# Thread: System of Equations Question Involving Fractions

1. ## 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!

2. Originally Posted by chianyingxuan
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 $\displaystyle \frac ab$ denote the original fraction and k the integer inquestion you'll get a system of 3 equations:

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

BTW: k = 8.

3. Wait sorry but could you explain further...like how did you get 8...

4. Originally Posted by chianyingxuan
Wait sorry but could you explain further...like how did you get 8...
I cross-multiplied the equations to get rid of the fractions:

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

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)