Rational Numbers 3-1

Warm Up Divide. 1. 36  3 2. 144  6 12 24 3. 68  17 4. 345  115 4 3 5. 1024  64 16

Learn to write rational numbers in equivalent forms.

Vocabulary rational number relatively prime

Decimals that terminate or repeat are rational numbers. A rational number is any number that can be written as a fraction , where n and d are integers and d  0. n d Decimals that terminate or repeat are rational numbers.

Numerator n d Denominator

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.

The same total area is shaded. You can often simplify fractions by dividing both the numerator and denominator by the same nonzero integer. You can simplify the fraction to by dividing both the numerator and denominator by 3. 12 15 4 5 12 of the 15 boxes are shaded. 4 of the 5 boxes are shaded. = 12 15 4 5 The same total area is shaded.

Example: Simplifying Fractions 5 10 5 = 1 • 5 10 = 2 • 5 ;5 is a common factor. A. 5 10 = 5 ÷ 5 10 ÷ 5 Divide the numerator and denominator by 5. 1 2 =

Example: Simplifying Fractions 16 80 16 = 1 • 16 80 = 5 • 16 ;16 is a common factor. B. Divide the numerator and denominator by 16. 16 80 = 16 ÷ 16 80 ÷ 16 1 5 =

Example: Simplifying Fractions –18 29 18 = 2 • 9 29 = 1 • 29 ;There are no common factors. C. –18 29 = –18 29 –18 and 29 are relatively prime.

Try This = = Simplify. 6 30 6 = 1 • 6 30 = 5 • 6 6 30 6 = 1 • 6 30 = 5 • 6 A. ;6 is a common factor. 6 30 = 6 ÷ 6 Divide the numerator and denominator by 6. 30 ÷ 6 1 5 =

Try This = = Simplify. 18 27 18 = 3 • 3 • 2 27 = 3 • 3 • 3 18 27 18 = 3 • 3 • 2 27 = 3 • 3 • 3 ;9 is a common factor. B. 18 27 = 18 ÷ 9 27 ÷ 9 Divide the numerator and denominator by 9. 2 3 =

Try This = – Simplify. 17 –35 17 = 1 • 17 35 = 5 • 7 17 –35 17 = 1 • 17 35 = 5 • 7 ;There are no common factors. C. 17 –35 = – 17 35 17 and –35 are relatively prime.

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.

Example: Writing Decimals as Fractions Write the decimal as a fraction in simplest form. A. –0.8 –8 is in the tenths place. –8 10 = Simplify by dividing by the common factor 2. –0.8 = – 4 5

Example: Writing Decimals as Fractions Write the decimal as a fraction in simplest form. B. 5.37 7 is in the hundredths place. 37 100 = 5 5.37

Example: Writing Decimals as Fractions Write the decimal as a fraction in simplest form. 2 is in the thousandths place. C. 0.622 622 1000 = Simplify by dividing by the common factor 2. 0.622 = 311 500

Try This = = – A. –0.4 –4 is in the tenths place. –4 10 Write the decimal as a fraction in simplest form. A. –0.4 –4 is in the tenths place. –4 10 = Simplify by dividing by the common factor 2. –0.4 = – 2 5

Try This = 8 = 8 B. 8.75 5 is in the hundredths place. 75 100 Write the decimal as a fraction in simplest form. B. 8.75 5 is in the hundredths place. 75 100 = 8 Simplify by dividing by the common factor 25. 8.75 = 8 3 4

Try This C. = = 5 is in the ten-thousandths place. 0.2625 2625 10,000 Write each decimal as a fraction in simplest form. 5 is in the ten-thousandths place. C. 0.2625 2625 10,000 = Simplify by dividing by the common factor 125. 0.2625 = 21 80

numerator denominator denominator numerator 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.

Example: Writing Fractions as Decimals Write the fraction as a decimal. 11 9 1 .2 A. 9 11 .0 The pattern repeats, so draw a bar over the 2 to indicate that this is a repeating decimal. –9 2 –1 8 2 The fraction is equivalent to the decimal 1.2. 11 9

Example: Writing Fractions as Decimals Write the fraction as a decimal. 7 20 .3 5 This is a terminating decimal. B. 20 7 .0 –0 7 –6 0 1 0 –1 0 The remainder is 0. The fraction is equivalent to the decimal 0.35. 7 20

Try This Write the fraction as a decimal. 15 9 1 .6 9 15 .0 15 9 1 .6 A. 9 15 .0 The pattern repeats, so draw a bar over the 6 to indicate that this is a repeating decimal. –9 6 –5 4 6 The fraction is equivalent to the decimal 1.6. 15 9

Try This Write the fraction as a decimal. 9 40 .2 2 5 40 9 .0 9 40 .2 2 5 B. 40 9 .0 This is a terminating decimal. –0 9 –8 0 1 0 – 8 2 – 2 The remainder is 0. The fraction is equivalent to the decimal 0.225. 9 40

Lesson Quiz: Part 1 1. 2. – 3. 0.27 4. –0.625 5. Write as a decimal Simplify. 18 42 3 7 15 21 5 7 1. 2. Write each decimal as a fraction in simplest form. 27 100 – 5 8 3. 0.27 4. –0.625 13 6 5. Write as a decimal 2.16

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