Answer to Question #322474 in Linear Algebra for Kenth Smith

Question

If the numerator of a fraction is doubled and the denominator is increased by 7, the value becomes 2/3; but if the denominator is doubled and the numerator is increased by 2, the value becomes 3/5. What is the original fraction?

Answer 1

Let $\cfrac{x}{y}-$ the original fraction, x - the numerator, y - the denominator.

$\begin{cases} \cfrac{2x}{y+7}=\cfrac{2}{3}\\ \cfrac{x+2}{2y}=\cfrac{3}{5} \end{cases}$

$\begin{cases} 6x=2y+14\\ 5x+10=6y \end{cases}$

$\begin{cases} y=3x-7\\ 5x-6(3x-7)+10=0 \end{cases}$

$5x-18x+42+10=0\\ -13x+52=0\\ x=4\\ y=3\cdot4-7=5$

$\cfrac{4}{5}-$ the original fraction.