Finding Original Fraction Based on Percentage Changes

Let the original fraction be represented algebraically as $\frac{x}{y}$.

Numerator Modification

The numerator $x$ is increased by 30%. The new numerator is calculated as:

$ x + (30\% \times x) = x + (0.30 \times x) = 1.30x $

Denominator Modification

The denominator $y$ is decreased by 35%. The new denominator is calculated as:

$ y - (35\% \times y) = y - (0.35 \times y) = 0.65y $

New Fraction Value

The resulting fraction after these modifications is $\frac{1.30x}{0.65y}$.

We are given that this new fraction equals $\frac{3}{15}$, which simplifies to $\frac{1}{5}$.

$ \frac{1.30x}{0.65y} = \frac{1}{5} $

Solving for the Original Fraction

First, simplify the numerical coefficients in the equation:

$ \frac{1.30}{0.65} = \frac{130}{65} = 2 $

Substitute this back into the equation:

$ 2 \times \frac{x}{y} = \frac{1}{5} $

To find the original fraction $\frac{x}{y}$, divide both sides by 2:

$ \frac{x}{y} = \frac{1}{5 \times 2} $

$ \frac{x}{y} = \frac{1}{10} $

Thus, the original fraction is $\frac{1}{10}$.