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§ 1.3 Fractions Numerators and Denominators A quotient of two numbers is called a fraction. The fraction 1 represents the 4 shaded part of the circle. 1 out of 1 4 pieces is shaded. 4 is read “one- fourth.” 1 numerator 4 denominator Martin-Gay, Beginning and Intermediate Algebra, 4ed 2 Simplifying Fractions To simplify fractions we can simplify the numerator and the denominator. 2 · 5 = 10 factors product A fraction is said to be simplified or in lowest terms when the numerator and denominator have no factors in common other than 1. 2 17 1 3 23 9 Martin-Gay, Beginning and Intermediate Algebra, 4ed 3 Prime and Composite Numbers A prime number is a natural number, other than 1, whose only factors are 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 The first 10 prime numbers A natural number, other than 1, that is not a prime number is called a composite number. Every composite number can be written as a product of prime numbers Martin-Gay, Beginning and Intermediate Algebra, 4ed 4 Product of Primes Example: Write the number 24 as a product of primes. 24 = 4 6 Write 24 as the product of any two whole numbers. 22 23 If the factors are not prime, they must be factored. 24 = 2 2 2 3 When all of the factors are prime, the number has been completely factored. Martin-Gay, Beginning and Intermediate Algebra, 4ed 5 The Fundamental Principal of Fractions The Fundamental Principal of Fractions If a is a fraction and c is a nonzero real number, then b ac a bc b Example: 25 Write the fraction in lowest terms. 40 25 55 5 5 40 2 2 2 5 2 2 2 8 Martin-Gay, Beginning and Intermediate Algebra, 4ed 6 Multiplying Fractions To multiply two fractions, multiply numerator times numerator to obtain the numerator of the product. Multiply denominator times denominator to obtain the denominator of the product. Multiplying Fractions a c ac , if b 0 and d 0 b d bd 3 2 6 3 2 6 7 5 35 7 5 35 Martin-Gay, Beginning and Intermediate Algebra, 4ed 7 Multiplying Fractions 12 3 Example: Multiply. 17 24 12 3 12 3 36 Multiply numerators. 17 24 17 24 408 Multiply denominators. Simplify the product by dividing the numerator and the denominator by any common factors. 36 2 2 3 3 3 408 2 2 2 3 17 34 Martin-Gay, Beginning and Intermediate Algebra, 4ed 8 Dividing Fractions Two fractions are reciprocals of each other if their product is 1. 3 4 3 4 1 4 and are reciprocals. 3 4 3 Dividing Fractions a c a d , if b 0 and d 0 b d b c Martin-Gay, Beginning and Intermediate Algebra, 4ed 9 Dividing Fractions 3 1 Example: Divide. 4 4 3 1 3 4 12 3 4 4 4 1 4 Martin-Gay, Beginning and Intermediate Algebra, 4ed 10 Fractions with the Same Denominator To add or subtract fractions with the same denominator, combine numerators and place the sum or difference over the common denominator. 2 1 3 4 4 4 Adding and Subtracting Fractions with the Same Denominator a c ac , if b 0 b b b a c ac , if b 0 b b b Martin-Gay, Beginning and Intermediate Algebra, 4ed 11 Equivalent Fractions Equivalent fractions are fractions that represent the same quantity. 3 is shaded. 1 is shaded. 6 2 Equivalent fractions Martin-Gay, Beginning and Intermediate Algebra, 4ed 12 Equivalent Fractions 3 Example: Write as an equivalent fraction with a 4 denominator of 20. Since 4 · 5 = 20, multiply the fraction by 5 . 5 3 3 5 3 5 15 4 4 5 4 5 20 5 Multiply by or 1. 5 Martin-Gay, Beginning and Intermediate Algebra, 4ed 13 Fractions without the Same Denominator To add or subtract fractions without the same denominator, first write the fractions as equivalent fractions with a common denominator The least common denominator (LCD) is the smallest number both denominators will divide evenly into. 3 1 Example: Add. 3 3 9 1 4 4 8 6 8 3 24 6 4 24 LCD = 24 9 4 13 24 24 24 Martin-Gay, Beginning and Intermediate Algebra, 4ed 14 Fractions without the Same Denominator 5 7 Example: Subtract. 12 30 LCD = 60 5 5 25 7 2 14 12 5 60 30 2 60 25 14 11 60 60 60 Martin-Gay, Beginning and Intermediate Algebra, 4ed 15

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posted: | 11/16/2011 |

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