8 / 7 and 3.78 / 0.7 are equivalent. So the answer to 37.8 / 7 is thesame as the answer to 3.78 / 0.7.

3.78 / 0.7 � 5.4

60H

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 theequivalent problem is the same as the answer to your original problem.

Divide.

1. 784 / 0.7 � ? 2. 36.9 / 1.5 � ? 3. 4.68 / 0.03 � ? 4. 3.05 / 0.005 � ?Check your answers on page 424A.


5.47�3�7�.8�

� 35

2 8

� 2 8

0


Use U.S. traditional long division to rename �

38� 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.)

8�3�.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.

�38� � 0.375

U.S. Traditional Long Division: Renaming Fractions as Decimals

U.S. traditional long division can be used to rename fractions as decimals.

Use U.S. traditional long divisionto rename �1

91� as a decimal.

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.)

11�9�.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.

�191� � 0.818181… � 0.81�


.3758�3�.0�0�0�

� 2 4

60

� 56

40

� 40

0

.81818111�9�.0�0�0�0�0�0�� 8 8

20

� 11

90

� 88

20

� 11

90

� 88

20

� 11

9

Use long division to rename these fractions as decimals.

1. �23� 2. �1

31� 3. �

89� 4. �

56�

Check your answers on page 424A.

60I


u.s. traditional long division: single-digit divisors · 2009-10-21 · u.s. traditional long...

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