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Chapter 1

Whole Numbers; How

to Dissect and Solve

Problems

McGraw-Hill/Irwin Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.

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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

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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.

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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 + 10 + 1

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Whole-number place-value chart

Each place is worth 10 times the place to

its immediate right. 1,605,743,891,412

Trillions Billions Millions UnitsThousands

No decimal point shown because this is a whole number.

hu

nd

red

trillion

s

ten

trillion

s

trillion

s

co

mm

a (,)

hu

nd

red

billio

ns

ten

billio

ns

billio

ns

co

mm

a

hu

nd

red

millio

ns

ten

millio

ns

millio

ns

co

mm

a

hu

nd

red

tho

usa

nd

s

ten

tho

usa

nd

s

thousands

co

mm

a

hu

nd

red

s

ten

s

on

es

de

cim

al p

oin

t

1 , 6 0 5 , 7 4 3 , 8 9 1 , 4 1 2 .

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Writing numeric and verbal whole numbers

Trillions Billions Millions UnitsThousands

No decimal point shown because this is a whole number.

hu

nd

red

trillion

s

ten

trillion

s

trillion

s

co

mm

a (,)

hu

nd

red

billio

ns

ten

billio

ns

billio

ns

co

mm

a

hu

nd

red

millio

ns

ten

millio

ns

millio

ns

co

mm

a

hu

nd

red

tho

usa

nd

s

ten

tho

usa

nd

s

thousands

co

mm

a

hu

nd

red

s

ten

s

on

es

de

cim

al p

oin

t

1 , 6 0 5 , 7 4 3 , 8 9 1 , 4 1 2 .

One trillion, six hundred five billion, seven hundred forty-three

million, eight hundred ninety-one thousand, four hundred twelve

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AND

• The decimal point is where you say AND

$891,412.04

Eight hundred ninety-one thousand,

four hundred twelve (dollars).

Eight hundred ninety-one thousand,

four hundred twelve (dollars), AND

four cents.

$891,412

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• 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).

Rounding Numbers

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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

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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’tadd 1.)

9,342

• Change all the numbers to the right of the identified digit to zeros.

9,340

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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

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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

• As you can see, rounding all the way is not very accurate.

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• 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 millions thousands units

Converting Parts to a Regular, Whole Number

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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.

Sales: One

hundred ninety-

four million

dollars.

Profit: Twenty-

two million, five

hundred fifty-

six thousand

dollars.

Sales and

profit rounded

all the way.

Express each

verbal form in

numeric form.

Identify the

leftmost digit in

each number.

Round it.

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

Facts Solving Steps to Key

for? Take Points

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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.

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Adding Whole Numbers

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.

Example

2 1 1

1,362

5,913

8,924

6,594

22,793

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Alternative check

Add each column as a separate total and then combine. The end result is the same.

1,362

5,913

8,924

6,594

13

18

2 6

20

22,793

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Estimate Addition by Rounding All the Way

Example

2 1 1

1,362

5,913

8,924

6,594

22,793

Example

211

1,000

6,000

9,000

7,000

23,000

*Final answer

could have more

than one non-

zero since total is

not rounded all

the way.

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Subtracting Whole Numbers

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

Example

6 14 7 10

74,580 (Minuend)

-56,114 (Subtrahend)

18,466 Difference

Check

56,114

+18,466

74,580

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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

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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 12,000,000

Hershey Kisses Inventory Problem

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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

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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 - 879,100

Amount overestimated 20,900

Jackson Manufacturing Company Problem

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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 10

x3

30

100

100 + 100 + 100 + 100 = 400 x4

400

Usually you put the smaller number on the bottom.

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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.

Example

418 (Multiplicand)

x52 (Multiplier)

836 (Partial Product)

20 90 (Partial Product)

21,736 (Product)

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Checking and Estimating Multiplication

Check

52

x 418

416

52

20 8

21,736

Estimate

400

x 50

20,000

Check the multiplication process by

reversing the multiplicand and

multiplier and then multiplying.

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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.

65

x 42

130

260

27,300,000

Example

65000 (3 zeros)

x 420 (1 zeros)

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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

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Other Ways to Show Multiplication

• The asterisk also means multiplication.

*2 x 8 = 16

2 * 8 = 16

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Division

• Division tells us how many times one

number is contained in another number.

• How many 2s are contained in 10?

5 (10 divided by 2 is 5)

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Division Terminology

• How many times

one number

(Divisor) is

contained in

another number

(Dividend).

• The result is the

Quotient.

Example

18

Divisor 15 270 Dividend

15

120

120

0

Quotient

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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.

36 R 111 Quotient

Divisor 138 5,079 Dividend

414

939

828

111

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Estimating and Checking Division

36 R 111 Quotient

Divisor 138 5,079 Dividend

414

939

828

111

Check

138

x 36

828

4 14

4,968

+ 111

5,079Estimate

50

100 5,000

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Ways of Showing Division

(600 divided by 3)

3 600

600

3

600/3

600 3

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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.

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

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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.

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Calculating Annual Sales

3,500 sales per week

x52 weeks in a year

7000

17500

182,000 total sales per year

$182,000

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Calculating Total Annual Sales to Each Company

(Total Sales Divided by Number of Companies)

45500

4 182000

16

22

20

20

20

00

$45,500

per company

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Dunkin’ Donuts

Sales per week 3,500

Weeks in Year x 52

Total Annual Sales 182,000

Divide total annual sales by

number of companies

182,000/4

Total annual sales per

company

45,500

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THE END