To find the perimeters of the two triangles, we first calculate the perimeter of the smaller triangle with side lengths 6, 6, and 9 units.
The perimeter \( P_1 \) of the smaller triangle is calculated as follows: \[ P_1 = 6 + 6 + 9 = 21 \text{ units} \]
Next, we calculate the perimeter of the larger triangle with side lengths 12, 12, and 18 units.
The perimeter \( P_2 \) of the larger triangle is calculated as follows: \[ P_2 = 12 + 12 + 18 = 42 \text{ units} \]
Now that we have both perimeters, we calculate the ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle: \[ \text{Ratio} = \frac{P_1}{P_2} = \frac{21}{42} \]
To simplify this fraction, we divide both the numerator and the denominator by their greatest common divisor, which is 21: \[ \frac{21}{42} = \frac{1}{2} \]
Thus, the ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle is: \[ \boxed{\frac{1}{2}} \]