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3-1 3-1 Rational RationalNumbers Numbers Warm Up Problem of the Day Lesson Presentation Pre-Algebra Pre-Algebra 3-1 Rational Numbers Warm Up Divide. 1. 36 3 3. 68 17 12 4 5. 1024 64 16 Pre-Algebra 2. 144 6 4. 345 115 24 3 3-1 Rational Numbers Problem of the Day An ice cream parlor has 6 flavors of ice cream. A dish with two scoops can have any two flavors, including the same flavor twice. How many different double-scoop combinations are possible? 21 Pre-Algebra 3-1 Rational Numbers Learn to write rational numbers in equivalent forms. Pre-Algebra 3-1 Rational Numbers Vocabulary rational number relatively prime Pre-Algebra 3-1 Rational Numbers A rational number is any number that can n be written as a fraction , where n and d d are integers and d 0. Decimals that terminate or repeat are rational numbers. Pre-Algebra 3-1 Rational Numbers Numerator Pre-Algebra n d Denominator 3-1 Rational Numbers The goal of simplifying fractions is to make the numerator and the denominator relatively prime. Relatively prime numbers have no common factors other than 1. Pre-Algebra 3-1 Rational Numbers You can often simplify fractions by dividing both the numerator and denominator by the same nonzero integer. You can simplify 12 4 the fraction to by dividing both the 15 5 numerator and denominator by 3. 12 of the 15 boxes are shaded. 12 15 4 of the 5 boxes are shaded. = 4 5 The same total area is shaded. Pre-Algebra 3-1 Rational Numbers Additional Example 1A: Simplifying Fractions Simplify. 5 A. 10 5 =1•5 10 = 2 • 5 5÷5 5 = 10 ÷ 5 10 1 = 2 Pre-Algebra ; 5 is a common factor. Divide the numerator and denominator by 5. 3-1 Rational Numbers Additional Example 1B: Simplifying Fractions Simplify. B. 16 80 16 = 1 • 16 ;16 is a common factor. 80 = 5 • 16 16 ÷ 16 16 = 80 ÷ 16 80 1 = 5 Pre-Algebra Divide the numerator and denominator by 16. 3-1 Rational Numbers Additional Example 1C: Simplifying Fractions Simplify. C. –18 29 18 = 2 • 9 29 = 1 • 29 –18 –18 = 29 29 Pre-Algebra ;There are no common factors. –18 and 29 are relatively prime. 3-1 Rational Numbers Try This: Example 1A Simplify. A. 6 30 6 = 1 • 6 ;6 is a common factor. 30 = 5 • 6 6÷6 6 = 30 ÷ 6 30 1 = 5 Pre-Algebra Divide the numerator and denominator by 6. 3-1 Rational Numbers Try This: Example 1B Simplify. 18 = 3 • 3 • 2 ;9 is a common factor. 27 = 3 • 3 • 3 B. 18 27 18 = 18 ÷ 9 27 ÷ 9 27 = Pre-Algebra 2 3 Divide the numerator and denominator by 9. 3-1 Rational Numbers Try This: Example 1C Simplify. C. 17 –35 17 = 1 • 17 ;There are no common factors. 35 = 5 • 7 17 17 17 and –35 are =– –35 35 relatively prime. Pre-Algebra 3-1 Rational Numbers To write a finite decimal as a fraction, identify the place value of the farthest digit to the right. Then write all of the digits after the decimal points as the numerator with the place value as the denominator. Pre-Algebra 3-1 Rational Numbers Additional Example 2A: Writing Decimals as Fractions Write the decimal as a fraction in simplest form. A. –0.8 –0.8 = –8 10 4 =– 5 Pre-Algebra –8 is in the tenths place. Simplify by dividing by the common factor 2. 3-1 Rational Numbers Additional Example 2B: Writing Decimals as Fractions Write the decimal as a fraction in simplest form. B. 5.37 37 7 is in the hundredths place. 5.37 = 5 100 Pre-Algebra 3-1 Rational Numbers Additional Example 2C: Writing Decimals as Fractions Write the decimal as a fraction in simplest form. C. 0.622 622 0.622 = 1000 311 = 500 Pre-Algebra 2 is in the thousandths place. Simplify by dividing by the common factor 2. 3-1 Rational Numbers Try This: Example 2A Write the decimal as a fraction in simplest form. A. –0.4 –4 –0.4 = 10 2 =– 5 Pre-Algebra –4 is in the tenths place. Simplify by dividing by the common factor 2. 3-1 Rational Numbers Try This: Example 2B Write the decimal as a fraction in simplest form. B. 8.75 75 5 is in the hundredths place. 8.75 = 8 100 3 = 8 4 Pre-Algebra Simplify by dividing by the common factor 25. 3-1 Rational Numbers Try This: Example 2C Write each decimal as a fraction in simplest form. C. 0.2625 2625 0.2625 = 10,000 21 = 80 Pre-Algebra 5 is in the ten-thousandths place. Simplify by dividing by the common factor 125. 3-1 Rational Numbers To write a fraction as a decimal, divide the numerator by the denominator. You can use long division. numerator denominator denominator numerator When writing a long division problem from a fraction, put the numerator inside the “box,” or division symbol. It may help to write the numerator first and then say “divided by” to yourself as you write the division symbol. Pre-Algebra 3-1 Rational Numbers Additional Example 3A: Writing Fractions as Decimals Write the fraction as a decimal. A. 11 9 The fraction Pre-Algebra 1 .2 9 11 .0 –9 20 –1 8 2 The pattern repeats, so draw a bar over the 2 to indicate that this is a repeating decimal. 11 is equivalent to the decimal 1.2. 9 3-1 Rational Numbers Additional Example 3B: Writing Fractions as Decimals Write the fraction as a decimal. 7 B. 20 0.3 5 This is a terminating decimal. 20 7.0 0 –0 70 –6 0 1 00 –1 0 0 0 The remainder is 0. The fraction Pre-Algebra 7 is equivalent to the decimal 0.35. 20 3-1 Rational Numbers Try This: Example 3A Write the fraction as a decimal. A. 15 9 The fraction Pre-Algebra 1 .6 9 15 .0 –9 60 –5 4 6 The pattern repeats, so draw a bar over the 6 to indicate that this is a repeating decimal. 15 is equivalent to the decimal 1.6. 9 3-1 Rational Numbers Try This: Example 3B Write the fraction as a decimal. 9 B. 40 0.2 2 5 This is a terminating decimal. 40 9.0 0 0 –0 90 –8 0 1 00 – 80 200 – 2 00 0 The remainder is 0. 9 The fraction is equivalent to the decimal 0.225. 40 Pre-Algebra 3-1 Rational Numbers Lesson Quiz: Part 1 Simplify. 18 1. 42 3 7 15 2. 21 5 7 Write each decimal as a fraction in simplest form. 5 27 – 3. 0.27 4. –0.625 8 100 13 5. Write as a decimal 6 Pre-Algebra 2.16 3-1 Rational Numbers Lesson Quiz: Part 2 6. Tommy had 13 hits in 40 at bats for his baseball team. What is his batting average? (Batting average is the number of hits divided by the number of at bats, expressed as a decimal.) 0.325 Pre-Algebra