A map has a scale of \( 10 \mathrm{~mm}: 2 \mathrm{~m} \). Express this ratio in the form \( 1: k \). \( \begin{array}{l}1: 200 \\ 1: 20 \\ 1: 100 \\ 1: 50\end{array} \)

To express the scale \( 10 \mathrm{~mm}: 2 \mathrm{~m} \) in the form \( 1: k \), we first need to convert both measurements to the same unit. 1. Convert \( 2 \mathrm{~m} \) to millimeters: \( 2 \mathrm{~m} = 2000 \mathrm{~mm} \). 2. The scale now reads \( 10 \mathrm{~mm}: 2000 \mathrm{~mm} \). 3. To express this as \( 1: k \), divide both sides by \( 10 \): \[ 1 \mathrm{~mm} : 200 \mathrm{~mm}. \] So, the ratio simplifies to \( 1: 200 \). Therefore, the correct answer is \( 1: 200 \).