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Math Help - Writing a Fraction in Simplest Form

  1. #16
    Junior Member Euclid Alexandria's Avatar
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    Lightbulb

    After a week I thought I'd give this another try. I'm using Soroban's formula to reproduce the mathematical joke,

    \frac{1\!\!\!\not{6}}{\not{6}4} = \frac{1}{4}

    While incorrectly solved, it still gives the correct answer. Any hints are appreciated on where I'm going wrong.

    Soroban's formula:

    We want digits a, b, c so that: \frac{10a + b}{10b + c} \:= \:\frac{a}{c}

    Solve for c:\;\;c\:=\:\frac{10ab}{9a + b}

    Step 1

    \frac{10a + b}{10b + c} = \frac{a}{c} = \frac{a}{c}

    = \frac{10 \cdot 1 + 6}{10 \cdot 4 + 4} = \frac{16}{44}

    Now, \frac{a = 16}{c = 44}

    Step 2

    c = \frac{10ab}{9a + b} = \frac{10 \cdot 16 \cdot 16}{9 \cdot 16 + 16} = \frac{960}{144}

    = \frac{\not2 \cdot \not2 \cdot \not2 \cdot \not2 \cdot 2 \cdot 2 \cdot \not3 \cdot 5}{\not2 \cdot \not2 \cdot \not2 \cdot \not2 \cdot \not3 \cdot 3} = \frac{20}{3}
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  2. #17
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by Euclid Alexandria
    After a week I thought I'd give this another try. I'm using Soroban's formula to reproduce the mathematical joke,

    \frac{1\!\!\!\not{6}}{\not{6}4} = \frac{1}{4}

    While incorrectly solved, it still gives the correct answer. Any hints are appreciated on where I'm going wrong.

    Soroban's formula:

    We want digits a, b, c so that: \frac{10a + b}{10b + c} \:= \:\frac{a}{c}

    Solve for c:\;\;c\:=\:\frac{10ab}{9a + b}

    Step 1

    \frac{10a + b}{10b + c} = \frac{a}{c} = \frac{a}{c}
    Error #1
    = \frac{10 \cdot 1 + 6}{10 \cdot 4 + 4} = \frac{16}{44}
    you say b=6 in the numerator, but you change that to b=4 in the denominator!

    Now, \frac{a = 16}{c = 44}

    Step 2

    c = \frac{10ab}{9a + b} = \frac{10 \cdot 16 \cdot 16}{9 \cdot 16 + 16} = \frac{960}{144}

    = \frac{\not2 \cdot \not2 \cdot \not2 \cdot \not2 \cdot 2 \cdot 2 \cdot \not3 \cdot 5}{\not2 \cdot \not2 \cdot \not2 \cdot \not2 \cdot \not3 \cdot 3} = \frac{20}{3}
    This is not incorrect, however, to produce the mathematical joke you can't have a b or c more than a single-digit number.
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  3. #18
    Junior Member Euclid Alexandria's Avatar
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    Cool The punchline rears its worn out head

    Thanks, Quick. Your tip about the single digit number for c sent me in the right direction. Actually, I see numerous errors now. I also substituted 16 for b in my step 2. And steps 1 and 2 are themselves erroneous. I repeatedly followed Soroban's steps in the order that he presented them in, rather than the order that was intended to be followed.

    To solve for c we use

    \frac{10ab}{9a+b}

    where a and b are any single digits we want to test. We want the result of the test to be a whole number, rather than a fraction. For instance, going through various combinations, if

    a = 1 and b = 2

    \frac{10 \cdot 1 \cdot 2}{9 \cdot 1 + 2} = \frac{20}{11}

    or if a = 1 and b = 3

    \frac{10 \cdot 1 \cdot 3}{9 \cdot 1 + 3} = \frac{30}{12} = \frac{\not2 \cdot \not3 \cdot 5}{\not2 \cdot 2 \cdot \not3} = \frac{5}{2}

    or if a = 1 and b = 4

    \frac{10 \cdot 1 \cdot 4}{9 \cdot 1 + 4} = \frac{40}{13}

    and so on, then our formula fails the test. However, if a = 1 and b = 6, then

    c = \frac{10 \cdot 1 \cdot 6}{9 \cdot 1 + 6} = \frac{60}{15}

    = \frac{2 \cdot 2 \cdot \not3 \cdot \not5}{\not3 \cdot \not5} = 4

    and our formula passes the test, because we have a single digit for c.

    We then apply our solution for c to the formula

    \frac{10a+b}{10b+c} = \frac{a}{c}

    This formula results in the butt of the joke, because

    \frac{10 \cdot 1 + 6}{10 \cdot 6 + 4} = \frac{16}{64} = \frac{\not2 \cdot \not2 \cdot \not2 \cdot \not2}{\not2 \cdot \not2 \cdot \not2 \cdot \not2 \cdot 2 \cdot 2} = \frac{1}{4}

    Which means that

    \frac{1\not6}{\not64} = \frac{1}{4}

    might be an incorrect method of simplifying the fraction, but it certainly does not give the wrong answer.

    Another fraction in Soroban's list,

    \frac{1\!\!\!\not{9}}{\not{9}5} = \frac{1}{5}

    might show an incorrect way of simplifying, but if we solve for c

    \frac{10 \cdot 1 \cdot 9}{9 \cdot 1 + 9} = \frac{90}{18}

    = \frac{\not2 \cdot 5 \cdot \not3 \cdot \not3}{\not2 \cdot \not3 \cdot \not3} = 5

    and apply the formula

    \frac{10 \cdot 1 + 9}{10 \cdot 9 + 5} = \frac{19}{95}

    \frac{19}{5 \cdot 19} = \frac{1}{5}

    And so on. Well I really drilled the humor out of the joke, but at least I get the joke now.

    Don't all clap at once.
    Last edited by Euclid Alexandria; January 4th 2007 at 03:44 PM. Reason: Corrections
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  4. #19
    MHF Contributor Quick's Avatar
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    good job

    (you did do some errors in the begining of your post )
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