12 The area of a farm is represented on two maps, map A and map B. The area of this farm on Map A is sixteen times that of the area represented on Map B. Map A has a scale of $1 : 12\,500$. Find the scale of Map B in the form $1 : n$. Answer ___

1. Convert the area comparison into a linear scale comparison.

Area on Map A is $16$ times area on Map B.

Hence the linear dimensions on Map A are
$\sqrt{16}=4$
times the corresponding linear dimensions on Map B.

2. Relate the two map scales using the linear scale factor.

Map A has scale $1:12\,500$.

For the same actual distance, Map A's distance is $4$ times Map B's distance, so Map B's scale denominator is $4$ times Map A's denominator.

3. Calculate the scale of Map B.

$n=4\times 12\,500=50\,000$

Therefore the scale of Map B is
$1:50\,000.$