Use the image to answer the question.

To find the fraction that expresses the relationship between the areas of the original square and the dilated square, we first need to calculate the areas of both squares.

1. Area of the original square (side length = 4 units): \[ \text{Area}_{\text{original}} = \text{side}^2 = 4^2 = 16 \text{ square units} \]

2. Area of the dilated square (side length = 6 units): \[ \text{Area}_{\text{dilated}} = \text{side}^2 = 6^2 = 36 \text{ square units} \]

Now we will write the fraction to express the relationship between the area of the dilated square (numerator) and the area of the original square (denominator): \[ \text{Fraction} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{36}{16} \]

Next, we simplify the fraction: \[ \frac{36}{16} = \frac{9}{4} \]

Thus, in terms of lowest terms, the fraction expressing the relationship between the area of the original square and the dilated square is: \[ \frac{9}{4} \]

Since the options provided do not include \(\frac{9}{4}\), let’s consider how we might express the relationship with the original area in the denominator: \[ \text{Fraction} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{36}{16} \] To express this relationship as the inverse (which seems to be implied by the options), we can write: \[ \frac{16}{36} = \frac{4}{9} \]

Therefore, the final fraction expressing the relationship in the lowest terms, with the numerator as the area of the original square and the denominator as the area of the dilated square, is: \[ \frac{4}{9} \]

So the answer is: \(\text{Start Fraction 4 over 9 End Fraction}\)