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Thread: N positive integer problem with a + exponent

  1. #1
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    N positive integer problem with a + exponent

    Here it is a problem that is driving me nuts...
    Note: ^ means the exponent, in this case the exponent is "+".
    If n is a positive integer, then n^+ denotes a number such that n < n^+ < n+1.

    So decide which of the following options is greater (or if both ofthem are equal or if it is impossible to determine it).

    Option A: 20^+ / 4^+

    Option B: 5^+

    What I did: 20^+ / 4^+ SO, (20/4)^+ , SO (5)^+ , so I concluded that both of them are equal, but the correct answer is " It canīt be determined which option is greater". Could anyone explain me why??
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  2. #2
    Super Member Anonymous1's Avatar
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    We have:

    $\displaystyle 20< 20^+ < 21$

    $\displaystyle 4< 4^+ < 5$

    $\displaystyle 5<5^+<6$

    Clearly we cannot divide these inequalities.

    $\displaystyle \frac{20< 20^+ < 21}{4< 4^+ < 5} = ???????$

    But we can determine bounds.

    How do we obtain the smallest possible value of $\displaystyle \frac{20^+}{4^+}? $ We minimize the numerator and maximize the denominator. The numerator's smallest value is close to $\displaystyle 20,$ and the denominator's largest value is close to $\displaystyle 5.$ So, the fractional value is $\displaystyle \frac{20^+}{4^+}> \frac{20}{5}=4$.

    Now, How do we obtain the largest possible value of $\displaystyle \frac{20^+}{4^+}? $ We maximize the numerator and minimize the denominator. We would end up with $\displaystyle \frac{20^+}{4^+} < \frac{21}{4} = 5.25.$

    So, $\displaystyle 4<\frac{20^+}{4^+}<5.25,$ Which could be greater or less than $\displaystyle 5 <5^+<6$ depending on which value $\displaystyle 5^+$ were to take on.
    Last edited by Anonymous1; Mar 24th 2010 at 10:37 AM.
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  3. #3
    Super Member Deadstar's Avatar
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    Quote Originally Posted by artabro View Post
    Here it is a problem that is driving me nuts...
    Note: ^ means the exponent, in this case the exponent is "+".
    If n is a positive integer, then n^+ denotes a number such that n < n^+ < n+1.

    So decide which of the following options is greater (or if both of them are equal or if it is impossible to determine it).

    Option A: 20^+ / 4^+

    Option B: 5^+

    What I did: 20^+ / 4^+ SO, (20/4)^+ , SO (5)^+ , so I concluded that both of them are equal, but the correct answer is " It canīt be determined which option is greater". Could anyone explain me why??
    Okay, here's how I'm looking at it.
    The + sign means you add on some fractional part that is greater than 0 but less than 1 onto n.

    The problem I'm seeing with the question is that, if we take two numbers, say 3 and 4,

    Does $\displaystyle 3^+$ and $\displaystyle 4^+$ mean that the SAME fractional part gets added on to both 3 and 4? Or does it mean some random fractional part gets added on?

    If its the first option... Than option B is greater since for option A you have...
    $\displaystyle \frac{21}{5} < \frac{20^+}{4^+} < \frac{20}{4} = 5$

    while for option B you have
    $\displaystyle 5 < 5^+ < 6$

    Clearly option B is larger.

    If the + sign means you add on a random fractional part as I think it probably is...

    Then you cant conclude that either is bigger.

    As an example...
    Take $\displaystyle 20^+ = 20.99$, $\displaystyle 4^+ = 4.01$ and $\displaystyle 5^+$ to be 5.01

    Than for option A you get
    roughly 5.23
    and for B you get
    5.01
    Hence A is bigger

    Now take $\displaystyle 20^+ = 20.01$, $\displaystyle 4^+ = 4.99$ and $\displaystyle 5^+$ to be 5.99

    Then for option A you get
    4.01
    and for B you get
    5.99
    So B is bigger.
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  4. #4
    Super Member Anonymous1's Avatar
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    Actually, this question reminds me of techniques used in elementary $\displaystyle Real$ $\displaystyle Analysis.$
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