Rational Numbers

3-1

Divide.

12 24

34

16

1. 36 3 2. 144 6

3. 68 17 4. 345 115

5. 1024 64

Warm Up

Learn to write rational numbers in equivalent forms.

rational number

relatively prime

Vocabulary

A rational number is any number that can be written as a fraction , where n and d are integers and d 0.

nd

Decimals that terminate or repeat are rational numbers.

Numerator nd 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.

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 of the 15 boxes are shaded.

4 of the 5 boxes are

shaded.

The same total area is shaded.

1215

45

=1215

45

5 10

5 = 1 • 5 10 = 2 • 5

;5 is a common factor.

Divide the numerator and denominator by 5.

1 2

=

510

Simplify.

= 5 ÷ 5 10 ÷ 5

A.

Example: Simplifying Fractions

16

80

16 = 1 • 16 80 = 5 • 16

;16 is a common factor.

1 5

=

1680

Divide the numerator and denominator by 16.= 16 ÷

16 80 ÷ 16

B.

Simplify.

Example: Simplifying Fractions

= –18 29

–18 29

18 = 2 • 9 29 = 1 • 29

;There are no common factors.

–18 and 29 are relatively prime.–18 29

C.

Simplify.

Example: Simplifying Fractions

6 30

6 = 1 • 630 = 5 • 6

;6 is a common factor.

Divide the numerator and denominator by 6.

1 5

=

630 = 6 ÷ 6

30 ÷ 6

A.

Simplify.

Try This

18

27

;9 is a common factor.

2 3

=

1827 =18 ÷ 9

27 ÷ 9

B.

Divide the numerator and denominator by 9.

Simplify.

18 = 3 • 3 • 227 = 3 • 3 • 3

Try This

= – 17 35

17 –35

17 = 1 • 17 35 = 5 • 7

;There are no common factors.

17 and –35 are relatively prime.

17 –35

C.

Simplify.

Try This

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.

–0.8

A. –0.8 –8 is in the tenths place.

Simplify by dividing by the common factor 2.

–8 10

=

= – 45

Write the decimal as a fraction in simplest form.

Example: Writing Decimals as Fractions

5.37

B. 5.37 7 is in the hundredths place.

37 100

= 5

Write the decimal as a fraction in simplest form.

Example: Writing Decimals as Fractions

0.622

C. 0.622 2 is in the thousandths place.

622 1000

=

= 311 500

Simplify by dividing by the common factor 2.

Write the decimal as a fraction in simplest form.

Example: Writing Decimals as Fractions

–0.4

A. –0.4 –4 is in the tenths place.

Simplify by dividing by the common factor 2.

–4 10

=

Write the decimal as a fraction in simplest form.

= – 25

Try This

8.75

B. 8.75 5 is in the hundredths place.

75 100

= 8

= 8 3 4

Simplify by dividing by the common factor 25.

Write the decimal as a fraction in simplest form.

Try This

0.2625

0.2625 5 is in the ten-thousandths place.

2625 10,000

=

= 21 80

Simplify by dividing by the common factor 125.

C.

Write each decimal as a fraction in simplest form.

Try This

denominator numerator

To write a fraction as a decimal, divide the numerator by the denominator. You can use long division.

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.

numeratordenominator

9 11 The pattern repeats, so draw a bar over the 2 to indicate that this is a repeating decimal.

1

–9

.2

2

0

.0

2

11 9

–1 8

A.

Write the fraction as a decimal.

The fraction is equivalent to the decimal 1.2.11 9

Example: Writing Fractions as Decimals

This is a terminating decimal.20 7

.30 5

The remainder is 0.

7 20

–07

1 0

0

0

0

.0

0–6 0

–1 0 0

B.

Write the fraction as a decimal.

The fraction is equivalent to the decimal 0.35.7 20

Example: Writing Fractions as Decimals

9 15 The pattern repeats, so draw a bar over the 6 to indicate that this is a repeating decimal.

1

–9

.6

6

0

.0

6

15 9

–5 4

Write the fraction as a decimal.

A.

The fraction is equivalent to the decimal 1.6.15 9

Try This

40 9 This is a terminating decimal..20 2

The remainder is 0.

9 40

–09

1 0

0

0

.0

0–8 0

– 8 02 0

0

0

5

0– 2 00

B.

Write the fraction as a decimal.

The fraction is equivalent to the decimal 0.225.9 40

Try This

Simplify.

1. 2.

Write each decimal as a fraction in simplest form.

3. 0.27 4. –0.625

5. Write as a decimal 2.16

18 42

3 7

15 21

5 7

27 100

– 5 8

13 6

Lesson Quiz: Part 1

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.)

6.

0.325

Lesson Quiz: Part 2