Transcript TGBasMathP4_01_05

Whole Numbers
Copyright © Cengage Learning. All rights reserved.
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S E C T I O N 1.5
Dividing Whole Numbers
Copyright © Cengage Learning. All rights reserved.
Objectives
1.
Write the related multiplication statement for a
division.
2.
Use properties of division to divide whole
numbers.
3.
Perform long division (no remainder).
4.
Perform long division (with a remainder).
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Objectives
5.
Use tests for divisibility.
6.
Divide whole numbers that end with zeros.
7.
Estimate quotients of whole numbers.
8.
Solve application problems by dividing whole
numbers.
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Dividing Whole Numbers
Division of whole numbers is used by everyone. For
example, to find how many 6-ounce servings a chef can get
from a 48-ounce roast, he divides 48 by 6.
To split a $36,000 inheritance equally, a brother and sister
divide the amount by 2. A professor divides the 35 students
in her class into groups of 5 for discussion.
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Write the related multiplication
statement for a division
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Write the related multiplication statement for a division
To divide whole numbers, think of separating a quantity into
equal-sized groups.
For example, if we start with a set of 12 stars and divide
them into groups of 4 stars, we will obtain 3 groups.
A set of 12 stars.
There are 3 groups of 4 stars.
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Write the related multiplication statement for a division
We can write this division problem using a division
symbol , a long division symbol
or a fraction bar
We call the number being divided the dividend and the
number that we are dividing by is called the divisor. The
answer is called the quotient.
Division symbol
Long division symbol
Fraction bar
We read each form as “12 divided by 4 equals (or is) 3.”
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Write the related multiplication statement for a division
Recall that multiplication is repeated addition. Likewise,
division is repeated subtraction. To divide 12 by 4, we ask,
“How many 4’s can be subtracted from 12?”
Subtract 4 one time.
Subtract 4 a second time.
Subtract 4 a third time.
Since exactly three 4’s can be subtracted from 12 to get 0,
we know that 12  4 = 3.
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Write the related multiplication statement for a division
Another way to answer a division problem is to think in
terms of multiplication.
For example, the division 12  4 asks the question, “What
must I multiply 4 by to get 12?” Since the answer is 3, we
know that
12  4 = 3 because 3  4 = 12
We call 3  4 = 12 the related multiplication statement for
the division 12  4 = 3.
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Write the related multiplication statement for a division
In general, to write the related multiplication statement for a
division, we use:
Quotient  divisor = dividend
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Example 1
Write the related multiplication statement for each division.
a.
b.
c.
Strategy:
We will identify the quotient, the divisor, and the dividend in
each division statement.
Solution:
Dividend
a. 10  5 = 2 because
Quotient
Divisor
2  5 = 10.
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Example 1 – Solution
b.
c.
because 4  6 = 24.
because 7  3 = 21.
cont’d
4 is the quotient, 6 is the
divisor, and 24 is the dividend.
7 is the quotient, 3 is the divisor,
and 21 is the dividend.
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2
Use properties of division to
divide whole numbers
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Use properties of division to divide whole numbers
Recall that the product of any whole number and 1 is that
whole number.
We can use that fact to establish two important properties
of division. Consider the following examples where a whole
number is divided by 1:
8  1 = 8 because 8  1 = 8.
because 4  1 = 4.
because 20  1 = 20.
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Use properties of division to divide whole numbers
These examples illustrate that any whole number divided
by 1 is equal to the number itself.
Consider the following examples where a whole number is
divided by itself:
6  6 = 1 because 1  6 = 6.
because 1  9 = 9.
because 1  35 = 35.
These examples illustrate that any nonzero whole number
divided by itself is equal to 1.
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Use properties of division to divide whole numbers
Recall that the product of any whole number and 0 is 0. We
can use that fact to establish another property of division.
Consider the following examples where 0 is divided by a
whole number:
0  2 = 0 because 0  2 = 0.
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Use properties of division to divide whole numbers
because 0  7 = 0.
because 0  42 = 0.
These examples illustrate that 0 divided by any nonzero
whole number is equal to 0.
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Use properties of division to divide whole numbers
We cannot divide a whole number by 0. To illustrate why,
we will attempt to find the quotient when 2 is divided by 0
using the related multiplication statement shown below.
Division statement
Related multiplication statement
?  0 = 0.
There is no number that gives
2 when multiplied by 0.
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Use properties of division to divide whole numbers
Since does not have a quotient, we say that division of
2 by 0 is undefined. Our observations about division of 0
and division by 0 are listed below.
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Perform long division
(no remainder)
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Perform long division (no remainder)
A process called long division can be used to divide larger
whole numbers.
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Example 2
Divide using long division: 2,514  6. Check the result.
Strategy:
We will write the problem in long-division form and follow a
four-step process: estimate, multiply, subtract, and bring
down.
Solution:
To help you understand the process, each step of this
division is explained separately. Your solution need only
look like the last step.
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Example 2 – Solution
cont’d
We write the problem in the form
The quotient will
appear above the long division symbol. Since 6 will not
divide 2,
we divide 25 by 6.
Next, we multiply 4 and 6, and subtract their product, 24,
from 25, to get 1.
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Example 2 – Solution
cont’d
Now we bring down the next digit in the dividend, the 1,
and again estimate, multiply, and subtract.
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Example 2 – Solution
cont’d
To complete the process, we bring down the last digit in
the dividend, the 4, and estimate, multiply, and subtract
one final time.
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Example 2 – Solution
cont’d
To check the result, we see if the product of the quotient
and the divisor equals the dividend.
The check confirms that 2,514  6 = 419.
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Perform long division (no remainder)
We can see how the long division process works if we write
the names of the place value columns above the quotient.
The solution for Example 2 is shown below in more detail.
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Perform long division (no remainder)
The extra zeros (shown in the steps highlighted in red and
blue) are often omitted.
We can use long division to perform divisions when the
divisor has more than one digit. The estimation step is often
made easier if we approximate the divisor.
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Perform long division
(with a remainder)
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Perform long division (with a remainder)
Sometimes, it is not possible to separate a group of objects
into a whole number of equal-sized groups.
For example, if we start with a set of 14 stars and divide
them into groups of 4 stars, we will have 3 groups of 4 stars
and 2 stars left over. We call the left over part the
remainder.
A set of 14 stars.
There are 3 groups of 4 stars.
There are 2
stars left over.
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Perform long division (with a remainder)
In the next long division example, there is a remainder. To
check such a problem, we add the remainder to the product
of the quotient and divisor. The result should equal the
dividend.
(Quotient  divisor) + remainder = dividend
Recall that the
operation within the
parentheses must be
performed first.
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Example 4
Divide:
Check the result.
Strategy:
We will follow a four-step process: estimate, multiply,
subtract, and bring down.
Solution:
Since 23 will not divide 8, we divide 83 by 23.
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Example 4 – Solution
cont’d
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Example 4 – Solution
cont’d
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Example 4 – Solution
cont’d
The quotient is 36, and the remainder is 4. We can write
this result as 36 R 4.
To check the result, we multiply the divisor by the quotient
and then add the remainder. The result should be the
dividend.
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Example 4 – Solution
cont’d
Check:
Quotient
(36
Divisor

23) +
Remainder
4
= 828 + 4
= 832
Dividend
Since 832 is the dividend, the answer 36 R 4 is correct.
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5
Use tests for divisibility
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Use tests for divisibility
We have seen that some divisions end with a 0 remainder
and others do not. The word divisible is used to describe
such situations.
Since 27  3 = 9, with a 0 remainder, we say that 27 is
divisible by 3. Since 27  5 = 5 R 2, we say that 27 is not
divisible by 5.
There are tests to help us decide whether one number is
divisible by another.
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Use tests for divisibility
There are tests for divisibility by a number other than 2, 3,
4, 5, 6, 9, or 10, but they are more complicated.
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Example 6
Is 534,840 divisible by:
a. 2 b. 3 c. 4 d. 5
e. 6
f. 9
g. 10
Strategy:
We will look at the last digit, the last two digits, and the sum
of the digits of each number.
Solution:
a. 534,840 is divisible by 2, because its last digit 0 is
divisible by 2.
b. 534,840 is divisible by 3, because the sum of its digits is
divisible by 3.
5 + 3 + 4 + 8 + 4 + 0 = 24 and 24  3 = 8
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Example 6 – Solution
cont’d
c. 534,840 is divisible by 4, because the number formed by
its last two digits is divisible by 4.
40  4 = 10
d. 534,840 divisible by 5, because its last digit is 0 or 5.
e. 534,840 is divisible by 6, because it is divisible by 2
and 3. (See parts a and b.)
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Example 6 – Solution
cont’d
f. 534,840 is not divisible by 9, because the sum of its digits
is not divisible by 9. There is a remainder.
24  9 = 2 R 6
g. 534,840 is divisible by 10, because its last digit is 0.
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Divide whole numbers that end
with zeros
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Divide whole numbers that end with zeros
There is a shortcut for dividing a dividend by a divisor when
both end with zeros. We simply remove the ending zeros in
the divisor and remove the same number of ending zeros in
the dividend.
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Example 7
Divide: a. 80  10
b. 47,000  100 c.
Strategy:
We will look for ending zeros in each divisor.
Solution:
There is one zero in the divisor.
a. 80  10 = 8  1
Remove one zero from the dividend and the divisor, and divide.
=8
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Example 7 – Solution
cont’d
There are two zeros in the divisor.
b. 47,000  100 = 470  1
Remove two zeros from the dividend and the divisor, and divide.
= 470
c. To find
we can drop one zero from the divisor and the dividend
and perform the division
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Example 7 – Solution
cont’d
Thus, 9,800  350 is 28.
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Estimate quotients of whole
numbers
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Estimate quotients of whole numbers
To estimate quotients, we use a method that approximates
both the dividend and the divisor so that they divide easily.
There is one rule of thumb for this method: If possible,
round both numbers up or both numbers down.
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Example 8
Estimate the quotient: 170,715  57
Strategy:
We will round the dividend and the divisor up and find
180,000  60.
Solution:
The dividend is
approximately
170,715  57
180,000  60 = 3,000
The divisor is
approximately
To divide, drop one
zero from 180,000
and from 60 and find
18,000  6.
The estimate is 3,000.
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Example 8 – Solution
cont’d
If we calculate 170,715  57, the quotient is exactly 2,995.
Note that the estimate is close: It’s just 5 more than 2,995.
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Solve application problems by
dividing whole numbers
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Solve application problems by dividing whole numbers
Application problems that involve forming equal-sized
groups can be solved by division.
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Example 9 – Managing a Soup Kitchen
A soup kitchen plans to feed 1,990 people. Because of
space limitations, only 144 people can be served at one
time. How many group seatings will be necessary to feed
everyone? How many will be served at the last seating?
Strategy:
We will divide 1,990 by 144.
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Example 9 – Solution
We translate the words of the problem to numbers and
symbols.
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Example 9 – Solution
cont’d
Use long division to find 1,990  144.
The quotient is 13, and the remainder is 118.
This indicates that fourteen group seatings are needed: 13
full-capacity seatings and one partial seating to serve the
remaining 118 people.
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