Activity 1: FINDING DECIMAL EQUIVALENTS In the course of converting fractions to their decimal equivalents, you should have observed the following: Some fractions convert to terminating decimals; others, to repeating decimals. In some fractions that convert to a repeating decimal, the digits begin repeating immediately; in others, one or more digits appear between the decimal point and the repetend. In this activity, you will investigate the answers to the following questions: Is there a way to predict whether the decimal equivalent of a given fraction terminates or repeats? If the decimal equivalent of a fraction repeats, how many digits will appear between the decimal point and the repetend? In this activity, the term fraction refers to a fraction expressed in simplest form. 1. Complete the table on the following page. Then use the data in the table to answer the following questions. If necessary, check fractions other than those in the table. a. Which prime numbers appear in the prime factorization of the denominators of those fractions whose decimal equivalents terminate?

2.

b.

Does the prime factorization of the denominator of any of the fractions whose decimal equivalents repeat contain only the primes in part a? If so, which fractions?

c.

How can you predict whether the decimal equivalent of a given fraction terminates or repeats?

3.

a.

For the fractions whose decimal equivalents terminate, how is the number of digits in the decimal equivalent of the fraction related to the exponents in the prime factorization of the denominator?

b.

Does the relationship in part a hold for the number of digits between the decimal point and the repetend in those cases where the decimal equivalent of the fraction repeats?

c.

How can you predict the number of digits between the decimal point and the repetend in the decimal equivalent of a fraction?

MATH 3443

Modeling: Real Numbers and Statistics

Page 43

Fraction

Decimal Equivalent

Repeating or Terminating

Prime Factorization of Denominator 31 (or 3) 21 x 32 x 51

in Repetend

No. of Digits between decimal & Repetend 0 1

2

/3

0. 6

Repeating Repeating

1 1

after decimal if terminating -

29

/90 /99

17

19

/3500 /200 /6 /8 Terminating 3

17

1

1

7

/15

3

/125 /140 /100 /400 /5000 /7 0.03 Terminating 2 -

1

3

17

119

1

99

/260 /35 /20 /24 /600

2

7

5

121

Page 44

Modeling: Real Numbers and Statistics

MATH 3443

Activity 2: DECIMAL PATTERNS 1. Use a calculator to convert each of the following fractions to its decimal equivalent. If your calculator rounds off answers be careful not to be fooled by the result it displays. 1 3 6

a.

/9 = /9 =

b.

/9 = /9 =

c.

/9 = /9 =

d. 2.

4

e.

8

f.

5

Using the results in problem 1, answer the following questions: a. b. c. How many digits appear in the repetend of the decimal equivalent of each fraction? How many 9’s are in the denominator of each fraction? What is the relationship between the numerator of each fraction and its decimal equivalent?

3.

Using only the answers in the problem 2, predict the decimal equivalent of the following fractions. a. 2

/9 =

b.

7

/9 =

c.

9

/9 =

4.

Convert 1/3 and 2/3 to their decimal equivalents. What is the sum of your answers? What does 1/3 + 2/3 equal? What do you conclude from your answers to problems 3c, 4, and 5?

5.

6.

7.

Using the result from problem 6, find the fraction equivalent to each of the following decimals: (HINT: 0.09 = 0.1 x 0. 9 )

a.

0.09 =

b.

0.009 =

c.

0.0009 =

c.

0.49 =

e.

0129 = .

f

05769 = .

MATH 3443

Modeling: Real Numbers and Statistics

Page 45

8.

Use a calculator to convert each of the following fractions to its decimal equivalent. 13 16

a.

/99 = /99 =

b.

/99 =

c. 5/99 = f.

36

d....