Math 115

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.

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  1. You posted this questin twice.

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  2. 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

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  3. 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.

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