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Chapter 3: Rational and Real Numbers

Chapter 3: Rational and Real Numbers. Regular Math. Section 3.1: Rational Numbers. A rational number is any number that can be written as a fraction. Relatively prime numbers have no common factors other than 1. Simplify. Try these on your own…. Example 1: Simplifying Fractions.

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Chapter 3: Rational and Real Numbers

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  1. Chapter 3: Rational and Real Numbers Regular Math

  2. Section 3.1: Rational Numbers • A rational number is any number that can be written as a fraction. • Relatively prime numbers have no common factors other than 1.

  3. Simplify. Try these on your own… Example 1: Simplifying Fractions

  4. Example 2: Writing Decimals as Fractions • Write each decimal as a fraction in simplest form. • 0.5 • -2.37 • 0.8716 • Try these on your own… • -0.8 • -4/5 • 5.37 • 5 37/100 • 0.622 • 311/500

  5. Write each fraction as a decimal. 5/4 1.25 1/6 0.16666 Try these on your own… 11/9 1.22222 7/20 0.35 Example 3: Writing Fractions as Decimals

  6. Section 3.2: Adding and Subtracting Rational Numbers • Example 1: In the 2001 World Championships 100-meter dash, it took Maurice Green 0.132 seconds to react to the starter pistol. His total race time, including this reaction time, was 9.82 seconds. How long did it take him to run the actual 100 meters?

  7. Try this one on your own… • In August 2001 at the World University Games in Beijing, China, Jimyria Hicks ran the 200-meter dash in 24.08 seconds. Her best time at the U.S. Senior National Meet in June of the same year was 23.35 seconds. How much faster did she run in June? • She ran 0.73 seconds faster in June.

  8. Example 2: Using a Number Line to Add Rational Numbers • -0.4 + 1.3 • -5/8 + (-7/8) • Try these on your own… • 0.3 +(-1.2) • -0.9 • 1/5 + 2/5 • 3/5

  9. Example 3: Adding and Subtracting Fractions with Like Denominators • Add or Subtract. • 6/11 + 9/11 • -3/8 – 5/8 • Try these on your own… • -2/9 – 5/9 • -7/9 • 6/7 + (-3/7) • 3/7

  10. Example 4: Evaluating Expressions with Rational Numbers • Try these on your own… • 12.1 – x for x = -0.1 • 12.2 • 7/10 + m for m = 3 1/10 • 3 4/5 • Evaluate each expression for the given value of the variable. • 23.8 + x for x = -41.3 • -1/8 + t for t = 2 5/8

  11. Section 3.3: Multiplying Rational Numbers

  12. Example 1: Multiplying a Fraction and an Integer • Multiply. Write each answer in simplest form. • 6 (2/3) • -4 (2 3/5) • Try these on your own… • -8(6/7) • -6 6/7 • 2(5 1/3) • 10 2/3

  13. Example 2: Multiplying Fractions • Multiply. Write each answer in simplest form. • Try these on your own…

  14. Example 3: Multiplying Decimals • Multiply. • -2.5(-8) • -0.07(4.6) • Try these on your own… • 2(-0.51) • -1.02 • (-0.4)(-3.75) • 1.5

  15. Example 4: Evaluating Expressions with Rational Numbers • Evaluate -5 1/2t for each value of t. • t = -2/3 • t = 8 • Try these one on your own… • Evaluate -3 1/8x for each value of x. • x = 5 • -15 5/8 • x = 2/7 • -25/28

  16. Section 3.4: Dividing Rational Numbers • A number and its reciprocal have a product of 1.

  17. Example 1: Dividing Fractions • Try these on your own… • Divide. Write each answer in simplest forms.

  18. Example 2: Dividing Decimals • Divide. • 2.92 / 0.4 • 7.3 • Try this one on your own… • 0.384 / 0.24 • 1.6

  19. Example 3: Evaluating Expressions with Fractions and Decimals • Evaluate each expression for the given value of the variable. • 7.2/n for n = 0.24 • M / (3/8) for M = 7 1/2

  20. Try these on your own… • Evaluate each expression for the given value of the variable. • 5.25/n for n = 0.15 • 35 • K / (4/5) for K = 5 • 6 1/4

  21. Example 4: Problem Solving • You pour 2/3 cup of sports drink into a glass. The serving size is 6 ounces, or ¾ cup. How many servings will you consume? How many calories will you consume?

  22. Try this one on your own… • A cookie recipe calls for ½ cup of oats. You have ¾ cup of oats. How many batches of the cookies can you bake? • You can bake 1 ½ batches of the cookies.

  23. Section 3.5: Adding and Subtracting with Unlike Denominators • Add or subtract. • 2/3 + 1/5 • 3 2/5 + (-3 ½) • Try these on your own… • 1/8 + 2/7 • 23/56 • 1 1/6 + 5/8 • 1 19/24

  24. Example 2: Evaluating Expressions with Rational Numbers • Evaluate n – 11/16 for n = -1/3. • Try this one on your own… • Evaluate t – 4/5 for t = 5/6. • 1/30

  25. Example 3: Consumer Application • A folkloric dance skirt pattern calls for 2 2/5 yards of 45-inch-wide material to make the ruffle and 9 1/3 yards to make the skirt. The material for the skirt and ruffle will be cut from a bolt that is 15 ½ yards long. How many yards will be left on the bolt?

  26. Try this one on your own… • Two dancers are making necklaces from ribbon for their costumes. They need pieces measuring 13 ¾ inches and 12 7/8 inches How much ribbon will be left over after the pieces are cut from 36-inch length? • There will be 9 3/8 inches left.

  27. Section 3.6: Solving Equations with Rational Numbers • Example 1: Solving Equations with Decimals • Solve. • y – 12.5 = 17 • -2.7p = 10.8 • t/7.5 = 4

  28. Try these on your own… • Solve. • M + 4.6 = 9 • M = 4.4 • 8.2p = -32.8 • p = -4 • x/1.2 = 15 • x = 18

  29. Example 2: Solving Equations with Fractions • Solve. • x + 1/5 = -2/5 • x – ¼ = 3/8 • 3/5(w) = 3/16 • Try these on your own… • n + 2/7 = -3/7 • n = -5/7 • y – 1/6 = 2/3 • y = 5/6 • 5/6(x) = 5/8 • x = 3/4

  30. Example 3: Solving Word Problems Using Equations • Try this one on your own… • Mr. Rios wants to prepare a casserole that requires 2 ½ cups of milk. If he makes the casserole, he will have only ¾ cup of milk left for his breakfast cereal. How much milk does Mr. Rios have? • Mr. Rios has 3 ¼ cups of milk. • In 1668 the Hope diamond was reduced from its original weight by 45 1/6 carats to a diamond weighing 67 1/8 carat. How many carats was the original diamond?

  31. Section 3.7: Solving Inequalities with Rational Numbers • Solving Inequalities with Decimals • Try these on your own…

  32. Example 2: Solving Inequalities with Fractions • Solve. • Try these on your own…

  33. Example 3: Problem Solving Application • With first-class mail, there is an extra cost in any of these cases: • The length is greater than 11 ½ inches. • The height is greater than 6 1/8 inches. • The thickness is greater than ¼ inch. • The length divided by the height is less than 1.3 or greater than 2.5 • The height of an envelope is 4.5 inches. What are the minimum and maximum lengths to avoid an extra charge?

  34. Try this one on your own… • With first-class mail, there is an extra cost in any of these cases: • The length is greater than 11 ½ inches. • The height is greater than 6 1/8 inches. • The thickness is greater than ¼ inch. • The length divided by the height is less than 1.3 or greater than 2.5 • The height of an envelope is 3.8 inches. What are the minimum and maximum lengths to avoid an extra charge. • The length of the envelope must be between 4.94 inches and 9.5 inches to avoid extra charges.

  35. Section 3.8: Squares and Square Roots • The principal square root is the non-negative square root. • A perfect square is a number that has integers as its square roots.

  36. Example 1: Finding the Positive and Negative Square Roots of a Number • Find the two square roots of each number. • 64 • 1 • 121

  37. Try these on your own… • Find the two square roots of each number. • 49 • + or - 7 • 100 • + or - 10 • 225 • + or - 15

  38. Example 2: Computer Application • The square computer icon (pg. 147) contains 676 pixels. How many pixels tall is the icon? • Try this one on your own… • A square window has an area of 169 square inches. How wide is the window? • The window is 13 inches wide.

  39. Example 3: Evaluating Expressions Involving Square Roots • Evaluate each expression. • Try these on your own…

  40. Section 3.9: Finding Square Roots • Estimating Square Roots of Numbers… • Try these on your own…

  41. Example 2: Problem Solving Application • You want to install a square skylight that has an area of 300 square inches. Calculate the length of each side and the length of trim you will need, to the nearest tenth of an inch.

  42. Try this one on your own… • You want to sew a fringe on a square tablecloth with an area of 500 square inches. Calculate the length of each side of the tablecloth and the length of fringe you will need to the nearest tenth of an inch. • The length of each side of the table is about 22.4 inches. • You will need about 89.6 inches of fringe.

  43. Example 3: Using a Calculator to Estimate the Value of a Square Root • Use a calculator to find . Round to the nearest tenth. • Try this one on your own… • Use a calculator to find . Round to the nearest tenth. • 22.4

  44. Section 3.10: The Real Numbers • Irrational numbers can only be written as decimals that do not terminate or repeat. • The set of real numbers consists of the set of rational numbers and the set of irrational numbers. • The Density Property of real numbers states that between any two real numbers is another real number.

  45. Example 1: Classifying Real Numbers • Write all the names that apply to each number. • Try these on your own…

  46. Example 2: Determine the Classification of All Numbers • Try these on your own… • State if the number is rational, irrational, or not a real number.

  47. Example 3: Applying the Density Property of Real Numbers • Find a real number between 2 1/3 and 2 2/3. • Try this one on your own… • Find a real number between 3 2/5 and 3 3/5. • 3 ½

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