4-1 Terminating and Repeating Decimals Write the fraction or mixed number as a decimal. Use bar notation if needed. 1. SOLUTION: Think: = 0.5. So, ANSWER: 0.5 2. SOLUTION: Think of the mixed number as a sum. Write the mixed number without the negative sign. Write as a decimal. Think: So, = 0.16. Add the negative sign. So, = –4.16. ANSWER: –4.16 3. SOLUTION: eSolutions Manual - Powered by Cognero So, = 0.125. Page 1 = –4.16. So, ANSWER: and Repeating Decimals 4-1 Terminating –4.16 3. SOLUTION: So, = 0.125. ANSWER: 0.125 4. SOLUTION: So, = 0.1875. ANSWER: 0.1875 5. SOLUTION: Think: So, = –0.66. ANSWER: –0.66 eSolutions Manual - Powered by Cognero Page 2 So, = 0.1875. ANSWER: and Repeating Decimals 4-1 Terminating 0.1875 5. SOLUTION: Think: So, = –0.66. ANSWER: –0.66 6. SOLUTION: So, = –0.425. ANSWER: –0.425 7. SOLUTION: Think of the mixed number as a sum. Write as a decimal. eSolutions - Powered by Cognero So, Manual = 0.875. Page 3 So, = –0.425. ANSWER: and Repeating Decimals 4-1 Terminating –0.425 7. SOLUTION: Think of the mixed number as a sum. Write So, So, as a decimal. = 0.875. = 5.875. ANSWER: 5.875 8. SOLUTION: Think of the mixed number as a sum. Write as a decimal. eSolutions Manual - Powered by Cognero So, = 0.375. Page 4 So, = 5.875. ANSWER: and Repeating Decimals 4-1 Terminating 5.875 8. SOLUTION: Think of the mixed number as a sum. Write So, So, as a decimal. = 0.375. = 9.375. ANSWER: 9.375 9. SOLUTION: So, . ANSWER: eSolutions Manual - Powered by Cognero 10. Page 5 So, = 9.375. ANSWER: and Repeating Decimals 4-1 Terminating 9.375 9. SOLUTION: So, . ANSWER: 10. SOLUTION: So, . ANSWER: 11. SOLUTION: eSolutions Manual - Powered by Cognero Page 6 So, . ANSWER: 4-1 Terminating and Repeating Decimals 11. SOLUTION: So, . ANSWER: 12. SOLUTION: Think of the mixed number as a sum. Write So, as a decimal. . eSolutions Manual - Powered by Cognero So, . Page 7 So, . ANSWER: 4-1 Terminating and Repeating Decimals 12. SOLUTION: Think of the mixed number as a sum. Write as a decimal. So, . So, . ANSWER: Write the decimal as a fraction or mixed number in simplest form. 13. –0.2 SOLUTION: The final digit, 2, is in the tenths place. So, –0.2 = . ANSWER: eSolutions Manual - Powered by Cognero 14. 0.55 SOLUTION: Page 8 So, . ANSWER: 4-1 Terminating and Repeating Decimals Write the decimal as a fraction or mixed number in simplest form. 13. –0.2 SOLUTION: The final digit, 2, is in the tenths place. So, –0.2 = . ANSWER: 14. 0.55 SOLUTION: The final digit, 5, is in the hundredths place. So, 0.55 = . ANSWER: 15. 5.96 SOLUTION: Think of the decimal as a sum. 5.96 = 5 + 0.96 Write 0.96 as a fraction. The final digit, 6, is in the hundredths place. So, 0.96 = . eSolutions Manual - Powered by Cognero So, 5.96 = . Page 9 ANSWER: 4-1 Terminating and Repeating Decimals 15. 5.96 SOLUTION: Think of the decimal as a sum. 5.96 = 5 + 0.96 Write 0.96 as a fraction. The final digit, 6, is in the hundredths place. So, 0.96 = . So, 5.96 = . ANSWER: 16. The screen on Brianna’s new phone is 2.85 centimeters long. What mixed number represents the length of the phone screen? SOLUTION: Think of the decimal as a sum. 2.85 = 2 + 0.85 Write 0.85 as a fraction. The final digit, 5, is in the hundredths place. So, 0.85 = So, . represents this length. ANSWER: eSolutions Manual - Powered by Cognero 17. Page 10 praying mantis is an interesting insect that can rotate its head 180 degrees. Suppose a praying mantis is 10.5 centimeters long. What mixed number represents this length? ANSWER: 4-1 Terminating and Repeating Decimals 16. The screen on Brianna’s new phone is 2.85 centimeters long. What mixed number represents the length of the phone screen? SOLUTION: Think of the decimal as a sum. 2.85 = 2 + 0.85 Write 0.85 as a fraction. The final digit, 5, is in the hundredths place. So, 0.85 = So, . represents this length. ANSWER: 17. praying mantis is an interesting insect that can rotate its head 180 degrees. Suppose a praying mantis is 10.5 centimeters long. What mixed number represents this length? SOLUTION: Think of the decimal as a sum. 10.5 = 10 + 0.5 Write 0.5 as a fraction. The final digit, 5, is in the tenths place. So, 0.5 = . So, 10 represents this length. ANSWER: Problems 18. Persevere eSolutions Manual - with Powered by CogneroSuppose you buy a 1.25-pound package of ham at $5.20 per pound. a. What fraction of a pound did you buy? Page 11 So, 10 represents this length. ANSWER: 4-1 Terminating and Repeating Decimals 18. Persevere with Problems Suppose you buy a 1.25-pound package of ham at $5.20 per pound. a. What fraction of a pound did you buy? b. How much money did you spend? SOLUTION: a. Think of the decimal as a sum. 1.25 = 1 + 0.25 Write 0.25 as a fraction. The final digit, 5, is in the hundredths place. So, 0.25 = . The fraction of a pound you bought is or . b. Multiply. So you spent $6.50 ANSWER: a. or b. $6.50 19. Identify Structure Write a fraction that is equivalent to a terminating decimal between 0.5 and 0.75. SOLUTION: Think of a decimal value between 0.5 and 0.75. Sample answer: One terminating decimal that is between 0.5 and 0.75 is 0.6. Change 0.6 to a fraction. The final digit, 6, is in the tenths place. eSolutions Manual - Powered by Cognero So, 0.6 = . Page 12 a. or 4-1 Terminating and Repeating Decimals b. $6.50 19. Identify Structure Write a fraction that is equivalent to a terminating decimal between 0.5 and 0.75. SOLUTION: Think of a decimal value between 0.5 and 0.75. Sample answer: One terminating decimal that is between 0.5 and 0.75 is 0.6. Change 0.6 to a fraction. The final digit, 6, is in the tenths place. So, 0.6 = . ANSWER: Sample answer: 20. Persevere with Problems Fractions in simplest form that have denominators of 2, 4, 8, 16, and 32 produce terminating decimals. Fractions with denominators of 6, 12, 18, and 24 produce repeating decimals. What causes the difference? Explain. SOLUTION: Sample answer: Write the prime factorization of the first set of numbers. 2=2 4 = 2 × 2 8 = 2 × 2 × 2 16 = 2 × 2 × 2 × 2 32 = 2 × 2 × 2 × 2 × 2 Write the prime factorization of the second set of numbers. 6 = 2 × 3 12 = 2 × 2 × 3 18 = 2 × 3 × 3 24 = 2 × 2 × 2 × 3 The first set of numbers can be factored into primes of 2s. The second set of numbers can be factored into primes of 2s and 3s; the factor of 3 produces the repeating decimal. ANSWER: Sample answer: The first set of numbers can be factored into primes of 2s. The second set of numbers can be factored into primes of 2s and 3s; the factor of 3 produces the repeating decimal. 21. Persevere with Problems The value of pi (π) is 3.1415926… . The mathematician Archimedes believed that π was between and . Was Archimedes correct? Explain your reasoning. SOLUTION: Sample answer: Write the mixed numbers as decimals. Think of the mixed number as a sum. Write as a decimal. eSolutions Manual - Powered by Cognero Page 13 Think of the mixed number as a sum. 4-1 Terminating and Repeating Decimals Write as a decimal. So, . Think of the mixed number as a sum. Write So, as a decimal. . eSolutions Manual - Powered by Cognero Page 14 4-1 Terminating and Repeating Decimals So, . So, ≈ 3.14286 and ≈ 3.14085; Since 3.1415927... is between and , Archimedes was correct. ANSWER: Sample answer: ≈ 3.14286 and ≈ 3.14085; Since 3.1415927... is between and , Archimedes was correct. 22. Reason Inductively A unit fraction is a fraction that has 1 as its numerator. Write the four greatest unit fractions that are repeating decimals. Then write each fraction as a decimal. SOLUTION: The smaller the denominator, the larger the fraction. Of the first 10 numbers, fractions with a denominator of 1, 2, 4, 5, 8, and 10 all terminate. That leaves 3, 6, 7, and 9 as denominators. Divide each fraction to determine the decimal. For : So, . For : So, . For : eSolutions Manual - Powered by Cognero Page 15 So, . 4-1 Terminating and Repeating Decimals For : So, . For : So, . ANSWER: 23. Model with Mathematics Write a real-world scenario in which it would be appropriate to write a value in fraction form. SOLUTION: Sample answer: Jason was building a cabin. He cut a board to the length of feet long. ANSWER: Sample answer: Jason was building a cabin. He cut a board to the length of feet long. Write the fraction or mixed number as a decimal. Use bar notation if needed. 24. SOLUTION: eSolutions Manual - Powered by Cognero Think: Page 16 Sample answer: Jason was building a cabin. He cut a board to the length of feet long. ANSWER: 4-1 Terminating and Repeating Decimals Sample answer: Jason was building a cabin. He cut a board to the length of feet long. Write the fraction or mixed number as a decimal. Use bar notation if needed. 24. SOLUTION: Think: So, = 0.8. ANSWER: 0.8 25. SOLUTION: Think of the mixed number as a sum. Write the mixed number without the negative sign. Write as a decimal. Think: So, = 0.05. Add the negative sign. So, = –7.05. ANSWER: –7.05 26. SOLUTION: eSolutions Manual - Powered by Cognero So, . Page 17 So, = –7.05. ANSWER: and Repeating Decimals 4-1 Terminating –7.05 26. SOLUTION: So, . ANSWER: 27. SOLUTION: Think of the mixed number as a sum. Write So, So, as a decimal. . . ANSWER: eSolutions Manual - Powered by Cognero 28. The fraction of a dime that is made up of copper is Page 18 . Write this fraction as a decimal. So, . ANSWER: 4-1 Terminating and Repeating Decimals 27. SOLUTION: Think of the mixed number as a sum. Write So, So, as a decimal. . . ANSWER: 28. The fraction of a dime that is made up of copper is . Write this fraction as a decimal. SOLUTION: So, is 0.75. ANSWER: 0.75 Write the decimal as a fraction or mixed number in simplest form. 29. –0.9Manual - Powered by Cognero eSolutions SOLUTION: The final digit, 9, is in the tenths place. Page 19 So, is 0.75. ANSWER: and Repeating Decimals 4-1 Terminating 0.75 Write the decimal as a fraction or mixed number in simplest form. 29. –0.9 SOLUTION: The final digit, 9, is in the tenths place. So, –0.9 = . ANSWER: 30. 0.34 SOLUTION: The final digit, 4, is in the hundredths place. So, 0.34 = . ANSWER: 31. 2.66 SOLUTION: Think of the decimal as a sum. 2.66 = 2 + 0.66 Write 0.66 as a fraction. The final digit, 6, is in the hundredths place. So, 0.66 = So, 2.66 = . . eSolutions Manual - Powered by Cognero ANSWER: Page 20 ANSWER: 4-1 Terminating and Repeating Decimals 31. 2.66 SOLUTION: Think of the decimal as a sum. 2.66 = 2 + 0.66 Write 0.66 as a fraction. The final digit, 6, is in the hundredths place. So, 0.66 = . So, 2.66 = . ANSWER: Write the following as an improper fraction. 32. –13 SOLUTION: ANSWER: 33. SOLUTION: Think of the mixed number as a sum. There are 7 × 3, or 21 thirds in the whole number 7. The sum of 21 thirds and one third is 22 thirds. So, . ANSWER: eSolutions Manual - Powered by Cognero 34. –3.2 Page 21 ANSWER: 4-1 Terminating and Repeating Decimals 33. SOLUTION: Think of the mixed number as a sum. There are 7 × 3, or 21 thirds in the whole number 7. The sum of 21 thirds and one third is 22 thirds. So, . ANSWER: 34. –3.2 SOLUTION: The final digit, 2, is in the tenths place. So, –3.2 = . ANSWER: 35. Be Precise Nicolás practiced playing the cello for 2 hours and 18 minutes. Write the time Nicolás spent practicing as a decimal. SOLUTION: Because there are 60 minutes in one hour, 2 hours and 18 minutes can be represented by the mixed number , or . Think of the mixed number as a sum. Write as a decimal. Think: eSolutions Manual - Powered by Cognero So, Nicolás practiced for 2.3 hours. ANSWER: Page 22 ANSWER: 4-1 Terminating and Repeating Decimals 35. Be Precise Nicolás practiced playing the cello for 2 hours and 18 minutes. Write the time Nicolás spent practicing as a decimal. SOLUTION: Because there are 60 minutes in one hour, 2 hours and 18 minutes can be represented by the mixed number , or . Think of the mixed number as a sum. Write as a decimal. Think: So, Nicolás practiced for 2.3 hours. ANSWER: 2.3 h 36. The table shows the lengths of four hiking trails. Select the appropriate decimal equivalent of each trail length. SOLUTION: Write the fraction corresponding to each hiking trail as a decimal. Lakeview: Think of the mixed number as a sum. You know that So, . . eSolutions Manual - Powered by Cognero Forest Lane: Think of the mixed number as a sum. Page 23 You know that . 4-1 Terminating and Repeating Decimals So, . Forest Lane: Think of the mixed number as a sum. Write So, So, as a decimal. . . Sparrow Stroll: Think of the mixed number as a sum. Write So, So, as a decimal. . . Mountain Climb: Think of the mixed number as a sum. Write as a decimal. eSolutions Manual - Powered by Cognero Page 24 So, . Mountain Climb: Think of the mixed number as a sum. 4-1 Terminating and Repeating Decimals Write as a decimal. So, . So, . ANSWER: 37. Zoe went to lunch with a friend. After tax, her bill was $12.05. Which of the following rational numbers is equivalent to this amount? Select all that apply. SOLUTION: Determine if each rational number is the same as 12.05 by changing it's form. For : The final digit, 5, is in the hundredths place. Write 12.05 as mixed number with a denominator of 100. Then simplify. eSolutions Manual - Powered by Cognero So, 12.05 = . Page 25 4-1 Terminating and Repeating Decimals 37. Zoe went to lunch with a friend. After tax, her bill was $12.05. Which of the following rational numbers is equivalent to this amount? Select all that apply. SOLUTION: Determine if each rational number is the same as 12.05 by changing it's form. For : The final digit, 5, is in the hundredths place. Write 12.05 as mixed number with a denominator of 100. Then simplify. So, 12.05 = . For : Change For , so is not equivalent. : Change into an improper fraction: 20 ×12 + 1 = 241, so For . into a mixed number, which is . Therefore, these are equivalent. : Simplify : So, these are equivalent. ANSWER: Round the decimal to the tenths place. 38. 5.69 SOLUTION: eSolutions Manual - Powered by Cognero The number to the right of the tenths place is 5. Since 9 ≥ 5 the 6 in the tenths place is rounded up to 7. So, 5.69 ≈ 5.7. Page 26 ANSWER: 4-1 Terminating and Repeating Decimals Round the decimal to the tenths place. 38. 5.69 SOLUTION: The number to the right of the tenths place is 5. Since 9 ≥ 5 the 6 in the tenths place is rounded up to 7. So, 5.69 ≈ 5.7. ANSWER: 5.7 39. 0.05 SOLUTION: The number to the right of the tenths place is 5. Since 5 ≥ 5 the 0 in the tenths place is rounded up to 1. So, 0.05 ≈ 0.1. ANSWER: 0.1 40. 98.99 SOLUTION: The number to the right of the tenths place is 9. Since 9 ≥ 5 the 9 in the tenths place is rounded up which changes the next place value. So, 98.99 ≈ 99.0. ANSWER: 99.0 Graph and label the fraction on the number line below. 41. SOLUTION: There are 12 intervals on the number line between 0 and 1. Find an equivalent fraction to 12. with a denominator of Place a dot on the 6th hash mark from 0 on the number line and label it. ANSWER: 42. eSolutions Manual - Powered by Cognero SOLUTION: There are 12 intervals on the number line between 0 and 1. Find an equivalent fraction to Page 27 with a denominator of 4-1 Terminating and Repeating Decimals 42. SOLUTION: There are 12 intervals on the number line between 0 and 1. Find an equivalent fraction to 12. with a denominator of Place a dot on the 9th hash mark from 0 on the number line and label it. ANSWER: 43. SOLUTION: There are 12 intervals on the number line between 0 and 1. Find an equivalent fraction to 12. with a denominator of Place a dot on the 8th hash mark from 0 on the number line and label it. ANSWER: 44. The table shows the discount on athletic shoes at two stores selling sporting equipment. Which store is offering the greater discount? Explain. SOLUTION: Write each number as a decimal. 25% = = 0.25 eSolutions Manual - Powered by Cognero Since 0.25 > 0.20, 25% > ANSWER: Page 28 ANSWER: 4-1 Terminating and Repeating Decimals 44. The table shows the discount on athletic shoes at two stores selling sporting equipment. Which store is offering the greater discount? Explain. SOLUTION: Write each number as a decimal. 25% = = 0.25 Since 0.25 > 0.20, 25% > ANSWER: Go Time; Sample answer: eSolutions Manual - Powered by Cognero = 0.20 and 25% = 0.25; Since 0.25 > 0.20, 25% > . Page 29