Pre - Algebra Bench Mark 7

Know Rational Numbers Are Either Terminating Or Repeating DecimalsWarm Up

Write as a decimal.

1.

2.

3.

4.

12

75

94

38

(answer: 0.5)

(answer: 1.4)

(answer: 2.25)

(answer: 0.375)

Pre - Algebra Bench Mark 7

Know Rational Numbers Are Either Terminating Or Repeating Decimals

Vocabulary:

rational number

terminating decimal

repeating decimal

period

vinculum

number that can be expressed as a ratio of two integers

number when expressed as a decimal has a finite decimal expansion; also called a regular number

number when expressed as a decimal has a ‘final’ set of digits which repeat an infinite number of times; also called a recurring decimal

the number of recurring digits in a repeating decimal; also called the repetend

a horizontal line placed over the recurring digits in a repeating decimal

Pre - Algebra Bench Mark 7

Know Rational Numbers Are Either Terminating Or Repeating Decimals

It is obvious that terminating decimals are rational numbers. For example,

0.75 = =75100

34

0.625 = =6251000

58

and 0.6 = =610

35

It is much less evident that repeating decimals are also rational numbers.

For example,

So we can see that repeating decimals are also rational numbers.

19

= 0.1111, and = 0.363636411

16

= 0.1666

Pre - Algebra Bench Mark 7

Know Rational Numbers Are Either Terminating Or Repeating Decimals

The real challenge will be to write a fraction in simplest form given a repeating decimal.

To illustrate, lets make up a repeating decimal such as 0.606060.

cancel each other in the subtraction.

We will name our number a = 0.606060

and multiply it by a power of 10,

then subtract the original a and the new numberso that the repeating decimal parts

a = .606060

10a = 6.06060The decimals do not line up so try multiplying by 100.

The decimals line up and the repeating parts will cancel.100a = 60.6060

100a = 60.606060a = 0.606060-

99a = 60 Solve for a and reduce.

a = 6099

2033

= 2033

and multiply it by a power of 10,

So, 0.606060 =

Pre - Algebra Bench Mark 7

Know Rational Numbers Are Either Terminating Or Repeating DecimalsExample

Find the period of the repeating part of .111

Divide. 11 1.0.0

09

- 99

1 0

0

0

00

0

0

- 991

00

0

9

- 991

9

We can see that 2 digits (0 and 9) are repeating so the period is 2.

Pre - Algebra Bench Mark 7

Know Rational Numbers Are Either Terminating Or Repeating DecimalsPractice

a. Convert 0.4444 to a fraction in simplest form.

b. Find the period of the repeating part of .32

33

c. Write as a decimal.

Classify the decimal as repeating or terminating.

1311

d. Convert 1.0555 to a fraction in simplest form.

(answer: )

(answer: 2 )

(answer: 1.1818; repeating)

49

1918

(answer: )

Pre - Algebra Bench Mark 7

Know Rational Numbers Are Either Terminating Or Repeating DecimalsQuiz

1. Write as a decimal.

Classify the decimal as repeating or terminating.

78

2. Convert 1.5555 to a fraction in simplest form.

3. Find the period of the repeating part of .

23

4. Convert 0.98019801 to a fraction in simplest form.

5. Find the period of the repeating part of .

85101

(answer: 0.875; terminating)

(answer: )149

(answer: 1)

(answer: )99101

(answer: 4)