To find the area of the original square and the dilated square, we can use the formula for the area of a square, which is given by:
\[ \text{Area} = \text{side length}^2 \]
First, let's calculate the area of the original square with side length 4 units:
\[ \text{Area of original square} = 4^2 = 16 \text{ square units} \]
Next, we calculate the area of the dilated square, which has side length 6 units:
\[ \text{Area of dilated square} = 6^2 = 36 \text{ square units} \]
Now that we have the areas, we will express the relationship between the area of the dilated square and the area of the original square as a fraction:
\[ \text{Fraction} = \frac{\text{Area of dilated square}}{\text{Area of original square}} = \frac{36}{16} \]
Next, we need to simplify this fraction. We can do so by finding the greatest common divisor (GCD) of 36 and 16.
The GCD of 36 and 16 is 4. We can divide both the numerator and denominator by 4:
\[ \frac{36 \div 4}{16 \div 4} = \frac{9}{4} \]
Thus, the fraction that expresses the relationship between the area of the dilated square and the original square in lowest terms is:
\[ \boxed{\frac{9}{4}} \]