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
Fraction Decimal Form Terminate? Prime factorization of the Denominator
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
8
1
7
1
9
1
10
1
11
1
12
State a general rule describing which fractions have decimal forms that terminate and which have decimal forms that repeat.
To find the decimal form of the fractions, we need to divide the numerator by the denominator. If the division process terminates and there is no remainder, then the decimal form of the fraction is terminating. If there is a remainder, the division process repeats indefinitely, and the decimal form of the fraction is repeating.
Now, let's determine the decimal form and termination for each fraction in the table, while also finding the prime factorization of the denominator:
Fraction | Decimal Form | Termination? | Prime Factorization of Denominator
1/2 | 0.5 | Terminates | 2^1
1/3 | 0.333... | Repeats | 3^1
1/4 | 0.25 | Terminates | 2^2
1/5 | 0.2 | Terminates | 5^1
1/6 | 0.166... | Repeats | 2^1 * 3^1
1/7 | 0.142857... | Repeats | 7^1
1/8 | 0.125 | Terminates | 2^3
1/9 | 0.111... | Repeats | 3^2
1/10 | 0.1 | Terminates | 2^1 * 5^1
1/11 | 0.090909... | Repeats | 11^1
1/12 | 0.083333... | Repeats | 2^2 * 3^1
Based on the results, we can observe that fractions with denominators that have prime factorizations consisting only of 2's and/or 5's (e.g., 2^a * 5^b) will have terminating decimal forms.
On the other hand, fractions with denominators that have prime factorizations including primes other than 2 or 5 will have repeating decimal forms.
In summary, the general rule is that fractions with denominators that contain only powers of 2 and/or 5 will have terminating decimal forms, while those with other prime factors in the denominator will have repeating decimal forms.