Lesson 1.4.2

1. Using Rounded Numbers Lesson 1.4.2

2. Lesson 1.4.2 Using Rounded Numbers California Standards: Mathematical Reasoning 2.1 Use estimation to verify the reasonableness of calculated results. Mathematical Reasoning 2.6 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. What it means for you: You’ll learn when it’s a good idea to use rounding, and how to check your answers using rounded numbers. • Key words: • rounding • place value • accuracy • check • reasonable • estimate

3. “about 200 miles” This Lesson will tell you more about using rounded numbers. “50,000 people” …126.5 cm” You’ll think about how much certain numbers should be rounded. “It’s about… …127 cm” …130 cm” You’ll also see how rounded numbers are useful for checking your work. 24.6 × 3.97 = 97.662 25 × 4 = 100 Lesson 1.4.2 Using Rounded Numbers

4. “Add 6.25 lb of chemical X” “I weigh 100 lb.” “Use 0.5 lb flour.” Lesson 1.4.2 Using Rounded Numbers Sometimes You Need to Choose How Much to Round People roundnumbers to different place valuesdepending on what the numbers are being used for.

5. Lesson 1.4.2 Using Rounded Numbers Example 1 Below are three situations connected with the distance between Town A and Town B. Match each situation to the most suitable level of rounding. Situation Distances A road sign in Town A shows the distance to Town B. A road sign in Town A shows the distance to Town B. 203.56 miles 203.56 miles Jada lives in Town B. She tells a friend from another state how far away she lives from Town A. Jada lives in Town B. She tells a friend from another state how far away she lives from Town A. 204 miles 204 miles 200 miles 200 miles A mapping company is making an accurate map of the area. A mapping company is making an accurate map of the area. Solution A road signwouldn’t give distances to the nearest hundredth of a mile. The road sign would give the distance as 204 miles.This is accurate enough, and can be read easily by a passing driver. The friend would only want a rough ideaof how far away Town A is. Jada could tell her friend that she lives 200 milesfrom Town A. The exact distance doesn’t matterto someone who lives far away. The mapping companyneeds to know exact distancesto draw an accurate map. They would use the figure of 203.56 miles. Solution follows…

6. Lesson 1.4.2 Using Rounded Numbers Guided Practice Exercises 1–2 give three figures: one exact and two rounded numbers. Choose which you think is most suitable, and explain your choice. (In each case, the first figure given is the exact answer.) 1. Ana Lucia writes in a history essay: Thomas Jefferson lived to the age of 83 / 80 / 100. 2. On a form, Gavin gives his height as: 160.67 cm / 161 cm / 200 cm. 83 years. Either of the rounded figures would be misleading. 161 cm. Nobody would need to know his height to the nearest 0.01 cm. To round up to 200 cm would suggest Gavin is much taller than he really is. Solution follows…

7. Lesson 1.4.2 Using Rounded Numbers Guided Practice Exercises 3–4 give three figures: one exact and two rounded numbers. Choose which you think is most suitable, and explain your choice. (In each case, the first figure given is the exact answer.) 3. A school records how many students are in school each day. Today there are 3914 / 3900 / 4000 students attending. 4. Mr. Anderson returns from a vacation in England with £10 left over from his extra cash. £1 is worth $1.85965. He changes the money at the bank and receives $18.5965 / $18.60 / $19. 3914 students. The school will want an accurate record of student numbers, for example in case there is a fire, so only the exact number will do. $18.60. The bank can’t give him 0.65 cents, and wouldn’t round to the nearest dollar. Solution follows…

8. 1552 + 2676 = 4228 1550 + 2680 = 4230 1600 + 2700 = 4300 2000 + 3000 = 5000 Lesson 1.4.2 Using Rounded Numbers The Amount of Rounding Affects the Accuracy If you use rounding to estimate asum, be careful how much you round. Rounding to higher place valuesusually gives an estimate farther from the actual answerthan rounding to lower place values.

9. 3000+ 50008000 Lesson 1.4.2 Using Rounded Numbers Example 2 Lucas wants to add 3439 and 5482. He doesn’t need an exact answer, so he decides to use rounding. Lucas's work is shown on the right. How could he have found a more accurate answer? Solution Lucas rounded to the nearest thousand, so he got an estimate of 8000. If he had rounded to the nearest hundred, he would have gotten 3400 + 5500 = 8900, which is much closer to the actual answer of 8921. Solution follows…

10. Lesson 1.4.2 Using Rounded Numbers Guided Practice 5. Estimate 962 – 246 by rounding to the nearest hundred. 6. Estimate 962 – 246 by rounding to the nearest ten. 7. Calculate 962 – 246 exactly. Which of your two estimates was closer to the actual result? 1000 – 200 = 800 960 – 250 = 710 962 – 246 = 716Ex.6 gave a much closer estimate than Ex.5. Solution follows…

11. Lesson 1.4.2 Using Rounded Numbers Rounded Numbers Can Be Used to Check Work Many times you’ll want to check your workwithout doing the calculation all over again. Rounding is a way to see if your answer is reasonable.

12. 2300+ 5100 2343 + 5077 Lesson 1.4.2 Using Rounded Numbers Example 3 Calculate 2343 + 5077. Then check your work by rounding to the nearest hundred. Solution These are rounded to the nearest hundred Actual sum: Rounded sum: 7420 7400 The answer to the rounded sum is close to the answer to the actual sum, so the answer to the actual sum is reasonable. Solution follows…

13. Lesson 1.4.2 Using Rounded Numbers Example 4 Martin is trying to solve 29.6 × 9.8. He gets the answer 192.08. Check Martin’s answer by rounding to the nearest ten. Solution Rounded sum: 30 × 10 = 300 Martin’s answer is a long way from the rounded answer, so it looks like his answer of 192.08 might be wrong. In fact 29.6 × 9.8 = 290.08 This is much closer to the rounded estimate. Solution follows…

14. Lesson 1.4.2 Using Rounded Numbers Guided Practice In Exercises 8–11, check the answers given by rounding to the place value shown in parentheses. Say whether the answers are reasonable. 8. 1818 + 700 = 3918 (hundred) 9. 22 × 79 = 738 (ten) 10. 490 + 770 = 1260 (hundred) 11. 642 – 369 = 273 (ten) 1800 + 700 = 2500, this isn’t close to the answer, so 3918 is unreasonable 20 × 80 = 1600, this isn’t close to the answer, so 738 is unreasonable 500 + 800 = 1300, this is close to the answer, so 1260 is reasonable 640 – 370 = 270, this is close to the answer, so 273 is reasonable Solution follows…

15. Lesson 1.4.2 Using Rounded Numbers Guided Practice In Exercises 12–15, check the answers given by rounding to the place value shown in parentheses. Say whether the answers are reasonable. 12. 2.85 × 52.1 = 96.385 (one) 13. 68 × 47 = 5032 (ten) 14. 32.815 + 84.565 = 117.38 (ten) 15. 10.48 × 67.02 = 902.3696 (one) 3 × 52 = 156, this isn’t close to the answer, so 96.385 is unreasonable 70 × 50 = 3500, this isn’t close to the answer, so 5032 is unreasonable 30 + 80 = 110, this is close to the answer, so 117.38 is reasonable 10 × 67 = 670, this isn’t close to the answer, so 902.3696 is unreasonable Solution follows…

16. Lesson 1.4.2 Using Rounded Numbers Independent Practice In Exercises 1–6, use rounding to check the answers given, and say whether or not you think they are reasonable: 1. 6898 + 517 = 7415 2. 97 × 411 = 13,677 3. 547 × 695 = 527,855 4. 74,861 – 2940 = 65,621 5. 96 × 7973 = 526,218 6. 4362 – 1855 = 2507 Unreasonable Reasonable Unreasonable Unreasonable Unreasonable Reasonable Solution follows…

17. Lesson 1.4.2 Using Rounded Numbers Independent Practice 7. Darnell is trying to work out the answer to 52 + 871.Darnell thinks the answer is 923. He asks Zoe and Enrique if his answer looks about right. Zoe thinks the answer should be near to 1000. Enrique thinks the answer should be about 920. Why might Zoe and Enrique have gotten different answers? Zoe rounded to the nearest hundred, Enrique rounded to the nearest ten. Solution follows…

18. Lesson 1.4.2 Using Rounded Numbers Independent Practice 8. In the Olympic 100 meters final, the first three athletes finish the race in times of 9.89 seconds, 9.94 seconds, and 9.99 seconds, to the nearest hundredth of a second. Why would it not be a good idea to round these any further? To the nearest tenth, the first two athletes both ran 9.9 seconds. To the nearest second, the first three athletes all ran 10 seconds. Both of these are misleading as they make it sound like the race was a dead heat. Solution follows…

19. Lesson 1.4.2 Using Rounded Numbers Independent Practice 9. Town C is putting up a new sign showing its population.The population is 45,691 people, but this figure is changing all the time so the town decides to use a rounded figure. What figure do you think they should use: 45,690, 45,700, 46,000, or 50,000? Explain your answer. The figure 45,690 could be out of date almost as soon as the exact number, as the population could quickly rise or fall by 10. 50,000 is not very close to the real population so that would not be a sensible choice. Rounding to either the nearest hundred or the nearest thousand would be sensible choices. Solution follows…

20. Lesson 1.4.2 Using Rounded Numbers Round Up Remember that a roundednumber is usually not the same as the real figure. It only gives you a guide to how big the real number is.