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Multi-Part Lesson 12-1 Rational Numbers PART Main Idea Compare and order fractions, decimals, and percents. NGSSS MA.7.A.5.1 Express rational numbers as terminating or repeating decimals. glencoe.com A B C Compare and Order Rational Numbers INSECTS The most common types of insects are shown in the table. Insect 1. Which of the insects is the longest? 2. Is the fire ant or housefly longer? Explain. 3. Are more than or less than half of the insects shown longer than 0.6 inch? 4. Order the lengths from shortest to longest. Length (in.) Bumble bee _3 Cricket _1 Fire ant _1 Firefly _3 Housefly _1 Japanese beetle _1 Mosquito _5 4 1 3 4 4 2 8 Just as positive and negative integers can be represented on a number line, so can positive and negative rational numbers. You can use a number line to help you compare and order negative rational numbers. Compare Decimals Replace the with <, >, or = to make a true statement. -1.2 0.8 Graph the decimals on a number line. -1.4-1.2 -1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Since -1.2 is to the left of 0.8, -1.2 < 0.8. -2.40 -2.45 Graph the decimals on a number line. -2.50 -2.48 -2.46 -2.44 -2.42 -2.40 Since -2.40 is to the right of -2.45, -2.40 > -2.45. a. 3.1 -3.7 b. -4.5 -4.49 Compare Fractions Replace the with <, >, or = to make a true statement. _ _ -7 6 8 8 Since the denominators are the same, compare the numerators by graphing on a number line. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 7 _ Since -7 < 6, -_ < 6. 8 Least Common Multiple (LCM) Find the LCM of 8 and 16. 8: 8, 16, 24 16: 16, 32, 48 The LCM is 16. _ 8 _ -3 - 5 8 16 Rename the fractions using the least common denominator. 5 5 5×1 -_ = -_ = -_ 16 16 16 16 × 1 6 5 3 5 < -_ and -_ < -_ . Since -6 < -5, -_ 8 16 16 16 3 3×2 6 -_ = -_ = -_ 8×2 8 Check -6 -5 16 c. 16 -4 16 -3 16 -2 9 12 _ -_ 16 16 -1 0 16 7 4 d. -_ -_ 16 10 5 Compare Decimals and Fractions Replace the with <, >, or = to make a true statement. _ -0.51 - 8 15 8 as a decimal. Then graph both decimals on a Rename -_ 15 number line. -0.53 -0.51 -0.6 -0.55 -5 − 8 -_ = -0.53 -0.5 -0.45 -0.4 15 − Since -0.51 is to the right of -0.53 on the number line, 8 . -0.51 > -_ 15 Replace the with <, >, or = to make a true statement. e. -3_ -3.625 5 8 f. _3 0.413 7 To order rational numbers, first write them in the same form and then order the numbers. Order Rational Numbers SCIENCE Mr. Plum’s science class is growing plants under different conditions. The average plant growth during a week was 5.5 inches. The table shows the difference from the average for some students’ plants. Order the differences from least to greatest. Real-World Link The longest species of seaweed measures 197 feet. This kelp is the fastest growing species, growing more than 13 inches per day. Difference from Average Plant Growth Difference (in.) Student 1 3_ Ricky 4 Debbie -2.2 Suni 1.7 7 -1_ Leonora 10 Express each number as a decimal. 1 Ricky’s plant: 3_ = 3.25 Debbie’s plant: -2.2 Suni’s plant: 1.7 7 Leonora’s plant: -1_ = -1.7 4 10 Graph each number on a number line. 1.7 -2.2 -1.7 3.25 -3.5-3 -2.5 -2 -1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 7 1 From least to greatest, the differences are -2.2, -1_ , 1.7, and 3_ . 10 4 g. OCEANS The average depths of the three deepest oceans are -2.46 miles, -2.88 miles, and -2.44 miles. Order the depths from least deep to deepest. Examples 1–5 Replace the with <, >, or = to make a true statement. (pp. 677–678) 2. 9.7 -10.3 3. 7 4. -_ _ 7 5. -6.7 -6_ 5 6. -_ -0.94 12 Example 6 (p. 679) _5 -_3 1. -7.8 -7.82 5 12 10 8 8 6 7. MONEY Stephen recorded the following amounts in his checkbook: -$6.50, $7.00, -$6.75, and $7.25. Order these amounts from least to greatest. Order the following numbers from least to greatest. 1 8. -3_ , 3.3, -3_, 3.5 3 3 4 9 − 3 _ 2.8, -2_ , 3 1 , -2.2 4 8 = Step-by-Step Solutions begin on page R31. Extra Practice is on page EP32. Examples 1–5 (pp. 677–678) Replace the with <, >, or = to make a true statement. 10. -2.5 -2.9 11. -18.6 -18.06 12. -4.08 -4.7 5 4 13. -_ -_ 9 9 5 1 14. _ -_ 4 4 3 2 15. -_ -_ 1 16. -6_ -6.375 3 17. -_ -0.6 2 18. -9_ -9.3 3 Example 6 (p. 679) 7 5 7 5 19. STOCK MARKET The change in four stocks during a day are shown. Order the changes from least to greatest. Company Change A -4_ B 5.6 C -2_ 1 2 3 8 20. IN-LINE SKATING The average amount of time D Brent spent in-line skating for one week was 34 minutes. During the next week, the difference between the average time and actual time spent skating was 1.35 1 1 4.2 minutes, -5_ minutes, -2_ minutes, and 3.75 minutes. 3 2 Order these differences from least to greatest. Replace the with <, >, or = to make a true statement. − −− −− −−− − 4 21. -0.4 -0.45 22. -2.21 -2.212 23. -6.4 -6_ 9 Order the following numbers from least to greatest. − − 1 1 1 24. 2.1, -2.1, 2_ , -2 25. _ , -0.025, 0.2, -_ 8 2 4 26. _ , 60%, 0.65, _ 3 5 7 − 1 3 27. 125%, 1_, 1.25, 1_ 5 4 28. SNACKS The bar graph shows the profit or loss of the after-school snack stand. a. Write each profit as a positive number and each loss as a negative number. Profit 11 Loss B $8.00 $6.00 $4.00 $2.00 $0.00 -$2.00 -$4.00 -$6.00 -$8.00 Profit or Loss 2 3 4 Day b. Order the numbers from least to greatest. 29 MODEL ROCKETS For a technology competition, Julienne constructed a model rocket. The average height the rocket can reach is 545 feet. The table shows the differences between the average height and the actual height reached. Order the differences from least to greatest. 1 Trials Difference Between Average Height and Actual Height (ft) 1 -10.8 2 11.4 3 9.0 4 -9.7 30. REASONING Determine whether the following statement is always, sometimes, or never true. Give examples to justify your answer. C If x and y are both greater than zero and x > y, then -x < -y. 4 4 4 4 31. NUMBER SENSE Determine whether the fractions -_ , -_ , -_ , and -_ 5 are arranged in order from least to greatest. Explain. 35. 3 "# -0.4 -0.3 -0.2 -0.1 $% 0 0.1 0.2 0.3 0.4 A. point A SHORT RESPONSE Student Council’s goal was to raise $50 each week for 4 weeks to have enough money for the school dance. The table shows the difference between the goal and the actual amount raised. Order these amounts from least to greatest. B. point B Week C. point C Difference ($) D. point D 1 2 3 4 5.50 -6.25 7.80 -2.45 36. Which of the following numbers is less GRIDDED RESPONSE Which of the following numbers is the greatest? -0.73, 0.32, -0.21, 0.19 2 than -_ ? 3 F. 0.6 − H. 0.6 1 G. _ 5 I. -_ 3 Write each fraction as a decimal. 37. _5 9 6 (Lesson 12-1B) 2 39. _ 1 38. -_ 3 8 40. Find the value of x ÷ y if x = -15 and y = -3. (Lesson 11-3E) 41. BUSINESS The table shows the results of a company’s operations over a period of three months. Find the company’s overall profit or loss over this time period. (Lesson 11-2B) 8 MA.6.A.5.2, MA.7.A.5.1 33. Which point shows the approximate 1 ? location of -_ 34. 7 −− Explain why -0.33 is greater than -0.33. 32. NGSSS Practice 6 Month Profit April May June Loss $15,000 $19,000 $12,000