Chapter 1. Whole Numbers; How to Dissect and Solve Problems 1-1. McGraw-Hill/Irwin

Transcription

1 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.

2 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

3 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

4 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

5 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 Thousands Units decimal point ones tens hundreds comma thousands ten thousands hundred thousands comma millions ten millions hundred millions comma billions ten billions hundred billions comma (,) trillions ten trillions hundred trillions 1, 6 0 5, 7 4 3, 8 9 1, No decimal point shown because this is a whole number.

6 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 decimal point ones tens hundreds comma thousands ten thousands hundred thousands comma millions ten millions hundred millions comma billions ten billions hundred billions comma (,) trillions ten trillions hundred trillions 1, 6 0 5, 7 4 3, 8 9 1, No decimal point shown because this is a whole number.

7 AND The decimal point is where you say AND $891,412 Eight hundred ninety-one thousand, four hundred twelve (dollars). $891, Eight hundred ninety-one thousand, four hundred twelve (dollars), AND four cents. 1-7

8 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

9 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,

10 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 ,340

11 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,

12 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. 1-12

13 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 millions thousands units 1-13

14 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 Solving Steps to Key for? Take Points Sales: One hundred ninetyfour million dollars. Profit: Twentytwo million, five hundred fiftysix 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, > $200,000,000 Profit: Twenty-two million, five hundred fifty-six thousand dollars -> $22,556,000 --> $20,000,

15 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 Key Points Sales => 200,000,000 Profit =>500,000 Rounding all the way means only the leftmost digit will remain. All other digits become zeros.

16 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 ,362 5,913 8,924 6,594 22,

17 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, ,

18 Estimate Addition by Rounding All the Way Example ,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 nonzero since total is not rounded all the way. 1-18

19 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 ,580 (Minuend) -56,114 (Subtrahend) 18,466 Difference Check 56, ,466 74,

20 Facts Hershey Kisses Problem 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

21 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 12,000,

22 Jackson Manufacturing Company Problem 1-22 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

23 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 - 879,100 Amount overestimated 20,

24 Multiplication Multiplication is a shortcut to addition. For instance, if you multiply a number by 3, you are adding the number 3 times = x = 400 x4 400 Usually you put the smaller number on the bottom. 1-24

25 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) (Partial Product) 21,736 (Product) 1-25

26 Checking and Estimating Multiplication Check 52 x ,736 Estimate 400 x 50 20,000 Check the multiplication process by reversing the multiplicand and multiplier and then multiplying. 1-26

27 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. Example (3 zeros) x 420 (1 zeros) 65 x ,300,

28 Multiplying a Whole Number by (a Power of) Count the number of zeros in the power of 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

29 Other Ways to Show Multiplication The asterisk also means multiplication. * 2 x 8 = 16 2 * 8 =

30 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) 1-30

31 Division Terminology How many times one number (Divisor) is contained in another number (Dividend). Example 18 Quotient Divisor Dividend The result is the Quotient. 1-31

32 Sometimes there is something left over How many times one number (Divisor) is contained in another number (Dividend). 36 R 111 Quotient Divisor 138 5,079 Dividend The result is the Quotient. The R stands for remainder. 1-32

33 Estimating and Checking Division Check 138 x , R 111 Quotient Divisor 138 5,079 Dividend ,079 Estimate ,

34 Ways of Showing Division (600 divided by 3) /

35 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 1-35

36 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

37 Calculating Annual Sales 3,500 sales per week x52 weeks in a year ,000 total sales per year $182,

38 Calculating Total Annual Sales to Each Company (Total Sales Divided by Number of Companies) $45,500 per company 1-38

39 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 Total annual sales per company 182,000/4 45,

40 1-40 THE END