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: $\frac{5}{11}$
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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