Rational Numbers 2-1. Lesson Objectives. Vocabulary. Additional Examples. Write rational numbers in equivalent forms. rational number (p.
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1 LESSON 2-1 Rational Numbers Lesson Objectives Write rational numbers in equivalent forms Vocabulary rational number (p. 6) relatively prime (p. 6) Additional Examples Example 1 Simplify. A is a common factor Divide the numerator and denominator by. B There are no factors and 29 are prime. Example 2 Write each decimal as a fraction in simplest form. A..7 is in the hundredths place. 2 Holt McDougal Mathematics
2 LESSON 2-1 CONTINUED B is in the place. Simplify by by the common factor 2. Example Write each fraction as a decimal. A The pattern repeats, so draw a bar over the 2 to indicate that this is a decimal. The fraction 1 1 is equivalent to the decimal. 9 7 B. 2 0 This is a decimal. The remainder is. 7 The fraction 2 is equivalent to the decimal. 0 Check It Out! 1. Simplify Write the decimal as a fraction in simplest form..7. Write the fraction as a decimal Holt McDougal Mathematics
3 LESSON 2-2 Comparing and Ordering Rational Numbers Lesson Objectives Compare and order positive and negative rational numbers written as fractions, decimals, and integers Vocabulary least common denominator (LCD) (p. 72) Additional Examples Example 1 Compare. Write,, or. A Method 1 Multiply to find the denominator Multiply and to find a common denominator Write the fractions with a common denominator , so Holt McDougal Mathematics
4 LESSON 2-2 CONTINUED B. 2 4 Method 2 Find the least common denominator. :, 6, 9, 12, 1,... :, 10, 1,... List multiples of and. The LCM is. 2 2 Write the fractions with a common denominator , so 2 4 Example 2 Compare. Write,, or. A and 2 Write the fractions as , so B Write as a , so C Write as a , so Holt McDougal Mathematics
5 LESSON 2-2 CONTINUED Example The numbers 1 4,.4, 6.0, and 2. represent the percentage of change 4 in populations for four states. List these numbers in order from least to greatest. Place the numbers on a number line and read them from left to right The percent changes in population from least to greatest are.4, 2., 1 4, and Check It Out! 1. Compare. Write,, or Compare. Write,, or The numbers 1.2, 7,and 2. represent the percent of change in population for three states. List these numbers in order from least to greatest. 2 Holt McDougal Mathematics
6 B. 1 2 Move right units. LESSON 2- Adding and Subtracting Rational Numbers Lesson Objectives Add and subtract decimals and rational numbers with like denominators Additional Examples Example 1 In August 2001 at the World University Games in Beijing, China, Jimyria Hicks ran the 200-meter dash in 24.0 seconds. Her best time at the U.S. Senior National Meet in June of the same year was 2. seconds. How much faster did she run in June? Align the decimals. She ran second faster in June. Example 2 Use a number line to find each sum. A. 0. (1.2) Move right units From, move left units. You finish at, so 0. (1.2) From units., move right You finish at, so 1 2. Holt McDougal Mathematics
7 LESSON 2- CONTINUED Example Add or subtract. Write each answer in simplest form. A B Subtract numerators. Keep the denominator Add numerators. Keep the denominator. Example 4 Evaluate each expression for the given value of the variable. A x for x ( ) Substitute for x. 7 1 B. 1 m for m Think: 12.1 (0.1) Substitute for m (10 ) Add numerators. Keep the denominator. Check It Out! 1. Tom ran the 100-meter dash in 11. seconds last year. This year he improved his time by 0.6 seconds. How fast did Tom run the 100-meter dash this year? 4 Holt McDougal Mathematics
8 LESSON 2-4 Multiplying Rational Numbers Lesson Objectives Multiply fractions, mixed numbers, and decimals Additional Examples Example 1 Multiply. Write each answer in simplest form. A B. 2 1 Multiply. Simplify. 2 1 Multiply. Multiply. Simplify. Holt McDougal Mathematics
9 LESSON 2-4 CONTINUED Example 2 Multiply. Write each answer in simplest form. A ( 6) Multiply. ( 7) Multiply. 1 ( 6) Look for common : 2. ( 7) B Simplest form 2(9) (2) Multiply. Multiply. 2(9) (2) Look for common : 2,. 1 Simplest form Example Multiply. A. 2(0.1) 2 (0.1) Product is with 2 decimal places. B. (0.4)(.7) (0.4) (.7) Product is with decimal places. You can drop the zeros after the. 6 Holt McDougal Mathematics
10 LESSON 2-4 CONTINUED Example 4 1 Joy completes 2 of her painting each day. How much of her painting 0 does she complete in a 7-day week? (7) Write as an improper fraction Multiply Joy completes of her painting in a 7-day week. Check It Out! 1. Multiply. Write the answer in simplest form. 2. Multiply. Write the answer in simplest form.. Multiply..1(0.2) 1 4. Ken completes 1 of his work each day. How much work does he 0 complete in a -day week? 7 Holt McDougal Mathematics
11 LESSON 2- Dividing Rational Numbers Lesson Objectives Divide fractions and decimals Vocabulary reciprocal (p. 4) Additional Examples Example 1 Divide. Write each answer in simplest form. A Multiply by the. B No common Simplest form Write as an fraction Multiply by the No common R Example 2 Find Divide. Holt McDougal Mathematics
12 LESSON 2- CONTINUED Example Evaluate each expression for the given value of the variable. A. 2 for n 0.1 ṅ has 2 decimal places, so use. Divide. B. k 4 for k Example 4 PROBLEM SOLVING APPLICATION A cookie recipe calls for 1 2 cup of oats. You have cup of oats. How 4 many batches of cookies can you bake using all of the oats you have? 1. Understand the Problem The number of batches of cookies you can bake is the number of batches using the oats that you have. List the important information: The amount of oats is cup. One batch of cookies calls for 2. Make a Plan Set up an equation. cup of oats. amount of oats you have amount for one batch number of batches 9 Holt McDougal Mathematics
13 LESSON 2- CONTINUED. Solve Let n number of batches. 4 1 n 2 4 n 6, or 4 batches of the cookies. 4. Look Back One cup of oats would make batches so 1 1 is a 2 answer. Check It Out! 1. Divide. Write the answer in simplest form Find Evaluate the expression for the given value of the variable. 2 ḃ for b A ship will use 1 of its total fuel load for a typical round trip. If there 6 is of a total fuel load on board now, how many complete trips can be made? 40 Holt McDougal Mathematics
14 LESSON 2-6 Adding and Subtracting with Unlike Denominators Lesson Objectives Add and subtract fractions with unlike denominators Additional Examples Example 1 Add or Subtract. Method 1: A. 1 2 Find a common : 7 (7) Multiply by fractions equal to. Rewrite with a common. Simplify. Method 2: B Write as an fraction. Multiples of 6: 6; 12; 24; 0 List the of each Multiples of : ; 16; 24; 2 denominator and find the Multiply by fractions equal to Rewrite with a denominator. Simplify. 41 Holt McDougal Mathematics
15 LESSON 2-6 CONTINUED Example 2 Evaluate t 4 for t 6. t 4 4 Substitute for t. 6 4 Multiply by fractions equal to. Rewrite with a denominator: 6(). Simplify. Example Two dancers are making necklaces from ribbon for their costumes. They need pieces measuring 1 4 inches and 127 inches. How much ribbon will be left over after the pieces are cut from a 6-inch length? Add to find the length of the two pieces. 4 Write as improper fractions. The LCD is. 10 Rewrite with common denominators.,or The length of the two ribbons is 26 inches. Now find the amount of ribbon left over. 42 Holt McDougal Mathematics
16 LESSON 2-6 CONTINUED 6 26 Subtract the length of the two ribbons from the total length of ribbon. 6 Write as improper fractions. The LCD is. 21 Rewrite with common denominators.,or There will be inches left. Check It Out! 1. Add Evaluate 9 h for h Michael wants to run 10 miles in three days. On Monday, he ran 2 4 miles. On Tuesday, he ran 1 miles. How many miles does Michael need to run on Wednesday? 4 Holt McDougal Mathematics
17 LESSON 2-7 Solving Equations with Rational Numbers Lesson Objectives Solve equations with rational numbers Additional Examples Example 1 Solve. A. m m Subtract from both sides. m B..2p 2..2p 2. Divide both sides by. p x C x Multiply both sides by. x Example 2 Solve. A. n n 2 7 Subtract from both sides. 7 n 44 Holt McDougal Mathematics
18 LESSON 2-7 CONTINUED B. y y Add to both sides. y 6 6 Find a common ; 6. C. 6 x y Simplify. 6 6 x 6 4 Multiply both sides by. x Simplify. Example Mr. Rios wants to prepare a dessert, but only has 2 2 sugar. If each serving of the dessert has 2 many servings can he make for the party? Convert fractions: 2 2 (2) 2 Write an equation: tablespoons of tablespoon of sugar, how Amount of sugar in one serving Total number of servings Total amount of sugar 2 x Now solve the equation. 2 x 2 x Divide both sides by. Multiply by the reciprocal,. x, or Simplify. Mr. Rios can make servings. 4 Holt McDougal Mathematics
19 LESSON 2-7 CONTINUED Check It Out! 1. Solve. m x Rick s car holds 2 the amount of gasoline as his wife s van. If the car s gas tank can hold 1 gallons of gasoline, how much gasoline can the 2 tank in the minivan hold? 46 Holt McDougal Mathematics
20 LESSON 2- Solving Two-Step Equations Lesson Objectives Solve two-step equations Additional Examples Example 1 PROBLEM SOLVING APPLICATION The mechanic s bill to repair Mr. Wong s car was $60. The mechanic charges $4 an hour for labor, and the parts that were used cost $44. How many hours did the mechanic work on the car? 1. Understand the Problem List the important information: The answer is the number of the mechanic worked on the car. The parts cost $. The labor cost $ per hour. The total bill was $. Let h represent the hours the mechanic worked. Total bill Parts Labor h 2. Make a Plan Think: First the variable is multiplied by, and then is added to the result. Work backward to solve the equation. Undo the operations in reverse order: First subtract from both sides of the equation, and then divide both sides of the new equation by. 47 Holt McDougal Mathematics
21 LESSON 2- CONTINUED. Solve h Subtract to undo the addition. h Divide to undo multiplication. h The mechanic worked for hours on Mr. Wong s car. 4. Look Back If the mechanic worked hours, the labor would be $4(4.6) $207. The sum of the parts and the labor would be $ $ $. Example 2 Solve. A. n 7 22 Think: First the variable is by, and then 7 is. To isolate the variable, subtract, and then multiply by. n 7 22 Subtract to undo addition. n n 1 Multiply to undo division. n 4 Holt McDougal Mathematics
22 LESSON 2- CONTINUED Check n Substitute into the original equation B. y 4 9 Think: First is subtracted from the variable, and then the result is divided by.to isolate the variable, by, and then 4. y 4 9 y 4 9 Multiply to undo division. y 4 Add to undo subtraction. y Check It Out! 1. The mechanic s bill to repair your car was $0. The mechanic charges $ an hour for labor, and the parts that were used cost $27. How many hours did the mechanic work on your car? 2. Solve. n Holt McDougal Mathematics
23 CHAPTER 2 Chapter Review 2-1 Rational Numbers Simplify Write each decimal as a fraction in simplest form Comparing and Ordering Rational Numbers Compare. Write,, or The lengths of four student s pencils in Mr. Roberson s class are.7 inches,. inches, inches, and. inches. List these measurements in order 6 from least to greatest. 2- Adding and Subtracting Rational Numbers Evaluate each expression for the given value of the variable m for m t for t x for x Sammie and Lance both jog home after school every day. On Thursday, they made it home in 0.7 hour. Friday, it took them of an hour. How much longer did it take Sammie and Lance to jog home on Thursday than Friday? 2-4 Multiplying Rational Numbers Multiply. Write each answer in simplest form (6.7) (9.1) 22. Kylie needs to cut 6 pieces of yarn that are each 2 foot long. How much yarn does Kylie need? 0 Holt McDougal Mathematics
24 CHAPTER 2 REVIEW CONTINUED 2- Dividing Rational Numbers Divide. Write each answer in simplest form Evaluate each expression for the given value of the variable h for h for p 2 p for l 4.7 l 2-6 Adding and Subtracting with Unlike Demoninators Add or Subtract Sallie needs cup of milk for a recipe. She has 22 cups of milk. How much more milk does Sallie need? 2-7 Solving Equations with Rational Numbers Solve. s y g w Amy watched television for 2 4 hours. Charles watched television for 1 of the time Amy did. How long, in minutes, did Charles watch television? 2- Solving Two-Step Equations Solve. 9. d c 9 v u Gina bought a magazine subscription. The magazine company charged $1.7 for the subscription and $2.1 for each issue. If Gina paid $4., how many issues did she receive? 1 Holt McDougal Mathematics
25 CHAPTER 2 Big Ideas Answer these question to summarize the important concepts from Chapter 2 in your own words. 1. Explain how to write 2.4 as a fraction in simplest form. 2. Explain how to add 4 9 and Explain how to divide rational numbers in fraction form. 4. Explain how to solve the equation x For more review of Chapter 2: Complete the Chapter 2 Study Guide and Review on pages of your textbook. Complete the Ready to Go On quizzes on pages 94 and 106 of your textbook. 2 Holt McDougal Mathematics
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