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Math 2303 section 2.3 Section 2.3 Rational Numbers A rational number is a number that may be written in the form , where a and b are integers and b is nonzero. Integer a is called the numerator, and the integer b is called the denominator. Examples of rational numbers: The rational numbers are ratios of whole numbers. Equivalent fractions: two rational numbers are equivalent if and only if . Note that . Example 1: Determine whether or not the fractions and are equivalent. Reducing Fractions To reduce a fraction we must find the greatest common factor, gcf, and divide it out. The resulting fraction is in simplest form (lowest terms). Finding the gcf is too much work, start by dividing out common factors ( you may use the divisibility rules here), the fraction will become simpler. Example 2: Reduce to lowest terms. a. b. 1 Math 2303 section 2.3 c. Popper 5 question 1: Write the following fraction in the simplest form. a. b. c. d. Mixed Numbers and Improper Fractions The form is called a mixed number. The form where a > b is called an improper fraction. We can convert from a mixed numeral to an improper fraction and vice versa. Mixed numeral to an improper fraction: Example 3: Convert the following numerals to an improper fractions. a. 3 b. 1 c. 9 Popper 05 question 2: Write the following mixed numeral as an improper fraction. 6 a. b. c. d. 2 Math 2303 section 2.3 Example 4: Convert the following improper fractions to mixed numbers. a. b. c. (‐120)/(‐18) d. Popper 05 question 3: Write the following improper fraction in the mixed form. a. 5 b. 4 c. 5 d. 4 Multiplication of Fractions In order to multiply fractions, first make sure the fractions are in simplest form. Then multiply the numerators together and multiply the denominators together. If as any of the numbers are mixed numbers, first convert to an improper fraction then multiply described: Division of Fractions The reciprocal of a fraction, is fraction . In order to divide fraction change the division to multiplication and find the reciprocal of the second fraction (the one dividing). Then follow the rules for multiplying fractions. 3 Math 2303 section 2.3 Example 5: Evaluation the following: a. b. 2
c. 3 d. 3 2 e. 1 2 Popper 05 question 4 Express your answer as a fraction in simplest form. a. b. c. d. Popper 05 question 5: Express your answer as a fraction in simplest form. a. b. c. ‐ d. . 4 Math 2303 section 2.3 Addition and Subtraction of Fractions In order to add or subtract fractions, we first need a common denominator. Once we write each fraction in the same denominator, simply add or subtract the numerators and keep the denominator. Reduce the answer if possible. If any of the numbers are mixed numerals, first convert them to an improper fraction then add/subtract as described above. Find the lcm of the denominators if the denominators are not the same. Express each fraction as an equivalent fraction with the lcm as the denominator. Example 6: Perform the indicated operation: a. b. c. d. 2 e. 3
2 f. 1
2 5 Math 2303 section 2.3 g. Popper 06 question 1: Express your answer as a fraction in simplest form: 3
1 a. b. c. d. Popper 06 question 2: Express your answer as a fraction in simplest form: a. . b. c. d. Popper 06 question 3: Express your answer as a fraction in simplest form: 4
a. 1 b. 1 c. d. 5 Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number. 0.12 since upon the dividing, the division ended. Example of terminating decimal is Example of repeating decimal is 0.333333… = 0.3 since upon dividing, the division did not end and it is repeated. Converting Decimal Numbers to Fractions First recall that 0.1, 0.01 , 0.001 etc. Example 7: Convert the following terminating decimal numbers to a quotient of integers. a. 0.6 b. 0.0622 6 Math 2303 section 2.3 c. 1.02 d. 643.875 Converting Repeating Decimal Numbers to a Fractions Example 8: Convert the following decimals to a quotient of integers. a. 0.35 b. 0.628 Example 9: Convert the following decimals to a quotient of integers. a. 0.72222 …
0. 72 b. 0.005555 …
0.005 c. 0.034151515 …
0.03415 7 Math 2303 section 2.3 d. 12.14222222 …
12.142 e. 15.2421421 …
15.2421 Popper 06 question 4: Write the following repeating decimal as a fraction in simplest form: 0.24 a. b. c. d. Example 10: Express fractions as decimals. a. b. c. 8 Math 2303 section 2.3 d. Popper 06 question 5: Write the fraction as a repeating decimal: a. 0.4 b. 0.45 c. 0.45 d. 0.454 Solutions to popper 3 Popper 3 question 1: Which of the following numbers is a divisor of 36. A. 5 B. 8 C. 12 D. 15 Popper 3 question 2: Determine the prime factorization of 84. a. 2 x 3 x 7 b. 12 x 7 c. 2 x 3 x 7 d. 2 x 3 x 7 Popper 3 questions 3: Find the greatest common factor of 168 and 96. a. 6 b. 9 c. 16 d. 24 Popper 3 question 4: Find the least common multiple of 30 and 105. a. 150 b. 210 c. 420 d. 1050 Popper 3 question 5: Determine the value of : 6
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2 a. 0 b. 4 c. 12 d. 16 9