Two similar triangles have corresponding side lengths in the ratio 2:5. If the smaller triangle has area 16, what is the area of the…

For similar figures, areas scale with the square of the ratio of corresponding side lengths. The sides are in the ratio $2:5$, so the larger triangle is $\frac{5}{2}$ times as wide, and its area is $\left(\frac{5}{2}\right)^2 = \frac{25}{4}$ times larger. Therefore the larger area is $16 \times \frac{25}{4} = 100$. A common wrong answer is 40, which comes from scaling the area by the side ratio $\frac{5}{2}$ only. Another is $\frac{64}{25}$, from squaring the ratio upside down ($\frac{2}{5}$ instead of $\frac{5}{2}$) and then applying it to the smaller triangle. Lengths scale linearly, but areas scale quadratically.