The time period ($T$) of a spring-mass system undergoing simple harmonic motion is given by the formula $T = 2\pi \sqrt{\frac{m}{k}}$, where $m$ is the mass and $k$ is the spring constant.

1. Case 1: For mass $m$, the time period is $3\text{ s}$. $3 = 2\pi \sqrt{\frac{m}{k}}$

2. Case 2: When the mass is increased by $1\text{ kg}$, the new mass is $(m + 1)$ and the time period becomes $5\text{ s}$. $5 = 2\pi \sqrt{\frac{m+1}{k}}$

3. Divide the two equations: $\frac{3}{5} = \frac{2\pi \sqrt{m/k}}{2\pi \sqrt{(m+1)/k}}$ $\frac{3}{5} = \sqrt{\frac{m}{m+1}}$

4. Solve for $m$: Squaring both sides: $\frac{9}{25} = \frac{m}{m+1}$ $9(m + 1) = 25m$ $9m + 9 = 25m$ $16m = 9$ $m = \frac{9}{16} \text{ kg}$