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Whole Numbers
Division Algorithms
Different symbols may be used to indicate division. For example,
94
“94 divided by 6” may be written as 94 ! 6, 6!9
"4
" , 94 / 6, or "6".
♦ The number that is being divided is called the dividend.
Four ways to show
“123 divided by 4”
♦ The number that divides the dividend is called the divisor.
123 ! 4
123 / 4
4!1
"2
"3
"
123
""
4
♦ The answer to a division problem is called the quotient.
♦ Some numbers cannot be divided evenly. When this happens,
the answer includes a quotient and a remainder.
123 is the dividend.
4 is the divisor.
Partial-Quotients Method
In the partial-quotients method, it takes several steps to find
the quotient. At each step, you find a partial answer (called a
partial quotient). These partial answers are then added to
find the quotient.
Study the example below. To find the number of 6s in 1,010 first
find partial quotients and then add them. Record the partial
quotients in a column to the right of the original problem.
1,010 / 6 # ?
Write partial quotients in this column.
6!1
",0
"1
"0
"
$ 600
↓
100
410
$ 300
The first partial quotient is 100. 100 ∗ 6 # 600
Subtract 600 from 1,010. At least 50 [6s] are left in 410.
50
110
$ 60
Think: How many [6s] are in 1,010? At least 100.
The second partial quotient is 50. 50 ∗ 6 # 300
Subtract. At least 10 [6s] are left in 110.
10
50
The third partial quotient is 10. 10 ∗ 6 # 60
Subtract. At least 8 [6s] are left in 50.
$ 48
8
2
168
↑
↑
The fourth partial quotient is 8. 8 ∗ 6 # 48
Subtract. Add the partial quotients.
Remainder Quotient
168 R2
The answer is 168 R2. Record the answer as 6!1
",0
"1
"0
"
or write 1,010 / 6 → 168 R2.
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Whole Numbers
The partial-quotients method works the same whether you
divide by a 2-digit or a 1-digit divisor. It often helps to write
down some easy facts for the divisor.
Divide 600 by 22.
22!6
"0
"0
"
" 440 20 (20 [22s] in 600)
160
1 ∗ 22 # 22
" 110
5 (5 [22s] in 160)
2 ∗ 22 # 44
50
5 ∗ 22 # 110
" 44
2 (2 [22s] in 50)
10 ∗ 22 # 220
6 27
27 R6
Record the answer as 22!6
"0
"0
"
or write 600 / 22 → 27 R6.
Some facts for 22
(to help find partial quotients)
There are different ways to find partial quotients when you use
the partial-quotients method. Study the example below. The
answer is the same for each way.
In 1820, Charles de
Colmar invented a
calculating machine
called the Arithmometer
that could both multiply
and divide numbers.
It was the first massproduced calculator.
381 / 4 # ?
One way:
4!3
"8
"1
"
A second way:
4!3
"8
"1
"
" 200 50
" 200
181
181
" 120 30
" 160
61
21
" 40 10
" 20
21
1
" 20 5
1 95
The answer, 95 R1, is the
Divide.
1. 4!7
"1
"
50
40
A third way:
4!3
"8
"1
"
" 360 90
21
" 20 5
1 95
5
95
same for each way.
2. 735 / 5
3. 342 ! 4
4. 3!6
"7
"4
"
Check your answers on page 434.
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Decimals and Percents
Division of Decimals
Here is one way to divide decimals:
Step 1: Make a magnitude estimate of the quotient.
Step 2: Divide as if the divisor and dividend were
whole numbers.
Step 3: Use the magnitude estimate to place the decimal
point in the answer.
A magnitude estimate
is a rough estimate of
the size of an answer. A
magnitude estimate tells
whether an answer is in
the ones, tens, hundreds,
and so on.
97.24 / 26 ! ?
Step 1:
Make a magnitude estimate.
• Since 26 is close to 25 and 97.24 is close to 100, the answer
to 97.24 / 26 will be close to the answer to 100 / 25.
• Since 100 / 25 ! 4, the answer to 97.24 / 26 should be in
the ones. (In the ones means between 1 and 10.)
Step 2:
Divide, ignoring the decimal point.
26!9
""
7"
2"
4
" 7800
1924
" 1040
884
" 780
104
" 104
0
300
40
30
4
374
9724 / 26 ! 374
Step 3:
Decide where to place the decimal point.
According to the magnitude estimate,
the answer should be in the ones.
Sometimes a magnitude
estimate is on the
“borderline” and you
need to be more careful.
For example, a magnitude
estimate for 2,890 / 3.4
is 3,000 / 3 ! 1,000.
This answer is “in the
thousands.” But the exact
answer may be “in the
hundreds.” You should
place the deimal point so
that the answer is close
to 1,000.
Since 2,890 / 34 ! 85,
you should attach
one zero, followed by
a decimal point:
2,890 / 3.4 ! 850.
So, 97.24 / 26 ! 3.74.
Divide.
1. 148.8 / 6
2. 25.32 / 12
Check your answers on page 434.
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3. 4.55 / 3.5
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Decimals and Percents
The answers to decimal divisions do not always come out even.
When you divide as if the divisor and dividend were whole
numbers, there may be a non-zero remainder. If the remainder
is not zero:
1. Rewrite a remainder as a fraction:
♦ Make the remainder the numerator of the fraction.
♦ Make the divisor the denominator of the fraction.
2. Add this fraction to the quotient and round the sum to the
nearest whole number.
3. Then use the magnitude estimate to place the decimal point
in the answer.
The decimal division below does not come out even.
80.27 / 4 ! ?
Make a magnitude estimate.
• Since 80.27 is close to 80, 80.27 / 4 ! 80 / 4.
• Since 80 / 4 ! 20, the answer to 80.27 / 4 should be in
the tens. (In the tens means between 10 and 100.)
The symbol ! means
is about equal to.
Divide, ignoring the decimal point.
4"8
#0
#2
#7
#
" 8000
27
" 24
3
2000
6
2006
8027 / 4 ∑ 2006 R3. The quotient is 2006, and the remainder is 3.
3
Rewrite the remainder 3 as the fraction #4#.
3
Add this fraction to the quotient: 8027 / 4 ! 2006#4#.
Round this answer to the nearest whole number, 2007.
Decide where to put the decimal point. According to the
magnitude estimate, the answer should be in the tens.
So, 80.27 / 4 ! 20.07.
Divide.
1. 8.8 / 3
2. 86.4 / 24
3. 45.2 / 3
Check your answers on page 434.
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