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Math Help - Which of these fractions is bigger?

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    Which of these fractions is bigger?

    I am studying the book "Algebra" by Israel Gelfand and am having trouble with this problem:

    Which of the fractions \frac{12345}{54321} and \frac{12346}{54322} is bigger?

    I know I could calculate it by hand by finding a common denominator, but I'm fairly certain the book is not interested in rote calculation of this magnitude, so there must be a simpler way. For example, the previous problem was to determine whether \frac{10,001}{10,002} or \frac{100,001}{100,002} was bigger. By noting that both fractions are very close to one, but that the latter is the closest, you can easily determine which is larger. Is there something similar you can do for this one?

    Thanks!
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    Quote Originally Posted by Ragnarok View Post
    I am studying the book "Algebra" by Israel Gelfand and am having trouble with this problem:

    Which of the fractions \frac{12345}{54321} and \frac{12346}{54322} is bigger?

    I know I could calculate it by hand by finding a common denominator, but I'm fairly certain the book is not interested in rote calculation of this magnitude, so there must be a simpler way. For example, the previous problem was to determine whether \frac{10,001}{10,002} or \frac{100,001}{100,002} was bigger. By noting that both fractions are very close to one, but that the latter is the closest, you can easily determine which is larger. Is there something similar you can do for this one?

    Thanks!
    The easiest way to me is to see that if you start with a/b where 0<a<b and you keep replacing a/b with (a+1)/(b+1) you will get closer and closer to 1, therefore the second fraction is bigger.
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    Quote Originally Posted by Ragnarok View Post
    I am studying the book "Algebra" by Israel Gelfand and am having trouble with this problem:

    Which of the fractions \frac{12345}{54321} and \frac{12346}{54322} is bigger?

    I know I could calculate it by hand by finding a common denominator, but I'm fairly certain the book is not interested in rote calculation of this magnitude, so there must be a simpler way. For example, the previous problem was to determine whether \frac{10,001}{10,002} or \frac{100,001}{100,002} was bigger. By noting that both fractions are very close to one, but that the latter is the closest, you can easily determine which is larger. Is there something similar you can do for this one?

    Thanks!
    The fraction will remain the same if values in the same proportion are added to numerator and denominator.

    \frac{x}{y}=\frac{x+kx}{y+ky}=\frac{(k+1)x}{(k+1)y  }

    Example

    \frac{1}{5}=\frac{1+1}{5+5}=\frac{1+0.2}{5+1}

    since the values added are in the same proportion as in the original fraction.

    Hence if the proportions of the additional values are different, the fraction will change

    Example

    \frac{1+0.1}{5+1}<\frac{1}{5},\;\;\;\frac{1+0.3}{5  +1}>\frac{1}{5}

    Hence, for your fraction, the numerator is < the denominator.

    If we add values that are equal, then their proportion is greater than the proportions
    of the original numerator and denominator,
    hence the fraction increases.

    \frac{12345}{54321}<\frac{12345+1}{54321+1}
    Last edited by Archie Meade; September 26th 2010 at 10:45 AM. Reason: small typo
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    Thank you both so much! Cleared it right up for me.
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