Every fraction has a decimal equivalent that either terminates (for example,1/4=0.25 ) or repeats (for example,2/9=0.2 ). 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.

1/2
1/3
1/4
1/5
1/6
1/7
1/8
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.
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.

1/2=0.5

1/3=0.333333333333333
1/4=0.25
1/5=0.2
1/6=0.166666666666667
1/7=0.142857142857143
1/8=0.125
1/9=0.111111111111111
1/10=0.1
1/11=0.090909090909091
1/12=0.083333333333333

1/2=0.5

1/3=0.333333333333333
1/4=0.25
1/5=0.2
1/6=0.166666666666667
1/7=0.142857142857143
1/8=0.125
1/9=0.111111111111111
1/10=0.1
1/11=0.090909090909091
1/12=0.083333333333333
fractions with denominators of 3,6,7,9,11,12 (or multiples of these numbers) have decimal forms that repeat, while fractions with denominators of 2,4,5,8,10 (or multiples of these numbers)have decimal forms that terminate.

i predict that 1/25 has a terminating decimal
reason: 25 is a multiple of 5, which has a terminating decimal
1/25=.04 correct
Any power of 10 only has 2 and 5 as prime factors. So if the reduced fraction has a denominator with factors other than 2 or 5, those factors will always be there, and there will be no way to get the fraction expressed with a power of 10 for the denominator. So in conclusion: if the reduced form of the fraction has only 2 and 5 for prime factors in its denominator, then the decimal terminates.

To determine which fractions have decimal forms that terminate and which have decimal forms that repeat, we can examine the prime factorization of the fraction's denominator.

Let's complete the table by analyzing each fraction and its denominator:

1/2:
The denominator is 2, which has a prime factorization of 2^1. Since the only prime factor is 2 and not a combination of 2 and 5, the decimal form of 1/2 terminates.

1/3:
The denominator is 3, which has a prime factorization of 3^1. As there are no prime factors other than 2 and 5, the decimal form of 1/3 repeats.

1/4:
The denominator is 4, which can be written as 2^2. Since the prime factorization contains only powers of 2, the decimal form of 1/4 terminates.

1/5:
The denominator is 5, which has a prime factorization of 5^1. As there are no prime factors other than 2 and 5, the decimal form of 1/5 terminates.

1/6:
The denominator is 6, which can be written as 2^1 * 3^1. Since the prime factorization includes both 2 and 3, the decimal form of 1/6 repeats.

1/7:
The denominator is 7, which has a prime factorization of 7^1. As there are no prime factors other than 2 and 5, the decimal form of 1/7 repeats.

1/8:
The denominator is 8, which can be written as 2^3. Since the prime factorization contains only powers of 2, the decimal form of 1/8 terminates.

1/9:
The denominator is 9, which can be written as 3^2. Since the prime factorization includes only powers of 3 but not 2 or 5, the decimal form of 1/9 repeats.

1/10:
The denominator is 10, which can be written as 2^1 * 5^1. Since the prime factorization includes both 2 and 5, the decimal form of 1/10 terminates.

1/11:
The denominator is 11, which has a prime factorization of 11^1. As there are no prime factors other than 2 and 5, the decimal form of 1/11 repeats.

1/12:
The denominator is 12, which can be written as 2^2 * 3^1. Since the prime factorization includes a combination of 2 and 3, the decimal form of 1/12 repeats.

From the analysis above, we can state a general rule:
A fraction expressed as a simplified ratio of two integers will have a decimal form that terminates if and only if the prime factorization of its denominator consists only of powers of 2 and/or powers of 5.

Now, let's apply this rule to three new fractions as requested:

1/15:
The denominator is 15, which can be written as 3^1 * 5^1. Since the prime factorization includes both 3 and 5, the decimal form of 1/15 repeats. (Prediction: Repeating)

1/20:
The denominator is 20, which can be written as 2^2 * 5^1. Since the prime factorization includes both 2 and 5, the decimal form of 1/20 terminates. (Prediction: Terminating)

1/25:
The denominator is 25, which has a prime factorization of 5^2. Since there are no prime factors other than 2 and 5, the decimal form of 1/25 terminates. (Prediction: Terminating)

Now, let's confirm these predictions:

1/15 = 0.0666666... (Repeating)
1/20 = 0.05 (Terminating)
1/25 = 0.04 (Terminating)

The predictions made using the rule were accurate in classifying the decimal forms as either terminating or repeating.

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