Rounding, Estimation, and Order

1.4 Rounding, Estimation, and Order 1.4 OBJECTIVES 1. Round a whole number at any place value 2. Estimate sums and differences by rounding 3. Estimate distance 4. Use the symbols and It is a common practice to express numbers to the nearest hundred, thousand, and so on. For instance, the distance from Los Angeles to New York along one route is 2833 miles (mi). We might say that the distance is 2800 mi. This is called rounding, because we have rounded the distance to the nearest hundred miles. One way to picture this rounding process is with the use of a number line. Example 1 Rounding to the Nearest Hundred To round 2833 to the nearest hundred: 2833 2800 2900 Because 2833 is closer to 2800, we round down to 2800. CHECK YOURSELF 1 Round 587 to the nearest hundred. 587 500 600 Example 2 Rounding to the Nearest Thousand To round 28,734 to the nearest thousand: 28,734 28,000 29,000 Because 28,734 is closer to 29,000, we round up to 29,000. CHECK YOURSELF 2 Locate 1375 and round to the nearest hundred. 1300 1400 43

44 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS Instead of using a number line, we can apply the following rule. NOTE By a certain place, we mean tens, hundreds, thousands, and so on. NOTE This is called rounding up. NOTE This is called rounding down. Step by Step: Step 1 Step 2 Step 3 Rounding Whole Numbers Identify the place of the digit to be rounded. Look at the digit to the right of that place. a. If that digit is 5 or more, that digit and all digits to the right become 0. The digit in the place you are rounding to is increased by 1. b. If that digit is less than 5, that digit and all digits to the right become 0. The digit in the place you are rounding to remains the same. Example 3 Rounding to the Nearest Ten Round 587 to the nearest ten: Tens 5 8 7 The digit to the right of the tens place NOTE 587 is between 580 and 590. It is closer to 590, so it makes sense to round up. 5 8 7 is rounded to 590 580 590 587 We identify the tens digit. The digit to the right of the tens place, 7, is 5 or more. So round up. CHECK YOURSELF 3 Round 847 to the nearest ten. Example 4 Rounding to the Nearest Hundred Round 2638 to the nearest hundred: NOTE 2638 is closer to 2600 than to 2700. So it makes sense to round down. 2 6 38 is rounded to 2600 2600 2700 2638 CHECK YOURSELF 4 Round 3482 to the nearest hundred. We identify the hundreds digit. The digit to the right, 3, is less than 5. So round down.

ROUNDING, ESTIMATION, AND ORDER SECTION 1.4 45 Let s look at some further examples of using the rounding rule. Example 5 Rounding Whole Numbers (a) Round 2378 to the nearest hundred: 2 3 78 is rounded to 2400 We have identified the hundreds digit. The digit to the right is 7. Because this is 5 or more, the 7 and all digits to the right become 0. The hundreds digit is increased by 1. (b) Round 53,258 to the nearest thousand: 5 3,258 is rounded to 53,000 We have identified the thousands digit. Because the digit to the right is less than 5, it and all digits to the right become 0, and the thousands digit remains the same. (c) Round 685 to the nearest ten: 6 8 5 is rounded to 690 The digit to the right of the tens place is 5 or more. Round up by our rule. (d) Round 52,813,212 to the nearest million: 5 2,813,212 is rounded to 53,000,000 CHECK YOURSELF 5 (a) Round 568 to the nearest ten. (b) Round 5446 to the nearest hundred. Let s look at a case in which we round up a 9. NOTE Which number is 397 closer to? 390 397 400 Example 6 Rounding to the Nearest Ten Suppose we want to round 397 to the nearest ten. We identify the tens digit and look at the next digit to the right. 3 9 7 The digit to the right is 5 or more. If this digit is 9, and it must be increased by 1, replace the 9 with 0 and increase the next digit to the left by 1. So 397 is rounded to 400.

46 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS CHECK YOURSELF 6 Round 4961 to the nearest hundred. NOTE An estimate is basically a good guess. If your answer is close to your estimate, then your answer is reasonable. Whether you are doing an addition problem by hand or using a calculator, rounding numbers gives you a handy way of deciding if the answer seems reasonable. The process is called estimating. Let s illustrate with an example. Example 7 Estimating a Sum Begin by rounding to the nearest hundred NOTE Placing an arrow above the column to be rounded can be helpful. 456 500 235 200 976 1000 344 300 2011 2000 Estimate By rounding to the nearest hundred and adding quickly, we get an estimate or guess of 2000. Because this is close to the sum calculated, 2011, our answer seems reasonable. CHECK YOURSELF 7 Round each addend to the nearest hundred and estimate the sum. Then find the actual sum. 287 526 311 378 Estimation is a wonderful tool to use while you re shopping. Every time you go to the store, you should try to estimate the total bill by rounding the price of each item. If you do this regularly, both your addition skills and your rounding skills will improve. The same holds true when you eat in a restaurant. It is always a good idea to know approximately how much you are spending. Example 8 Estimating a Sum in a Word Problem Samantha has taken the family out to dinner, and she s now ready to pay the bill. The dinner check has no total, only the individual entries, as below: Soup $2.95 Soup $2.95 Salad 1.95 Salad 1.95 Salad 1.95 Lasagna 7.25 Spaghetti 4.95 Ravioli 5.95 What is the approximate cost of the dinner? Rounding each entry to the nearest whole dollar, we can estimate the total by finding the sum 3 3 2 2 2 7 5 6 $30

ROUNDING, ESTIMATION, AND ORDER SECTION 1.4 47 CHECK YOURSELF 8 Jason is doing the weekly food shopping at FoodWay. So far his basket has items that cost $3.99, $7.98, $2.95, $1.15, $2.99, and $1.95. Approximate the total cost of these items. Earlier in this section, we used the number line to illustrate the idea of rounding numbers. The number line also gives us an excellent way to picture the concept of order for whole numbers, which means that numbers become larger as we move from left to right on the line. For instance, we know that 3 is less than 5. On the number line NOTE 3 is less than or smaller than 5. 0 1 2 3 4 5 6 7 we see that 3 lies to the left of 5. We also know that 4 is greater than 2. On the number line NOTE 4 is greater than or larger than 2. 0 1 2 3 4 5 6 7 we see that 4 lies to the right of 2. Two symbols, for less than and for greater than, are used to indicate these relationships. Definitions: Inequalities For whole numbers, we can write NOTE The inequality always points at the smaller number. 1. 2 5 (read 2 is less than 5 ) because 2 is to the left of 5 on the number line. 2. 8 3 (read 8 is greater than 3 ) because 8 is to the right of 3 on the number line. Example 9 illustrates the use of this notation. Example 9 Indicating Order with or Use the symbols or to complete each statement. (a) 7 10 (b) 25 20 (c) 200 300 (d) 8 0 (a) 7 10 (b) 25 20 (c) 200 300 (d) 8 0 7 lies to the left of 10 on the number line. 25 lies to the right of 20 on the number line.

48 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS CHECK YOURSELF 9 Use one of the symbols or to complete each of the following statements. (a) 35 25 (b) 0 4 (c) 12 18 (d) 1000 100 CHECK YOURSELF ANSWERS 1. 600 2. 1375 Round 1375 up to 1400 1300 1400 3. 850 4. 3500 5. (a) 570; (b) 5400 6. 5000 7. 1500; 1502 8. $21 9. (a) 35 25; (b) 0 4; (c) 12 18; (d) 1000 100

Name 1.4 Exercises Section Date Round each of the following numbers to the indicated place. 1. 38, the nearest ten 2. 72, the nearest ten ANSWERS 1. 3. 253, the nearest ten 4. 578, the nearest ten 5. 696, the nearest ten 6. 683, the nearest hundred 7. 3482, the nearest hundred 8. 6741, the nearest hundred 2. 3. 4. 5. 6. 9. 5962, the nearest hundred 10. 4352, the nearest thousand 11. 4927, the nearest thousand 12. 39,621, the nearest thousand 13. 23,429, the nearest thousand 14. 38,589, the nearest thousand 7. 8. 9. 10. 11. 15. 787,000, the nearest ten 16. 582,000, the nearest hundred thousand thousand 12. 13. 17. 21,800,000, the nearest million 18. 931,000, the nearest ten thousand In exercises 19 to 40, estimate each of the sums or differences by rounding to the indicated place. Then do the addition or subtraction and use your estimate to see if your actual sum or difference seems reasonable. Round to the nearest ten. 14. 15. 16. 17. 18. 19. 58 20. 92 27 37 33 85 64 21. 87 22. 78 53 67 41 53 93 42 62 86 19. 20. 21. 22. 49

ANSWERS 23. 23. 83 24. 97 27 31 24. 25. 25. 33 26. 47 21 36 26. 27. 28. 29. Round to the nearest hundred. 27. 379 28. 967 1215 2365 528 544 738 30. 31. 32. 29. 1378 30. 3145 519 889 792 259 2041 692 2518 33. 34. 31. 679 32. 824 231 358 35. 36. 33. 915 34. 697 411 539 37. 38. 39. Round to the nearest thousand. 35. 2238 36. 3678 3925 4215 5217 2032 40. 37. 9137 38. 11,548 2315 3874 7643 14,435 3092 5398 39. 4822 40. 6120 2134 4890 50

ANSWERS Use the symbol or to complete each statement. 41. 4 8 42. 0 5 43. 500 400 44. 20 15 45. 100 1000 46. 3000 2000 Solve the following applications. 47. Lunch bills. Ed and Sharon go to lunch. The lunch check has no total but only lists individual items: Soup $1.95 Soup $1.95 Salad $1.80 Salad $1.80 Salmon $8.95 Flounder $6.95 Pecan pie $3.25 Vanilla ice cream $2.25 Estimate the total amount of the lunch check. 48. Consumer spending. Olivia will purchase several items at the stationery store. Thus far, the items she has collected cost $2.99, $6.97, $3.90, $2.15, $9.95, and $1.10. Approximate the total cost of these items. 49. Test scores. Oscar scored 78, 91, 79, 67, and 100 on his arithmetic tests. Round each score to the nearest ten to estimate his total score. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 50. Pizza production. Luigi s pizza parlor makes 293 pizzas on an average day. Estimate (to the nearest hundred) how many pizzas were made on a 3-day holiday weekend. 0 51. Numeration. A whole number rounded to the nearest ten is 60. (a) What is the smallest possible number? (b) What is the largest possible number? 52. Numeration. A whole number rounded to the nearest hundred is 7700. (a) What is the smallest possible number? (b) What is the largest possible number? 53. Numeration. A whole number rounded to the nearest thousand is 5000. (a) What is the smallest possible number? (b) What is the largest possible number? 54. Consumer spending. Amir bought several items at the hardware store: hammer, $8.95; screwdriver, $3.15; pliers, $6.90; wire cutters, $4.25; and sandpaper; $1.89. Estimate the total cost of Amir s bill. 51

ANSWERS 55. 56. 57. 58. 55. Clothes shopping. Mrs. Gonzalez went shopping for clothes. She bought a sweater for $32.95, a scarf for $9.99, boots for $68.29, a coat for $125.90, and socks for $18.15. Estimate the total amount of Mrs. Gonzalez s purchases. 56. Food shopping. Maritza went to the local supermarket and purchased the following items: milk, $2.89; butter, $1.75; bread, $1.10; orange juice, $1.25; cereal, $3.95; and coffee, $3.80. Approximate the total cost of these items. 57. A bag contains 60 marbles. The number of blue marbles, rounded to the nearest 10, is 40, and the number of green marbles in the bag, rounded to the nearest 10, is 20. How many blue marbles are in the bag? (List all answers that satisfy the conditions of the problem.) 58. Describe some situations in which estimating and rounding would not produce a result that would be suitable or acceptable. Review the instructions for filing your federal income tax. What rounding rules are used in the preparation of your tax returns? Do the same rules apply to the filing of your state tax returns? If not, what are these rules? Answers 1. 40 3. 250 5. 700 7. 3500 9. 6000 11. 5000 13. 23,000 15. 790,000 17. 22,000,000 19. Estimate: 120, actual sum: 118 21. Estimate: 330, actual sum: 336 23. Estimate: 50, actual difference: 56 25. Estimate: 10, actual difference: 12 27. Estimate: 2100, actual sum: 2122 29. Estimate: 4700, actual sum: 4730 31. Estimate: 500, actual difference: 448 33. Estimate: 500; actual difference: 504 35. Estimate: 11,000, actual sum: 11,380 37. Estimate: 22,000, actual sum: 22,187 39. Estimate: 3000, actual difference: 2688 41. 43. 45. 47. $29 49. 420 51. (a) 55; (b) 64 53. (a) 4500; (b) 5499 55. $255 57. 36, 37, 38, 39, 40, 41, 42, 43, 44 52