One plant is now $44$ centimeters tall and will grow at a rate of $3$ centimeters every $2$ years. A second plant is now $80$ centimeters tall and will grow at a rate of $5$ centimeters every $6$ years. In how many years will the plants be the same height?
1. Define the height functions for both plants:
   - The height of the first plant as a function of time \( t \) (in years) is given by:
     \[
     h_1(t) = 44 + \frac{3}{2}t
     \]
   - The height of the second plant as a function of time \( t \) (in years) is given by:
     \[
     h_2(t) = 80 + \frac{5}{6}t
     \]

2. Set the two height functions equal to each other to find the time \( t \) when both plants are the same height:
   \[
   44 + \frac{3}{2}t = 80 + \frac{5}{6}t
   \]

3. Solve the equation for \( t \):
   \[
   44 + \frac{3}{2}t = 80 + \frac{5}{6}t
   \]
   Subtract 44 from both sides:
   \[
   \frac{3}{2}t = 80 - 44 + \frac{5}{6}t
   \]
   Simplify the right-hand side:
   \[
   \frac{3}{2}t = 36 + \frac{5}{6}t
   \]
   Subtract \(\frac{5}{6}t\) from both sides:
   \[
   \frac{3}{2}t - \frac{5}{6}t = 36
   \]
   Find a common denominator for the fractions:
   \[
   \frac{9}{6}t - \frac{5}{6}t = 36
   \]
   Combine the fractions:
   \[
   \frac{4}{6}t = 36
   \]
   Simplify the fraction:
   \[
   \frac{2}{3}t = 36
   \]
   Multiply both sides by \(\frac{3}{2}\):
   \[
   t = 36 \cdot \frac{3}{2}
   \]
   Calculate the result:
   \[
   t = 54
   \]

4. Verify the solution:
   - In 54 years, the first plant grows:
     \[
     \frac{3}{2} \cdot 54 = 81 \text{ cm}
     \]
     So, the height of the first plant is:
     \[
     44 + 81 = 125 \text{ cm}
     \]
   - In 54 years, the second plant grows:
     \[
     \frac{5}{6} \cdot 54 = 45 \text{ cm}
     \]
     So, the height of the second plant is:
     \[
     80 + 45 = 125 \text{ cm}
     \]

Since both plants are 125 cm tall after 54 years, the solution is verified.

The final answer is \(\boxed{54}\) years.