# Math 115

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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.

• Math 115 -

Complete the table shown by converting each fraction to decimal form. The answer to the other questions should then become apparent.

The 1/x fractions that do NOT have terminating decimal forms are
1/3, 1/6, 1/7, 1/9, 1/11, 1/13, 1/14 etc. In all of these repeating cases, the denominator is either a prime number other than 5, or can be factored into a set of numbers that contain a prime number other than 5. My guess is that 5 is a special case because it is 1/2 of ten, the base of the decimal numbering system.

Show us your work and we will be glad to critique it.

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