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Chapter 1
Whole Numbers; How
to Dissect and Solve
Problems
1-1 McGraw-Hill/Irwin
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
Numbering System
• Decimal System (base 10)
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Can write any number using these
10 digits
• Decimal point (.) separates whole
numbers from decimal numbers
1-2
Decimal Point
 whole numbers . decimal numbers 
left of decimal point right of decimal point
Decimal Point
We will be concerned with whole numbers
in this chapter. They are to the left of the
decimal point.
1-3
Place Values
• In the next slide, you will see the place value
diagram of whole number groups.
• Each place is worth 10 times the place to its
right.
111
100
1-4
+ 10
+
1
Whole-number place-value chart
Each place is worth 10 times the place to
its immediate right.
Trillions
Billions
1,605,743,891,412
Millions
Thousands
Units
trillions
comma (,)
hundred billions
ten billions
billions
comma
hundred millions
ten millions
millions
comma
hundred thousands
ten thousands
thousands
comma
hundreds
tens
ones
decimal point
ten trillions
hundred trillions
1
,
6
0
5
,
7
4
3
,
8
9
1
,
4
1
2
.
No decimal point shown because this is a whole number.
1-5
Writing numeric and verbal whole numbers
One trillion, six hundred five billion, seven hundred forty-three
million, eight hundred ninety-one thousand, four hundred twelve
Trillions
Billions
Millions
Thousands
Units
trillions
comma (,)
hundred billions
ten billions
billions
comma
hundred millions
ten millions
millions
comma
hundred thousands
ten thousands
thousands
comma
hundreds
tens
ones
decimal point
ten trillions
hundred trillions
1
,
6
0
5
,
7
4
3
,
8
9
1
,
4
1
2
.
No decimal point shown because this is a whole number.
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AND
•
The decimal point is where you say AND
$891,412
Eight hundred ninety-one thousand,
four hundred twelve (dollars).
$891,412.04
Eight hundred ninety-one thousand,
four hundred twelve (dollars), AND
four cents.
1-7
Rounding Numbers
• Because they deal with such large figures,
government statistics and financial reports for
large organizations use rounded numbers.
• Rounded numbers are good for quick estimates
and are easier to remember than exact numbers.
• Example: 2,403,895,682 or 2.4 billion
• The more rounded a number is, the more
approximate it is (less exact).
• Numbers can be rounded to any identified digit
place value including the first (left most).
1-8
Rounding Whole Numbers, Example 1 (rounded up)
•
Identify the place value of the digit to where you want to round. (For example,
let’s round to the nearest hundred.)
9,362
•
Look at the number to the right of that digit—(the 6)
If the number to the right is 5 or higher, add 1 to the identified digit
(in our case the hundreds place, the 3).
If the number to the right is less than 5, do not change the identified
digit. (In our case, it was higher than 5.)
9,462
Change all the numbers to the right of the rounded, identified digit to
zeros.
9,400
1-9
Rounding Whole Numbers, Example 2 (rounded down)
•
Identify the place value of the digit you want to round to. (For
example, let’s round to the nearest 10.)
9,342
•
Look at the number to the right of that digit (the 2).
•
If the number you are looking at is 5 or higher, add 1 to the
identified digit. (Ours is 2, and not 5 or higher; therefore, we will
not add 1 to the identified digit.)
•
If the number to the right is less than 5, do not change the
identified digit. (In our case, it is less; therefore, we won’t
add 1.)
9,342
•
Change all the numbers to the right of the identified digit to
zeros.
9,340
1-10
Rounding all the Way, Example 1
•
In rounding all the way, you round to the first digit of the
number (leftmost).
•
Rounding a digit to a specific place value depends on
the degree of accuracy you need in your estimate.
•
Example:
24,800
(4 is less than 5; so we do not increase the first digit
by 1)
So, this number, when rounded all the way is:
20,000
1-11
Rounding all the Way, Example 2
• Again, in rounding all the way, you round to the first
digit of the number (leftmost).
• Example:
26,100
(6 is 5 or more, so we increase the first digit by 1)
So, this number, when rounded all the way is:
30,000
•
1-12
As you can see, rounding all the way is not very accurate.
Converting Parts to a Regular, Whole Number
•
We will convert 2.4 billion to a regular, whole number in the
following steps.
• Drop the decimal point and replace it with a comma.
2,4 billion
• Add the needed zeros to the right
2,400,000,000
billions
1-13
millions
thousands
units
How to Dissect and Solve a Word Problem
Tootsie Roll Industries sales reached one hundred ninety-four million
dollars and a record profit of twenty-two million, five hundred fifty-six
thousand dollars. Round the sales and profit figures all the way.
Facts
Sales: One
hundred ninetyfour million
dollars.
Profit: Twentytwo million, five
hundred fiftysix thousand
dollars.
Solving
for?
Sales and
profit rounded
all the way.
Steps to
Take
Express each
verbal form in
numeric form.
Identify the
leftmost digit in
each number.
Round it.
Key
Points
Rounding all
the way means
only the
leftmost digit
will remain. All
other digits
become zeros.
Sales: One hundred ninety-four million dollars. ----------->$194,000,000 -----------> $200,000,000
Profit: Twenty-two million, five hundred fifty-six thousand dollars -> $22,556,000 --> $20,000,000
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Facts
•Sales are one hundred ninety-four million dollars.
•Profit is five hundred twenty thousand dollars.
Solving for
•Sales and profit rounded all the way
Steps to take
•Express each verbal form in numeric form; then identify leftmost
digit in each number.
Sales => $194,000,000 Profit => $520,000
•Round the first (leftmost) digit, and change the rest of the digits to 0.
For sales, 9 is 5 or higher, so we add 1 to the leftmost digit.
For profit, 2 is less than 5, so we don’t add 1 to the leftmost digit.
Sales => 200,000,000
Profit =>500,000
Key Points
Rounding all the way means only the leftmost digit will remain. All other
digits become zeros.
1-15
Adding Whole Numbers
Example
1. Align the numbers according to
their place values
2. Add the units column. Write the
sum below the column. If the
sum is more than 9, write the
units digit and carry the tens
digit.
3.
Moving to the left, repeat Step 2
until all place values are added.
Small numbers in red are
amounts carried.
1-16
211
1,362
5,913
8,924
6,594
22,793
Alternative check
1,362
5,913
Add each column as
a separate total and
then combine. The
end result is the
same.
8,924
6,594
13
18
26
20
22,793
1-17
Estimate Addition by Rounding All the Way
Example
211
1-18
Example
211
1,362
1,000
5,913
6,000
8,924
9,000
6,594
7,000
22,793
23,000
*Final answer
could have more
than one nonzero since total is
not rounded all
the way.
Subtracting Whole Numbers
Example
1. Align the minuend and subtrahend by
place values
2. Begin the subtraction with the units
digits. Write the difference below the
column.
If the units digit in the minuend is smaller than the
digit in the subtrahend, borrow
1 from the tens digit in the minuend.
3. Moving to the left, repeat Step 2 until
all place values in the subtrahend are
subtracted
1-19
6 14
7 10
74,580 (Minuend)
-56,114 (Subtrahend)
18,466
Difference
Check
56,114
+18,466
74,580
Hershey Kisses Problem
•
Facts
• Produced 25 million
• Shipped 4 million to Japan
• Shipped 3 million to France
• Shipped 6 million to locations in the US
• Solving for
• Total Kisses left in inventory (none before production)
• Inventory balance rounded all the way
• Steps
• Total Kisses produced
• Total Kisses shipped
• Total Kisses left in inventory
• Key Points
• Minuend-Subtrahend = Difference
• Rounding all the way is rounding to left most digit
1-20
Hershey Kisses Inventory Problem
Shipped to Japan
4,000,000
Shipped to France
3,000,000
Shipped to US
6,000,000
Total shipped
13,000,000
Total Kisses produced
25,000,000
Subtract total shipped
- 13,000,000
Inventory
1-21
12,000,000
Jackson Manufacturing Company Problem
•
Facts
• Projected (estimated) 2003 sales: $900,000
• Sales to Major Clients = 510,000
• Sales to Other Clients = 369,100
• Solving for
• The amount that sales were over or under estimated
• Steps
• Keep in mind the total projected sales
• Add up the actual client sales figures
• Subtract total actual sales
• Difference will be the amount over/under estimated
• Key Points
• Projected sales is minuend
• Actual sales is the subtrahend
• Amount over/under estimated will be the difference
1-22
Jackson Manufacturing Company Problem
Sales to Major Clients
510,000
Sales to Other Clients
369,100
Total Actual Sales
879,100
Projected (estimated) sales
900,000
Total actual sales
Amount overestimated
1-23
- 879,100
20,900
Multiplication
• Multiplication is a shortcut to addition.
• For instance, if you multiply a number by 3, you are
adding the number 3 times.
10 + 10 + 10 = 30
100 + 100 + 100 + 100 = 400
10
x3
30
100
x4
400
Usually you put the smaller number on the bottom.
1-24
Multiplication of Whole Numbers
1. Align the multiplicand and multiplier at the
right.
2. Multiply the right digit of the multiplier by
the right digit of the multiplicand. Keep
multiplying as you move left through the
multiplicand, carrying where necessary.
3. Your partial product right digit or first digit
is placed directly below the digit in the
multiplier that you used to multiply.
4. Continue steps 2 and 3 until multiplication
process is complete. Add the partial
products to get the final product.
1-25
Example
418 (Multiplicand)
x52
(Multiplier)
836 (Partial Product)
20 90 (Partial Product)
21,736 (Product)
Checking and Estimating Multiplication
Check
Estimate
52
400
x 418
416
52
x
50
20,000
20 8
21,736
Check the multiplication process by
reversing the multiplicand and
multiplier and then multiplying.
1-26
Multiplication Shortcut with
Numbers ending in Zero
1. When zeros are at the end of
the multiplicand or the
multiplier, or both, disregard
the zeros and multiply.
2. Count the number of zeros in
the multiplicand and
multiplier, (4).
3. Then attach the number of
zeros counted in Step 2 to
your answer.
1-27
Example
65000 (3 zeros)
x 420 (1 zeros)
65
x 42
130
260
27,300,000
Multiplying a Whole Number
by (a Power of) 10
1. Count the number of zeros in the power of 10.
2. Attach that number of zeros to the right side of
the other whole number to obtain the answer.
3. Insert commas as needed
99 x 10
=
990 =
990 <----Add 1 Zero
99 x 100 = 9,900 = 9,900 <----Add 2 Zeros
99 x 1,000 = 99,000 = 99,000 <----Add 3 Zeros
1-28
Other Ways to Show Multiplication
•
The asterisk also means multiplication.
*
2 x 8 = 16
2 * 8 = 16
1-29
Division
• Division tells us how many times one
number is contained in another number.
• How many 2s are contained in 10?
5
1-30
(10 divided by 2 is 5)
Division Terminology
• How many times
one number
(Divisor) is
contained in
another number
(Dividend).
• The result is the
Quotient.
1-31
Divisor
Example
18
15 270
15
120
120
0
Quotient
Dividend
Sometimes there is something left over
• How many times
one number
(Divisor) is
contained in
another number
(Dividend).
• The result is the
Quotient.
• The R stands for
remainder.
1-32
Divisor
36 R 111
138 5,079
414
939
828
111
Quotient
Dividend
Estimating and Checking Division
Check
138
x
36
828
4 14
4,968
Divisor
36 R 111
Quotient
138 5,079
Dividend
414
939
828
111
+ 111
5,079
1-33
Estimate
50
100 5,000
Ways of Showing Division
(600 divided by 3)
3 600
600
3
600/3
600  3
1-34
Division Shortcut with Numbers
Ending in Zeros
1. Count the number of zeros in the divisor.
2. Drop the same number of zeros in the dividend
as in the divisor, counting from right to left.
1-35
95,000/10
=> 95,000 = 9,500
Drop 1 Zero
95,000/100
=> 95,000 =
950
Drop 2 Zeros
95,000/1,000 => 95,000 =
95
Drop 3 Zeros
Dunkin’ Donuts Case
Dunkin’ Donuts has 4 customers. It has total combined sales of
$3,500 per week. All four customers buy the same amount each
week. What is the total annual sales to each of these companies?
Facts
Sales per week are $3,500. There are only 4 customers (companies)
They all buy the same amount each week.
Solving For
Total annual sales to all four companies
Yearly sales per company
Steps to Take
Sales per week times weeks in a year (52). Total annual sales
divided by total companies will give the yearly sales per company.
1-36
Calculating Annual Sales
3,500 sales per week
x52 weeks in a year
7000
17500
182,000 total sales per year
$182,000
1-37
Calculating Total Annual Sales to Each Company
(Total Sales Divided by Number of Companies)
45500
4 182000
16
22
20
20
20
00
1-38
$45,500
per company
Dunkin’ Donuts
Sales per week
3,500
Weeks in Year
x 52
Total Annual Sales
Divide total annual sales by
number of companies
Total annual sales per
company
1-39
182,000
182,000/4
45,500
THE END
1-40