1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 1.5 Rounding and Estimating with Whole Numbers

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Know how to round whole numbers. o Estimate sums and differences by using rounded numbers. o Estimate products by using rounded numbers. o Estimate quotients by using rounded numbers.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rounding Whole Numbers Rounding Numbers To round a given number means to find another number close to the given number. The desired place of accuracy must be stated.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Round the following numbers to the indicated place of accuracy. a.43, tens Solution To round 43 to the nearest ten: Example 1: Rounding Whole Numbers We see that 43 is closer to 40 than to 50. Thus 43 rounds to 40 (to the nearest ten).

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Rounding Whole Numbers (cont.) b.762, hundreds Solution To round 762 to the nearest hundred: We see that 762 is closer to 800 than to 700. Thus 762 rounds to 800 (to the nearest hundred).

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Rounding Whole Numbers (cont.) c.5500, thousands Solution To round 5500 to the nearest thousand, round up to 6000.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Rounding Whole Numbers (cont.) We see that 5500 is the same distance from 5000 as it is from 6000. In situations like this, we round up to the larger number. Thus 5500 rounds to 6000 (to the nearest thousand).

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rounding Rule for Whole Numbers 1.Look at the single digit just to the right of the digit that is in the place of desired accuracy. 2.If this digit is 5 or greater, make the digit in the desired place of accuracy one larger and replace all digits to the right with zeros. All digits to the left remain unchanged unless a 9 is made one larger; then the next digit to the left is increased by 1. Rounding Whole Numbers

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rounding Whole Numbers Rounding Rule for Whole Numbers (cont.) 3.If this digit is less than 5, leave the digit that is in the place of desired accuracy as it is, and replace all digits to the right with zeros. All digits to the left remain unchanged.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Round the following numbers to the indicated place of accuracy. a.6849, hundreds Solution Example 2: Rounding Whole Numbers 6849 6800 Place of desired accuracy. Look at one digit to the right; 4 is less than 5. Leave 8 and fill in zeros. So 6849 rounds to 6800 (to the nearest hundred).

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Rounding Whole Numbers (cont.) b.3500, thousands Solution 3500 4000 Place of desired accuracy. Look at 5; 5 is 5 or greater. Increase 3 to 4 (one larger) and fill in zeros. So 3500 rounds to 4000 (to the nearest thousand).

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Rounding Whole Numbers (cont.) c.597, tens Solution 597 600 Place of desired accuracy. Look at 7; 7 is 5 or greater. Increase 9 to 10 (this changes the 5 to a 6). So 597 rounds to 600 (to the nearest ten).

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Rounding Whole Numbers (cont.) d.20,560, ten thousands Solution 20,560 20,000 Place of desired accuracy. Look at 0; 0 is less than 5. Leave 2 and fill in zeros. So 20,560 rounds to 20,000 (to the nearest ten thousand).

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. So 2709 rounds to 3000 (to the nearest thousand). Example 3: Rounding Whole Numbers A jar of jelly beans at a local candy store is filled with 2709 jelly beans and put on display in the store window. Round this figure to the nearest thousand. Solution To round 2709 to the nearest thousand: 2709 3000 Place of desired accuracy. Look at 7; 7 is 5 or greater. Increase 2 to 3 (one larger) and fill in zeros.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimating Sums and Differences To Estimate a Sum or Difference 1.Round each number to the place of the leftmost digit. 2.Perform the addition or subtraction with these rounded numbers.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Estimating a Sum of Whole Numbers Estimate the sum; then find the sum. Solution Note that in this example, numbers are rounded to different places because they are of different sizes. That is, the leftmost digit is not in the same place for all numbers.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Estimating a Sum of Whole Numbers (cont.) a.Estimate the sum by first rounding each number to the place of the leftmost digit and then adding. 1470 rounded value of 68 rounded value of 925 rounded value of 487 estimated sum

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. b.Now find the sum, knowing that the answer should be close to 1470. Example 4: Estimating a Sum of Whole Numbers (cont.) This sum is very close to 1470. 21

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 5: Estimating a Sum of Whole Numbers Estimate the sum; then find the sum.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. 5900 Completion Example 5: Estimating a Sum of Whole Numbers (cont.) Solution a.First, estimate the sum by rounding each number to the place of the leftmost digit and adding these rounded numbers. 700 estimated sum

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 5: Estimating a Sum of Whole Numbers (cont.) 6372 b.Now find the sum, and compare your answer with the estimated sum. They should be “close.” actual sum

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimate the difference; then find the difference. Solution a.Round each number to the place of the leftmost digit and subtract using these rounded numbers. Example 6: Estimating a Difference of Whole Numbers 2000 rounded value of 2783 rounded value of 975 estimated difference

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Estimating a Difference of Whole Numbers (cont.) b.Now we find the difference, knowing that the difference should be close to 2000. The difference is close to 2000.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Estimating a Difference of Whole Numbers At the beginning of July, your bank account balance is $3859. Over the course of the month, you spend $823. Estimate the remaining balance in your account. Then find the actual balance. Solution In order to find the balance, we must subtract the amount of money you spent throughout the month from the starting balance. First, estimate this difference.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Now, find the actual difference. This should be close to 3200. Thus, the remaining balance in your account is $3036. Example 7: Estimating a Difference of Whole Numbers (cont.) 3200 rounded value of 3859 rounded value of 823 estimated difference 3036

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimating Products Using Rounded Numbers To Estimate a Product 1.Round each number to the place of the leftmost digit. 2.Multiply the rounded numbers.

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Estimating Products of Whole Numbers Find the product 62 ⋅ 38, but first estimate the product by multiplying the numbers in rounded form. Solution First, estimate the product. 2400 rounded value of 62 rounded value of 38 estimated product

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Now, we find the product, keeping in mind that our answer should be close to 2400.  186 Example 8: Estimating Products of Whole Numbers (cont.) The actual product is close to 2400. 496 2356 1

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Estimating Products of Whole Numbers An apartment complex is planning to furnish all of its 93 apartments with new dishwashers. The owner can purchase the dishwashers for $267 each. Estimate the total cost of buying dishwashers for every apartment. Then find the actual total cost to the apartment complex. Solution In order to find the total cost to the apartment complex, we must multiply the number of dishwashers purchased by the cost of each dishwasher.

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Estimating Products of Whole Numbers (cont.) a.We begin by estimating the product 267 ⋅ 93. 27,000 rounded value of 267 rounded value of 93 estimated product

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. b.The product should be near 27,000. Thus, the actual total cost to the apartment complex is $24,831. Example 9: Estimating Products of Whole Numbers (cont.) 801  24 03 24,831

32. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimating Quotients Using Rounded Numbers To Estimate a Quotient 1.Round both the divisor and dividend to the place of the leftmost digit. 2.Divide with the rounded numbers. (When calculating an estimate, ignore the remainder if there is one.)

33. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimate the quotient 8875 ÷ 25 by using rounded values; then find the quotient. Solution a.Estimation: Example 10: Estimating Quotients of Whole Numbers estimated quotient 8875 ÷ 259000 ÷ 30

34. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Estimating Quotients of Whole Numbers (cont.) b.The quotient should be near 300. quotient

35. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimate the quotient 325 ÷ 42 by using rounded values; then find the quotient. Solution a.Estimation: Thus the estimated quotient is 7. Example 11: Estimating Quotients of Whole Numbers estimated quotient 325 ÷ 42300 ÷ 40

36. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Estimating Quotients of Whole Numbers (cont.) b.The quotient should be near 7. The quotient is 7 and the remainder is 31. In this case, the quotient is the same as the estimated value. The true remainder is different. quotient remainder

37. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimate the quotient 6461 ÷ 21. Then finish the division to find the quotient and remainder. Solution Completion Example 12: Estimating Quotients of Whole Numbers 3 1 147 14 07 estimate 300 quotient remainder

38. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 12: Estimating Quotients of Whole Numbers (cont.) Is your estimate close to the actual quotient?__________ What is the difference between your estimate and the actual quotient?__________ yes; estimate: 300, quotient: 307 difference: 7

39. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 13: Estimating Quotients of Whole Numbers A group of 11 friends bought tickets to a local jazz concert. The total price of all 11 tickets was $341. a.Estimate the cost of each ticket. Solution In order to find the cost of each person’s ticket, we must divide the total cost of the tickets by the number of tickets that were purchased.

40. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. We begin by estimating the quotient. Thus, the estimated cost of each ticket is $30. Example 13: Estimating Quotients of Whole Numbers (cont.) estimated quotient 341 ÷ 11300 ÷ 10

41. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 13: Estimating Quotients of Whole Numbers (cont.) b.Then, calculate the actual cost of each ticket. Solution The quotient should be near 30. So the actual cost of each ticket is $31. quotient remainder

42. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Round each number as indicated. 1.1832 (nearest ten) 2.14,751 (nearest hundred) 3.289,300 (nearest ten thousand) 4.Estimate the sum; then find the sum.

43. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems 5.Estimate the difference; then find the difference. 6. Estimate the product; then find the product. 7. Estimate the quotient; then find the quotient and remainder.

44. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.1830 2.14,800 3.290,000 4.estimate: 1100; sum: 1091 5.estimate: 7000; difference: 6295 6.estimate: 140,000; product: 152,234 7.estimate: 40; quotient 42 R17