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Whole Numbers; How to Dissect and Solve Word Problems

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Whole Numbers; How to Dissect and Solve Word Problems

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Whole Numbers; How to Dissect and Solve Word Problems

Kirkwood Community College

January 26, 2009

Presented by Sanh Tran, MBA, CPIM, CTL

Chapter 1

Whole Numbers: How to Dissect and Solve Problems

Learning Unit Objectives

Whole Numbers; How to Dissect and Solve Word ProblemsReading, Writing, and Rounding Whole Numbers

LU1.1

Use place values to read and write numeric and verbal whole numbers

Round whole numbers to the indicated position

Use blueprint aid for dissecting and solving a word problem

1-3

Learning Unit Objectives

Whole Numbers; How to Dissect and Solve Word ProblemsAdding and Subtracting Whole Numbers

LU1.2

Add whole numbers; check and estimate addition computations

Subtract whole numbers; check and estimate subtraction computations

1-4

Learning Unit Objectives

Whole Numbers; How to Dissect and Solve Word ProblemsMultiplying and Dividing Whole Numbers

LU1.3

Multiply whole numbers; check and estimate multiplication computations

Divide whole numbers; check and estimate computations

1-5

Decimal System

- U.S. numbering system: Decimal system
- Base 10 system
- Decimal point: A dividing point that separates the whole numbers from the decimal numbers.
- Example:
145.79

Whole-number place-value chart

Hundred Thousands

Hundred trillions

Ten trillions

Trillions

Ten Thousands

Hundred millions

Millions

Ten millions

Hundred billions

Ten billions

Billions

Ones

Comma

Comma

Comma

Thousands

Comma

Hundreds

Tens

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

Figure 1.1

1,605,743,891,412

Trillions

Billions

Millions

Thousands

Units

Decimal Point

Writing numeric and verbal whole numbers

Hundred Thousands

Hundred trillions

Ten trillions

Trillions

Ten Thousands

Hundred millions

Millions

Ten millions

Hundred billions

Ten billions

Billions

Ones

Comma

Comma

Comma

Thousands

Comma

Hundreds

Tens

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

Trillions

Billions

Millions

Thousands

Units

Decimal Point

Converting parts to a regular whole number

2,4

Convert 2.4 billion to a

regular whole number

Step 1. Drop decimal point and

insert a comma

Step 2. Add zeros so the leftmost digit

ends in the word name of the amount you

want to convert. Be sure to add commas

as needed.

2,400,000,000

Rounding Whole Numbers

9362

9462

Step 1. Identify the place

value of the digit you want

to round

Step 3. Drop all digits to the right

of the identified digit

Step 2. Identify the digit to the right.

If 5 or more, increase the identified digit

by 1, if less than 5 do not change

9400

Rounding all the way

9362

9362

Step 1. Identify leftmost

digit

Step 3. Change all other digits to

zero

Step 2. Identify the digit to the right.

If 5 or more, increase the identified digit

by 1, if less than 5 do not change

9000

How to Dissect and Solve a Word Problem

Organization and persistence

General Problem-Solving Procedure

- Step 1. State the problem(s)
- Step 2. Decide on the best methods to solve the problem(s)
- Step 3. Does the solution make sense?
- Step 4. Evaluate results

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 leftmost digit in each number.

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

Addition

- Addends: Numbers that are to be added together in an addition.
- Sum (Amount or total): The result of an addition.

Adding Whole Numbers

Example

2 1 1

1,362

5,913

8,924

6,594

22,793

3 Steps

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.

Alternative check

1,362

5,913

8,924

6,594

13

18

2 6

20

22,793

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

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.

Subtraction

- Minuend: The larger number to from which to subtract another number.
- Subtrahend: The number that is to be subtracted (taken away) from another number.
- Difference: The result of a subtraction.

Subtracting Whole Numbers

Example

12

3 2 12

4,327 (Minuend)

-1,340 (Subtrahend)

2,987 Difference

3 Steps

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

Check

2,987

+1,340

4,327

Multiplication

- Multiplicand: The top number that we want to multiply in a multiplication.
- Multiplier: The bottom number that is used to multiply another number.
- Product: The final answer (result) of a multiplication.

Multiplication of Whole Numbers

4 Steps

1. Align the multiplicand and multiplier at the right.

2. Multiplying the right digit of the multiplier with the right digit of the multiplicand. Keep multiplying as you move left through the multiplicand.

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)

x 52 (Multiplier)

836

20 90 (Partial Product)

21,736 (Product)

Checking and Estimating Multiplication

Check

52

x 418

416

52

20 8

21,736

Estimate

50

x 400

20,000

Check the multiplication process by reversing the multiplicand and multiplier and then multiplying

Multiplication Shortcut with Numbers Ending in Zero

3 Steps

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. Attach the number of zeros counted in Step 2 to your answer

Solution

65

x 42

130

260

27,300,000

Example

65000 (3 zeros)

x 420 (1 zeros)

(4 zeros)

Multiplying a Whole Number by a Power of 10

2 Steps

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. Insert commas as needed.

99 x 10 = 990 = 990 <----Add 1 Zero

99 x 100 = 9,900 = 9,900 <----Add 2 Zero

99 x 1,000 = 99,000 = 99,000 <----Add 3 Zero

Division

- Dividend: The number that will be divided by another number.
- Divisor: The number that is used to divide another number.
- Quotient: The result of a division.
- Partial quotient: Part of the result of an uneven division, excluding the remainder.
- Remainder: The leftover amount in an uneven division.

Division of Whole Numbers

How many times one number (Divisor) is contained in another number (Dividend). The result is the Quotient.

Example

18Quotient

Divisor15270Dividend

15

120

120

0

Division of Whole Numbers

How many times one number (Divisor) is contained in another number (Dividend). The result is the Quotient.

Example

36 R 111Quotient

Divisor138 5,079Dividend

4 14

939

828

111

Estimating and Checking Division

Example

36 R 111Quotient

Divisor138 5,079Dividend

4 14

939

828

111

Check

138

x 36

828

4 14

4,968

+ 111

5,079

Estimate

50

100 5,000

Division Shortcut with Numbers Ending in Zeros

2 Steps

1. Count the number of ending 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

Problem 1-53

- 4,528,000
- 45
- 28
- 25
- 30
- 30

905,600 average

- Website:Average daily unique visitor:
- Orbitz.com1,527,000
- Mypoints.com1,356,000
- Americangreetings.com 745,000
- Bizrate.com 503,000
- Half.com 397,000

Solution:

1,527,000

1,356,000

745,0000

503,000

+ 397,000

4,528,000 visitors

Problem 1-54:

Solution:

1. Calculate shares sold:

190+450+450+900= 1,990

2. Remaining shares Lee Wong owned:

5,000 shares bought

- 1,990 shares sold

---------------------------

3,010

3, Total values of Lee’s stock:

3,010 shares x $48=$144,480

Problem 1-56

- 90, 65, 85, 80, 75 and 90
↓

- Lowest grade: 65
- 90+85+80+75+90 = 420
- 5 different grades
- 420÷5 = 84 (average grade)

Problem 1-63:

Solution (a):

Total customers in the week:

90 + 70 + 65 + 310 = 535 customers

Total sales for the week:

535

x $9

$4,815

Solution (b):

52 weeks in a year

Total sales for the year:

$4,815 x 52 = $250,380

Problem 1-65:

Solution:

1.Calculate the total deductions:

$1,462 + $3,782 + $884 = 6,128

2. Calculate net pay:

$61,000

- 6,128

_______

$54,872

Problem 1-69:

Solution:

Expenses:

$350 + $44 + $160 + $60=614

Deposit: 1,200

$ 900

+ 1,200

$2,100 (Subtotal after deposit)

- 614 (Less subtotal for expenses)

$1,486

Reference

- Slater, J. (2008). Practical business math procedures (9th ed.). New York: McGraw-Hill/Irwin