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Math Help - Terminating fractions

  1. #1
    Senior Member
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    Terminating fractions

    I need some help with this please.

    Changing fractions into decimals.
    Here my work:

    4 5 = 0.8
    0.8 This decimal TERMINATES

    2 3 = 0.6666.....
    0.6666...... This decimal RECURS

    4 27 = 0.148148148148
    0.148148148.... Its pattern REPEATS.

    I'm stuck with these questions (below). Any help will be much much appreciated thanks!

    Which fractions will terminate and which will recur or repeat.
    Are there any other possibilities than terminate, recur or repeat?
    Can you explain WHY some groups of fractions terminate, recur or repeat?
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  2. #2
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by Natasha1 View Post
    I need some help with this please.

    Changing fractions into decimals.
    Here my work:

    4 5 = 0.8
    0.8 This decimal TERMINATES

    2 3 = 0.6666.....
    0.6666...... This decimal RECURS

    4 27 = 0.148148148148
    0.148148148.... Its pattern REPEATS.

    I'm stuck with these questions (below). Any help will be much much appreciated thanks!

    Which fractions will terminate and which will recur or repeat.
    Are there any other possibilities than terminate, recur or repeat?
    Can you explain WHY some groups of fractions terminate, recur or repeat?
    This is an interesting question. I plan to consider it more in depth, but here's the simple answer (that I think is true in all situations)

    First, the numbers being divided have to be integers (if they are decimals, then these rules might not work).

    Fractions that result in terminating decimals involve numbers being divided by: 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100... (the numbers go on). The more simple answer would be that the divisor MUST BE either 1, 2, 5 OR some multiple of only these three terms OR the fraction can be reduced such that the divisor follows one of the first two rules. (If the divisor is a multiple of any number other than these, such as 3, 7, 11, then the decimal is not terminating.)

    Fractions that result in a reacuring decimal (such as .3333... or .8888...) occur only when the divisor is 3 or 9, or the fraction can be reduced in such a way that the divsor will become 3 or 9. No other divisor (that I can think of) will result in a reacuring decimal.

    By a process of elimination, any fraction that does not follow either of the first two rules contains a decimal that is repeating. Examples would be a number divided by 12, 23, 98, etc.
    Last edited by ecMathGeek; March 25th 2007 at 07:02 PM. Reason: I didn't write in some of the terms that work...
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  3. #3
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    Hello, Natasha!

    This is tricky stuff if you've never seen it before.


    Which fractions will terminate and which will recur or repeat?

    If the denominator is of the form: (2^m)(5^n), the decimal will terminate,
    . . where m and n are nonnegative integers: 0, 1, 2, 3, ...

    If the denominator is 9, the decimal recurs.

    All other fractions will repeat.



    Are there any other possibilities than terminate, recur or repeat?

    Irrational numbers have decimals that are nonterminating and nonrepeating.
    . . . . . . . . . . ._
    For example, √2 .= .1.414213562... goes on and on and never repeats.



    Maybe someone else can answer the "why" of question 3.
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