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Fractions and Rational Expressions

Fractions and Rational Expressions. Name the fraction represented by a shaded region. Graph fractions on a number line. Simplify fractions. Write equivalent fractions. Use, <, >, or = to write a true statement. Write improper fractions as mixed numbers.

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Fractions and Rational Expressions

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  1. Fractions and Rational Expressions Name the fraction represented by a shaded region. Graph fractions on a number line. Simplify fractions. Write equivalent fractions. Use, <, >, or = to write a true statement. Write improper fractions as mixed numbers. Write mixed numbers as improper fractions. 5.1

  2. Objective 1 Name the fraction represented by a shaded region.

  3. Definition Fraction: A number that describes a part of a whole. We can describe the three lots that have been sold out of the five total lots using the fraction , which is read “three fifths.” Fractions have numerators (the number 3) and denominators (the number 5 in this example).

  4. Definitions Numerator: The number written in the top position of the fraction. Denominator: The number written in the bottom position of a fraction. The denominator, 5, is the number of equal-sized divisions. The numerator, 3, is the number of those division we are interested in working with. Numerator Denominator

  5. Definition Rational number: A number that can be expressed in the form , where a and b are integers and b ≠ 0. …is a rational number because 3 and 5 are integers and the denominator, 5, is not 0.

  6. Example 1 Name the fraction represented by the shaded region. a. b.

  7. Example 2 In a group of 35 people at a conference, 17 are wearing glasses. What fraction of the people at the conference are wearing glasses? What fraction are not wearing glasses?

  8. Objective 2 Graph fractions on a number line.

  9. Graph the fraction on a number line. Example 3 a. 0 1

  10. Objective 3 Simplify fractions.

  11. Definitions Simplify: The process of writing an equivalent expression with fewer symbols or smaller numbers. Simplest form: An equivalent expression written with the fewest symbols and the smallest numbers possible.

  12. Example 4 Simplify.

  13. Rule If the denominator of a fraction is 1, the fraction can be simplified to the numerator. In math language:

  14. Example 5 Simplify.

  15. Rule If the numerator of a fraction is 0, and the denominator is any number other than 0, the fraction can be simplified to 0. In math language:

  16. Example 6 Simplify.

  17. Rule If the denominator of a fraction is 0, and the numerator is any number other than 0, we say the fraction is undefined. In math language:

  18. Example 7 Simplify.

  19. Rules A fraction with the same nonzero numerator and nonzero denominator can be simplified to 1. In math language: If the numerator and denominator of a fraction are both 0, the fraction is indeterminate. In math language:

  20. Objective 4 Write equivalent fractions.

  21. Definition Equivalent fractions: Fractions that name the same number. 1

  22. Procedure To write an equivalent fraction, multiply or divide both the numerator and denominator by the same nonzero number.

  23. Example 8 Fill in the blank so that the fractions are equivalent.

  24. Objective 5 Use <, >, or = to write a true statement.

  25. We can easily compare fractions with the same denominator. If two fractions don’t have the same denominator, we can draw a picture and compare them. We could also write fractions so that they have a common denominator by using multiples.

  26. Definition Multiple: A number that is evenly divisible by a given number. Multiples of 2 are 2, 4, 6, 8, 10,… Multiples of 3 are 3, 6, 9, 12, 15,… Notice the common multiple for 2 and 3 is 6…it appears in both lists. To upscale 1/2, we multiply numerator and denominator by 3. To upscale 1/3, we, multiply numerator and denominator by 2.

  27. Procedure To compare two fractions: • Write equivalent fractions that have a common denominator. • Compare the numerators of the equivalent fractions.

  28. Example 10 Use <, >, or = to write a true statement. b. a.

  29. Objective 6 Write improper fractions as mixed numbers.

  30. Definition Improper fraction: A fraction in which the absolute value of the numerator is greater than or equal to the absolute value of the denominator.

  31. Definition Mixed number: An integer combined with a fraction When we say combined, we literally mean added. Note: 2 ¼ is read “two and one-forth” and means two wholes plus ¼ of another whole. How does this apply to negative mixed numbers? Note: The negative sign applies to both the integer and the fraction.

  32. Procedure To write an improper fraction as a mixed number: • Divide the denominator into the numerator. • Write the result in the following form.

  33. Example 11 Write the improper fraction as a mixed number. b. a.

  34. Objective 7 Write mixed numbers as improper fractions.

  35. Procedure To write a mixed number as an improper fraction: • Multiply the integer by the denominator. • Add the resulting product to the numerator to find the numerator of the improper fraction. • Keep the same denominator.

  36. Example 13 Write the mixed number as an improper fraction. b. a.

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