1. The problem asks for the ratio of the area of the smaller square to the area of the larger square given their side lengths. 2. Recall the formula for the area of a square: $$\text{Area} = \text{side}^2$$. 3. The smaller square has side length 6 ft, so its area is $$6^2 = 36$$ square feet. 4. The larger square has side length 9 ft, so its area is $$9^2 = 81$$ square feet. 5. The ratio of the areas is therefore $$36 : 81$$. 6. Simplify this ratio by dividing both numbers by their greatest common divisor, which is 9: $$\frac{\cancel{36}}{\cancel{9}} : \frac{\cancel{81}}{\cancel{9}} = 4 : 9$$. 7. So, the ratio of the area of the smaller square to the area of the larger square is \textbf{4:9}.