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3-4 Equivalent Fractions and Mixed Numbers California Standards NS2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g. to find a common denominator to add two fractions or to find the reduced form of a fraction). Also covered: NS1.1 Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers Vocabulary equivalent fractions Different expressions for the same nonzero number. (fractions that are equal to each other). improper fraction A fraction in which the numerator is greater than the denominator. mixed number A number that contains a whole number and a fraction. Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers Different fractions can name the same number. 3 5 Holt CA Course 1 = 6 10 = 15 25 3-4 Equivalent Fractions and Mixed Numbers In the diagram 3 = 6 = 15 . These are called 5 10 25 equivalent fractions because they are different expressions for the same nonzero number. To create fractions equivalent to a given fraction, multiply or divide the numerator and denominator by the same nonzero number. Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers Example 1: Finding Equivalent Fractions Find two fractions equivalent to 5 7. 5 2 = 10 72 14 Multiply the numerator and denominator by 2. 53 73 Multiply the numerator and denominator by 3. = 15 21 Remember! A fraction with the same numerator and 2 denominator, such as is equal to 1. 2 Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers 10 , and 15 are equivalent, The fractions 5 , 21 7 14 but only 5 is in simplest form. A fraction is in 7 simplest form when the greatest common divisor of its numerator and denominator is 1. Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers Example 2: Writing Fractions in Simplest Form 18 Write the fraction 24 in simplest form. Find the GCD of 18 and 24. 18 = 2 • 3 • 3 24 = 2 • 2 • 2 The GCD is 6 = 2 • 3. 3 18 = 18 ÷ 6 = 3 24 24 ÷ 6 4 Holt CA Course 1 • Divide the numerator and denominator by 6. 3-4 Equivalent Fractions and Mixed Numbers You can also simplify fractions by dividing by common factors until the numerator and denominator have no more common factors except one. To determine if two fractions are equivalent, simplify the fractions. Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers Additional Example 3A: Determining Whether Fractions are Equivalent Determine whether the fractions in each pair are equivalent. 4 and 28 6 42 4 Simplify both fractions 4÷2 2 6 = 6÷2= 3 and compare. 28 = 28 ÷ 14 = 2 42 ÷ 14 3 42 4 and 28 are equivalent because both are equal to 2 . 6 42 3 Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers Example 3B: Determining Whether Fractions are Equivalent Determine whether the fractions in each pair are equivalent. 6 and 20 10 25 Simplify both fractions and compare. 6 = 6÷2 = 3 10 ÷ 2 10 5 20 = 20 ÷ 5 = 4 25 ÷ 5 25 5 6 and 20 are not equivalent because their simplest 10 25 forms are not equal. Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers 8 is an improper 5 fraction. Its numerator is greater than its denominator. Holt CA Course 1 3 is a mixed 5 number. It contains both a whole number and a fraction. 1 8 = 13 5 5 3-4 Equivalent Fractions and Mixed Numbers Example 4: Converting Between Improper Fractions and Mixed Numbers A. Write 13 as a mixed number. 5 First divide the numerator by the denominator. 13 = 2 3 5 5 Use the quotient and remainder to write the mixed number. B. Write 7 2 as an improper fraction. 3 First multiply the denominator and whole number, and then add the numerator. + 2 = 3 7 + 2 = 23 Use the result to write the improper 3 3 3 fraction. Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers Check It Out! Example 1 Find two fractions equivalent to 6 . 12 6 2 = 12 12 2 24 Multiply the numerator and denominator by 2. 6 ÷2 3 = 12 ÷ 2 6 Divide the numerator and denominator by 2. Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers Check It Out! Example 2 15 Write the fraction 45 in simplest form. Find the GCD of 15 and 45. 15 = 3 • 5 45 = 3 • 3 The GCD is 15 = 3 • 5. 5 15 = 15 ÷ 15 = 1 45 45 ÷ 15 3 Holt CA Course 1 • Divide the numerator and denominator by 15. 3-4 Equivalent Fractions and Mixed Numbers Check It Out! Example 3A Determine whether the fractions in each pair are equivalent. 3 and 6 9 18 Simplify both fractions and compare. 3 3÷3 1 = = 9 9÷3 3 6 = 6 ÷6 = 1 18 ÷ 6 18 3 3 and 6 are equivalent because both are equal to 1 . 9 18 3 Holt CA Course 1 3-4 Equivalent Fractions and Mixed Numbers Check It Out! Example 4 A. Write 15 6 as a mixed number. First divide the numerator by the denominator. 15 = 2 3 = 2 1 2 6 6 Use the quotient and remainder to write the mixed number. B. Write 8 1 3 as an improper fraction. First multiply the denominator and whole number, and then add the numerator. + Use the result to 3 8 + 1 1 25 = 83 = write the improper 3 3 fraction. Holt CA Course 1