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Chapter 1

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Chapter 1

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  1. Chapter 1 Section 1

  2. 1 Fractions 1.1 Objectives Learn the definition of factor. Write fractions in lowest terms. Multiply and divide fractions. Add and subtract fractions. Solve applied problems that involve fractions. Interpret data in a circle graph. 2 3 4 5 6

  3. Natural numbers: 1, 2, 3, 4,…, Fractions: Example: The improper fraction can be written , a mixed number. Definitions Whole numbers: 0, 1, 2, 3, 4,…, Numerator Fraction Bar Denominator , , Proper fraction: Numerator is less than denominator and the value is less than 1. Improper fraction: Numerator is greater than or equal to denominator and the value is greater than or equal to 1. Mixed number: A combination of a whole number and a proper fraction. Slide 1.1-3

  4. Objective 1 Learn the definition of factor. Slide 1.1-4

  5. Learn the definition of factor. In the statement 3 × 6 = 18, the numbers 3 and 6 are called factors of 18. Other factors of 18 include 1, 2, 9, and 18. The number 18 in this statement is called the product. The number 18 is factoredby writing it as a product of two or more numbers. Examples: 6 · 3, 18 × 1, (2)(9), 2(3)(3) A raised dot • is often used instead of the × symbol to indicate multiplication because × may be confused with the letter x. Slide 1.1-5

  6. Learn the definition of factor. (cont’d) A natural number greater than 1 is prime if its products include only 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37… A natural number greater than 1 that is not prime is called a compositenumber. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21… Slide 1.1-6

  7. EXAMPLE 1 Write 90 as the product of prime factors. Factoring Numbers Solution: Starting with the least prime factor is not necessary. No matter which prime factor we start with, the same prime factorization will always be obtained. Slide 1.1-7

  8. Objective 2 Write fractions in lowest terms. Slide 1.1-8

  9. A fraction is in lowest terms when the numerator and denominator have no common factors other than 1. Write fractions in lowest terms. Basic Principle of Fractions If the numerator and denominator of a fraction are multiplied or divided by the same nonzero number, the value of the fraction remains unchanged. Slide 1.1-9

  10. Writing a Fraction in Lowest Terms Step 1:Write the numerator and the denominator as the product of prime factors. Write fractions in lowest terms. (cont’d) Step 2:Divide the numerator and the denominator by the greatest common factor, the product of all factors common to both. Slide 1.1-10

  11. EXAMPLE 2 Write in lowest terms. Writing Fractions in Lowest Terms Solution: = When writing fractions in lowest terms, be sure to include the factor 1 in the numerator or an error may result. Slide 1.1-11

  12. Objective 3 Multiply and divide fractions. Slide 1.1-12

  13. Multiply and divide fractions. Multiplying Fractions If and are fractions, then · = . That is, to multiply two fractions, multiply their numerators and then multiply their denominators. Some prefer to factor and divide out any common factors before multiplying. Slide 1.1-13

  14. EXAMPLE 3 Find each product, and write it in lowest terms. Multiplying Fractions Solution: or Slide 1.1-14

  15. Multiply and divide fractions. (cont’d) Dividing Fractions If and are fractions, then ÷ = . That is, to divide by a fraction, multiply by its reciprocal; the fraction flipped upside down. Slide 1.1-15

  16. EXAMPLE 4 Find each quotient, and write it in lowest terms. Dividing Fractions Solution: or Slide 1.1-16

  17. Objective 4 Add and subtract fractions. Slide 1.1-17

  18. Add and subtract fractions. Adding Fractions If and are fractions, then + = . To find the sum of two fractions having the same denominator, add the numerators and keep the same denominator. Slide 1.1-18

  19. EXAMPLE 5 Find the sum , and write it in lowest terms. Adding Fractions with the Same Denominator Solution: Slide 1.1-19

  20. Add and subtract fractions. (cont’d) Finding the Least Common Denominator If the fractions do not share a common denominator, the least common denominator (LCD) must first be found as follows: Step 1: Factor each denominator. Step 2:Use every factor that appears in any factored form. If a factor is repeated, use the largest number of repeats in the LCD. Slide 1.1-20

  21. EXAMPLE 6 Find each sum, and write it in lowest terms. Adding Fractions with Different Denominators Solution: or Slide 1.1-21

  22. Add and subtract fractions. (cont’d) Subtracting Fractions If and are fractions, then . To find the difference between two fractions having the same denominator, subtract the numerators and keep the same denominator. If fractions have different denominators, find the LCD using the same method as with adding fractions. Slide 1.1-22

  23. Subtracting Fractions Find each difference, and write it in lowest terms. EXAMPLE 7 Solution: or Slide 1.1-23

  24. Objective 5 Solve applied problems that involve fractions. Slide 1.1-24

  25. EXAMPLE 8 A gallon of paint covers 500 ft2. To paint his house, Tran needs enough paint to cover 4200 ft2. How many gallons of paint should he buy? Solution: Adding Fractions to Solve an Applied Problem Tran needs to buy 9 gallons of paint. Slide 1.1-25

  26. Objective 6 Interpret data in a circle graph. Slide 1.1-26

  27. EXAMPLE 9 Recently there were about 970 million Internet users world wide. The circle graph below shows the fractions of these users living in various regions of the world. Using a Circle Graph to Interpret Information Which region had the second-largest number of Internet Users? Estimate the number of Internet users in Europe. How many actual Internet users were there in Europe? Slide 1.1-27

  28. EXAMPLE 9 Using a Circle Graph to Interpret Information (cont’d) • Solution: • Europe Slide 1.1-28