A fraction becomes $\frac{1}{3}$ when $1$ is subtracted from the numerator and it becomes $\frac{1}{4}$ when $8$ is added to its denominator. Find the fraction.

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Step 1: Let the fraction be $\frac{x}{y}$.
Step 2: $\frac{x - 1}{y} = \frac{1}{3}$ and $\frac{x}{y + 8} = \frac{1}{4}$. This gives $3x - y = 3$ and $4x - y = 8$.
Step 3: Subtracting: $x = 5$, so $y = 12$.
Step 4: Fraction $= \frac{5}{12}$

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