Question : When 8 is added to the numerator of a fraction and 12 is added to its denominator, the fraction becomes $\frac{1}{2}$. When 2 is subtracted from its numerator and denominator, the fraction becomes $\frac{1}{8}$. Find the original fraction.
Option 1: $\frac{9}{19}$
Option 2: $\frac{3}{10}$
Option 3: $\frac{4}{7}$
Option 4: $\frac{5}{11}$
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Correct Answer: $\frac{3}{10}$
Solution : Let the original fraction be $\frac{x}{y}$ where $x$ and $y$ are the numerator and denominator, respectively. According to the question, $\frac{x+8}{y+12}=\frac{1}{2}$ ⇒ $2x+16=y+12$ ⇒ $2x+4=y$ (equation 1) When 2 is subtracted from its numerator and denominator, the fraction becomes $\frac{1}{8}$. $\frac{x–2}{y–2}=\frac{1}{8}$ ⇒ $8x-16=y-2$ ⇒ $y=8x-14$ (equation 2) Substitute the value of $y$ in equation 2, ⇒ $2x+4=8x-14$ ⇒ $8x-2x=14+4$ ⇒ $6x=18$ ⇒ $x=3$ Substitute the value of $x$ in equation 1, ⇒ $2\times 3+4=y$ ⇒ $y=6+4=10$ The original fraction = $\frac{x}{y}=\frac{3}{10}$ Hence, the correct answer is $\frac{3}{10}$.
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