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Math Help - Find largest square number - without using calculator

  1. #1
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    Find largest square number - without using calculator

    Hey All,

    This problem must be done without the use of a calculator. I am able to guesstimate my way to a close enough answer but need some help to improve this.

    If n is the largest square number such that s \leq n, find s when

    (i) n = 6.4 \times 10^{3}

    (ii) n = 6.4 \times 10^{6}

    I got (i) s = 6400 easily since its a perfect square,

    For (ii) I wrote the number as  64 \times 10^{4} \times 10

    Then nearest perfect square close to 10 is 9, so s \approx 8^{2} \times 100^{2} \times 3^{2}

    From there I figured that the number is between 2400^{2} and further got to s = 2500^{2} = 6250000

    The problem is that the required answer is 6395841 and the weight-age is only 1 mark. Using the calculator suggests this is the exactly correct answer.

    But since this is a non-calculator use problem, leads me to believe I am missing an obvious/simpler/faster way of getting there. What am I missing?

    Thanks again for all your help.
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  2. #2
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    Quote Originally Posted by mathguy80 View Post
    Hey All,

    This problem must be done without the use of a calculator. I am able to guesstimate my way to a close enough answer but need some help to improve this.

    If n <-- typo? Shouldn't that be s? is the largest square number such that s \leq n, find s when

    (i) n = 6.4 \times 10^{3}

    (ii) n = 6.4 \times 10^{6}

    I got (i) s = 6400 easily since its a perfect square,

    For (ii) I wrote the number as  64 \times 10^{4} \times 10

    Then nearest perfect square close to 10 is 9, so s \approx 8^{2} \times 100^{2} \times 3^{2}

    From there I figured that the number is between 2400^{2} and further got to s = 2500^{2} = 6250000

    The problem is that the required answer is 6395841 and the weight-age is only 1 mark. Using the calculator suggests this is the exactly correct answer.

    But since this is a non-calculator use problem, leads me to believe I am missing an obvious/simpler/faster way of getting there. What am I missing?

    Thanks again for all your help.
    I'm only guessing:

    6.4 \cdot 10^6 = 640 \cdot 100^2

    So 25 < a < 26 (see attachment)

    Use linear interpolation. According to my sketch a \approx 25 + \frac{15}{51}

    Since you multiply the apprximate value of a by 100 to get s you only have to calculate the first two digits of \frac{15}{51} = 0.29

    Thus a = 25.29~\implies~s=2529
    Attached Thumbnails Attached Thumbnails Find largest square number - without using calculator-quadratschaetzen.png  
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  3. #3
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    Awesome @earboth! A much clearer and very elegant solution. Learning something new everyday here. Thanks again!
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