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Math Help - Letters used in place of numbers

  1. #1
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    Letters used in place of numbers

    Letters can replace numbers in simple mathematical problems. If E and F represent digits and EF + F = FE, then F equals

    A. 3
    B. 6
    C. 8
    D. 9

    How best can I resolve this problem such that it will be clear to grade six pupils.

    Thanks
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  2. #2
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    Re: Letters used in place of numbers

    Quote Originally Posted by KayPee View Post
    Letters can replace numbers in simple mathematical problems. If E and F represent digits and EF + F = FE, then F equals
    A. 3 B. 6 C. 8 D. 9
    How best can I resolve this problem such that it will be clear to grade six pupils.
    I have no business commenting on six graders.
    But here goes: (10E+F)+F=10F+E~. so 9E-8F=0 is it clear now?
    Thanks from KayPee
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    Re: Letters used in place of numbers

    Edit: Plato beat me to it (plus I was wrong )
    Last edited by TwoPlusTwo; October 1st 2012 at 05:28 AM.
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    Re: Letters used in place of numbers

    Quote Originally Posted by Plato View Post
    I have no business commenting on six graders.
    But here goes: (10E+F)+F=10F+E~. so 9E-8F=0 is it clear now?
    I followed your approach and my last line was 9E = 8F

    why did you equate it to zero i.e. 9E - 8F = 0?
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    Re: Letters used in place of numbers

    Quote Originally Posted by KayPee View Post
    I followed your approach and my last line was 9E = 8F

    why did you equate it to zero i.e. 9E - 8F = 0?
    For all numbers x and y, x = y if and only if x - y = 0.
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    Re: Letters used in place of numbers

    Quote Originally Posted by emakarov View Post
    For all numbers x and y, x = y if and only if x - y = 0.
    Thanks I got i.e. 9E-8F = 0.

    So from that point how's the equation: 9E - 8F = 0 resolved to find F and possibly E as well and then CHECK whether these values make the equation true?
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    Re: Letters used in place of numbers

    Quote Originally Posted by emakarov View Post
    For all numbers x and y, x = y if and only if x - y = 0.
    Thanks I got i.e. 9E-8F = 0.

    So from that point how's the equation: 9E - 8F = 0 resolved to find F and possibly E as well and then CHECK whether these values make the equation true?
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    Re: Letters used in place of numbers

    Quote Originally Posted by KayPee View Post
    Thanks I got i.e. 9E-8F = 0.
    So from that point how's the equation: 9E - 8F = 0 resolved to find F and possibly E as well and then CHECK whether these values make the equation true?
    Each of E~\&~F belongs to the set \{0,1,2,3,4,5,6,7,8,9\}.
    Use simple inspection to determine the solution.
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  9. #9
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    Re: Letters used in place of numbers

    Quote Originally Posted by Plato View Post
    Each of E~\&~F belongs to the set \{0,1,2,3,4,5,6,7,8,9\}.
    Use simple inspection to determine the solution.
    So from simple inspection E= 8 and F=9.

    I think I clear now.Thanks

    I hope I can post here if I have any further questions related to the above question.
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    Re: Letters used in place of numbers

    Quote Originally Posted by Plato View Post
    Each of E~\&~F belongs to the set \{0,1,2,3,4,5,6,7,8,9\}.
    Use simple inspection to determine the solution.
    Hi Plato

    I've got another issue:

    I believe the range of possible answers:{0,1,2,3,4,5,6,7,8,9} were selected based on the equation: 9E = 8F.

    So if the equation had been something like

    1.9R = 9S
    As such R and S=1

    The possible range must be: {0,1,2,3,4,5,6,7,8,9}

    2.For 10x = 8y

    The range of possible answers will be: {0,1,2,3,4,5,6,7,8,9,10}


    I hope I'm right?
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    Re: Letters used in place of numbers

    Quote Originally Posted by KayPee View Post
    2.For 10x = 8y
    The range of possible answers will be: {0,1,2,3,4,5,6,7,8,9,10}
    I hope I'm right?
    10x = 8y has no solution.
    x~\&~y are digits. No three digit number equals a two digit number.
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    Re: Letters used in place of numbers

    Quote Originally Posted by KayPee View Post
    I believe the range of possible answers:{0,1,2,3,4,5,6,7,8,9} were selected based on the equation: 9E = 8F.
    The range of E and F was not selected based on the equation 9E = 8F. The problem statement says, "E and F represent digits." Therefore, 0\le E,F\le 9. In fact, since the problem statement uses two-digit numbers FE and EF, neither E nor F can be 0. Further, 9E = 8F implies that 8F is divisible by 9. But since the GCD(8, 9) = 1, (a generalization of) Euclid's lemma says that F must be divisible by 9. Since F is an integer between 1 and 9, F must be 9. It is also possible to go through all possible variants for F and see for which one 8F is divisible by 9.
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