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Advanced Arithmetic

  1. Jan 30, 2016 #1
    1. The problem statement, all variables and given/known data
    What is the minimum number of digits to the right of the decimal point needed to express the fraction abee710ba93bd9619bd3e0d2821d027f0278f31c.png as a decimal?
    a) 4
    b) 22
    c) 26
    d) 30
    e) 104

    2. Relevant equations


    3. The attempt at a solution
    One possible solution is: "We can rewrite the fraction as b6d710910e5aafbc58b027e85340bedc12d33e5e.png . Since the last digit of the numerator is odd, a 79069377f91364c2f87a64e5f9f562a091c8a6c1.png is added to the right if the numerator is divided by [PLAIN]https://latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png, [Broken] and this will continuously happen because [PLAIN]https://latex.artofproblemsolving.com/7/9/0/79069377f91364c2f87a64e5f9f562a091c8a6c1.png, [Broken] itself, is odd. Indeed, this happens twenty-two times since we divide by https://latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png twenty-two times, so we will need 1d111a5e00ee5ea6700bb629994dc629874c505b.png more digits. Hence, the answer is [PLAIN]https://latex.artofproblemsolving.com/8/9/7/897cab64b41a26c4bf59f579a975ec600cf2441b.png." [Broken]

    So I understand how they rewrote the fraction, however I'm totally lost by everything after that. I'm not exactly sure why a 5 is added to the right if the numerator is divided by 2 and how that has anything to do with the last digit being odd. Also, a calculator is not allowed on this. If someone could either clarify the explanation given or give a more simplified alternate solution, that would be appreciated.
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Jan 30, 2016 #2

    ehild

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    I think it is a bit more clear if you do the division by 104 at the end.

    123456789=123456788 + 1. Divide it by 2 : it becomes an integer + 0.5 that means one decimal.
    Divide that number by 2 gain, you get an integer, + 0.25, so you have two decimals. Do again: it is something + 0.125. Repeat: the number of decimals increases by one at every division by 2. The first term becomes odd after some divisions, but that will have less decimals than the division of 1.
    So how many decimals are there if you divide by 222 (that is, you do 22 divisions) ?
     
  4. Jan 30, 2016 #3
    Okay I'm kind of making sense out of it. If you divide 1 by 222 then you will have 22 decimals, and then if you divide 123456788 by 104 then you will have 4 decimals, and 22 + 4 = 26, I don't know though something about that process seems off, I feel like you can't just separate them like that, can you?
     
  5. Jan 30, 2016 #4

    ehild

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    Is 123456789 = 123456788 +1?
    Is (a+b)/2 = a/2 + b/2 true?
    try with a smaller number : How many decimals are there in 9/23?
     
  6. Jan 30, 2016 #5
    Yes, but I would have 123456788/(222 * 104) + 1/(222 * 104)
    and if I ONLY divided 123456788 by 104, then where did the 222 in the denominator go
    and if I ONLY divided 1 by 222, then where did the 104 in the denominator go, do you see what I mean?
     
  7. Jan 30, 2016 #6

    ehild

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    ##\frac {A}{b c}= \frac {A/b}{c} ##. If you have a product of two numbers in the denominator, you can divide by one number, then you divide the result by the other.
    Here, you divide by 222 first, then you divide the result by 104.
    If you evaluate ##\frac{36}{3*4}## you can do it by dividing 36 with 12 or first dividing 36 by 3 (you get 12) then dividing 12 by 4 - the result is 3 with both methods.
     
  8. Jan 30, 2016 #7

    ehild

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    Do you understand it now?
     
  9. Jan 30, 2016 #8
    Yes for the most part, thank you!
     
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