Mario is constructing a square dart board. It will consist of a smaller square centered in a larger square. The smaller square measures $4$ inches on each side. The ratio of the area of the smaller square to the area of the entire dart board is $\frac 49$. How long is the side of the larger square?
I want to find the side length of the larger square, so I'll call it $x$. The area of the larger square is $x^2$. The area of the smaller square is $4^2 = 16$. The ratio of the areas is $\frac 49$, so I can write an equation: $\frac{16}{x^2} = \frac 49$. To solve for $x$, I can cross-multiply: $16 \cdot 9 = 4 \cdot x^2$. Simplifying, I get $144 = 4x^2$. Dividing both sides by $4$, I get $36 = x^2$. Taking the square root of both sides, I get $x = \sqrt{36}$. Since $x$ is a positive length, I only consider the positive root: $x = 6$.

