Two similar rectangular fields have corresponding side lengths in the ratio 4:7. If the smaller field has area 32 square meters, what is…

For similar figures, areas scale with the square of the ratio of corresponding lengths. The lengths are in the ratio $4:7$, so the area ratio is $4^2 : 7^2 = 16:49$. The larger field has area $32 \times \frac{49}{16} = 2 \times 49 = 98$ square meters. A common wrong answer is 56, which comes from scaling the area by the length ratio $\frac{7}{4}$ only. Another is about 10.4, from using the ratio upside down, $32 \times \frac{16}{49}$, which gives a smaller area instead of a larger one. Areas scale with the square of the length ratio.