ACTIVITY 12: TERMINATE

OR REPEAT?
Every fraction has a decimal equivalent that either terminates (for example, ) or
repeats (for example, ). Work with a group to discover which fractions have terminating
decimals and which have repeating decimals. You may assume that the numerator
of each fraction you consider is and focus your attention on the denominator. As you complete
the table below, you will find that the key to this question lies with the prime factorization
of the denominator.
2
9
� 0.2
1
4
� 0.25
Prime Factorization
Fraction Decimal Form Terminate? of the Denominator
1
12
1
11
1
10
1
9
1
8
1
7
1
6
1
5
1
4
1
3
1
2
State a general rule describing which fractions have decimal forms that terminate and
which have decimal forms that repeat.
Now test your rule on at least three new fractions. That is, be able to predict whether a
fraction such as or has a terminating decimal or a repeating decimal. Then confirm
your prediction.
1
30
1
25
360

To determine whether a fraction will have a decimal form that terminates or repeats, you need to consider the prime factorization of the denominator.

First, let's examine the fractions in the table and their decimal forms:

1/2 = 0.5 (terminates)
1/3 = 0.33333... (repeats)
1/4 = 0.25 (terminates)
1/5 = 0.2 (terminates)
1/6 = 0.16666... (repeats)
1/7 = 0.142857142857... (repeats)
1/8 = 0.125 (terminates)
1/9 = 0.11111... (repeats)
1/10 = 0.1 (terminates)
1/11 = 0.090909... (repeats)
1/12 = 0.08333... (repeats)

Looking at the prime factorizations of the denominators, we can observe the following pattern:

1/2 = 2^(-1) (terminates)
1/3 = 3^(-1) (repeats)
1/4 = 2^(-2) (terminates)
1/5 = 5^(-1) (terminates)
1/6 = 2^(-1) * 3^(-1) (repeats)
1/7 = 7^(-1) (repeats)
1/8 = 2^(-3) (terminates)
1/9 = 3^(-2) (repeats)
1/10 = 2^(-1) * 5^(-1) (terminates)
1/11 = 11^(-1) (repeats)
1/12 = 2^(-2) * 3^(-1) (repeats)

From this pattern, we can observe the following general rule:

A fraction will have a decimal form that terminates if the prime factorization of the denominator only contains powers of 2 and/or powers of 5.
A fraction will have a decimal form that repeats if the prime factorization of the denominator contains any prime factors other than 2 and 5.

Let's test this rule on three new fractions and confirm the predictions:

1/30:
The prime factorization of 30 is 2 * 3 * 5. Since it contains the prime factor 3, the decimal form will repeat. Let's confirm:
1/30 = 0.033333... (repeats)

1/25:
The prime factorization of 25 is 5^2. It only contains powers of 5, so the decimal form will terminate. Let's confirm:
1/25 = 0.04 (terminates)

1/360:
The prime factorization of 360 is 2^3 * 3^2 * 5. Since it contains the prime factors 3 and 2, the decimal form will repeat. Let's confirm:
1/360 = 0.0027777... (repeats)

By using the prime factorization of the denominator, we can accurately predict whether a fraction will have a terminating or repeating decimal form.