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Actually ALL fractions are either terminating, or they are equivalent to repeating decimals.Try to carry out the long division, by hand, using ANY fraction; for example, 1/7. At each step, there will be a remainder. If this remainder happens to be zero, the division stops (the decimal is terminating).

However, if it doesn't stop, there can only be (in this example, when dividing by 7), six different options for the remainder; therefore, sooner or later, you MUST get a remainder that you already had before; therefore, the pattern repeats.

Note: The fractions (in simplest terms) that are equivalent to terminating decimals are exactly those which have a denominator whose only prime factors are 2 and 5. This is because those are the prime factors that make up the number 10 - the base of our decimal number system.

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7y ago

If a fraction in its simplest form has a denominator which contains any prime factor other than 2 or 5 then it will be equivalent to a repeating decimal. For example, 2/3, 5/21, 6/85.

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Q: Why are some fractions equivalent to repeating decimals?
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Can fractions always be written as decimals?

You can always convert a fraction to a decimal. For some fractions, you'll get terminating decimals. For example, 1/8 = 0.125. For other fractions, you get repeating decimals, such as 1/7 = 0.142857 142857 142857...To convert the fraction to a decimal, just divide the numerator by the denominator, for example on a calculator.


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Some decimals are non-repeating numbers, and some aren't.


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In a non-zero fraction is written in its simplest form, then if the denominator has any prime factor other than 2 or 5 it will not go into any numerator evenly. This leads to repeating decimals.