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Math Help - System of Equations Question Involving Fractions

  1. #1
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    Smile System of Equations Question Involving Fractions

    Hi people I have a math question that I can't solve in the way my teacher wants me to...it is:

    A fraction, after being simplified, is 2/3. If an integer is added to both the numerator and its denominator of this fraction, it becomes 8/11. If one is added to this integer, and the new integer is subtracted from both the numerator and the denominator of this fraction, it becomes 5/9. Find this fraction.

    Well I got the answer 24/36, but my teacher needs the "algebraic approach". Can anyone help? Thanks!
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  2. #2
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    Quote Originally Posted by chianyingxuan View Post
    Hi people I have a math question that I can't solve in the way my teacher wants me to...it is:

    A fraction, after being simplified, is 2/3. If an integer is added to both the numerator and its denominator of this fraction, it becomes 8/11. If one is added to this integer, and the new integer is subtracted from both the numerator and the denominator of this fraction, it becomes 5/9. Find this fraction.

    Well I got the answer 24/36, but my teacher needs the "algebraic approach". Can anyone help? Thanks!
    Let \frac ab denote the original fraction and k the integer inquestion you'll get a system of 3 equations:

    \left|\begin{array}{rcl}\dfrac ab&=&\dfrac23 \\ \\ \dfrac{a+k}{b+k}&=&\dfrac8{11} \\ \\ \dfrac{a-(k+1)}{b-(k+1)}&=&\dfrac59 \end{array}\right.

    BTW: k = 8.
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  3. #3
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    Wait sorry but could you explain further...like how did you get 8...
    Last edited by chianyingxuan; February 28th 2010 at 01:09 AM.
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  4. #4
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    Quote Originally Posted by chianyingxuan View Post
    Wait sorry but could you explain further...like how did you get 8...
    I cross-multiplied the equations to get rid of the fractions:

    <br />
\left|\begin{array}{rcl}\dfrac ab&=&\dfrac23 \\ \\ \dfrac{a+k}{b+k}&=&\dfrac8{11} \\ \\ \dfrac{a-(k+1)}{b-(k+1)}&=&\dfrac59 \end{array}\right.<br />
...... \implies ...... <br />
\left|\begin{array}{rcl}3a-2b&=&0 \\ \\ 11a-8b+3k&=&0 \\ \\ 9a-5b-4k&=&4 \end{array}\right.<br />

    I don't know which method you prefer to solve a system of simultaneous equations. I used the Gaussian algorithm which yielded (a, b, k) = (24, 36, 8)
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