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Hmmm recurring numbers

  1. Sep 18, 2004 #1
    Hihi... I am stumped at this question... I know there is a techique in doin this question... But i forget oredi... Help plz...


    Express this recurring number at a fraction of a/b....
    Recurring number ---> 0.1454545454545......

    Plz explain to me the technique used.... No calculators allowed... :tongue2:
     
  2. jcsd
  3. Sep 18, 2004 #2

    Zurtex

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    The general trick is to equalise it x multiply by 10^(the number of digits recuring) then take away the orriginal number.

    Using your problem as an example.

    x = 0.1454545454545...

    2 digits are recurring so multiply it be 100

    100x = 14.54545454545...

    Take away the orriginal number:

    100x - x = 14.54545454545... - 0.1454545454545...

    Take this a digit at a time:

    99x = 14.40000000000...

    99x = 14.4

    That should be a little easier to solve now :smile:
     
  4. Sep 18, 2004 #3
    Stupid me... :rofl:

    Thankx.... lol... So its this easy... :surprised

    Hehe... Thankx for your help mate.... :blushing: :biggrin:
     
  5. Sep 18, 2004 #4

    HallsofIvy

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    Slightly different way:

    Because there is that "1" before the recurring "45", first multiply by 10:

    10x= 1.454545...

    Now multiply that by 100: 1000 x= 145.454545... and subtract

    1000x- 10x= 900x= 144 so x= 144/900

    Of course, that gives exactly the same result.
     
  6. Sep 18, 2004 #5
    Oh, no. Minor snag here. 1000x - 10x = 990x. So the answer really is 144/990. :smile:

    You may simplify 144/990, of course.
     
  7. Sep 18, 2004 #6
    I like the fact that there are many different ways for solving problems. Here's another method. Not that it is really different: it just differs slightly from all the others.

    x = 0.1454545...
    10x = 1.454545...

    10x - 1 = 0.454545...

    We can work out 0.454545... to be 45/99 (100z - z = 45).

    10x - 1 = 45/99
    10x = (99 + 45)/99
    10x = 144/99
    x = 144/990
     
  8. Sep 18, 2004 #7

    mathwonk

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    also can do it by "geometric series", really the same again, but done once for all:

    i.e. .1 + .045454545...

    is .1 Plus the geometric series with initial term a= .045 and ratio r= 1/100, so the sum

    is a + ar + ar^2 +......= a/(1-r) i.e. .045/(99/100) = (4.5)/99, so the answer is

    .1 + this, as before.

    I really do not like this answera s the others's answers are more elementary. but at least it shows how to algebraize their methods.
     
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