Ratio/scaling (High school)

Question

Have a question from my textbook which is causing some confusion. Q. A map has scale $1:7500$. A town has an area of $37.5~\text{cm}^2$ on the map. What is the real area of the town in $\text{km}^2$?

Can I please request as much info as possible on the scaling from lengths to areas etc? I believe it involves squaring or square rooting to find the scale factor etc but not sure and unable to find specific info in the chapter it is from.

Thanks a bunch in advance.

Answer 1

Note that

• $1$ cm con the map $\to 7500$ cm$=75$m real
• $1$ cm$^2$ con the map $\to 7500^2$ cm$^2=75^2$ m$^2$ real

then

• $37.5$ cm$^2\to37.5\cdot 75^2$ m$^2=210937.5$m$^2\approx0.211$ km$^2$

Answer 2

$$ \left. \begin{align} 37.5\,\,\text{cm}^2 \approx (6.1237\,\,\text{cm})^2 \leftrightarrow ((6.1237\times 7500)\,\,\text{cm})^2 & = (459.2775\,\,\text{m})^2 \\ & = (459.2793268\,\,\text{m})^2 \end{align} \right\} \text{Which is right?} $$ On the first line above, I rounded the square root of $37.5$ to $6.1237$ and went on from there; on the second line I did no rounding beyond what a calculator forced on me. If one then rounds to three places after the decimal point, the one ending with $9$ is right: $459.279.$

However, here is a more accurate way: \begin{align} & 37.5\,\,\text{cm}^2 \leftrightarrow 37.5\,\,\text{cm}^2\times 7500^2 = 2109375000\,\,\text{cm}^2 = \frac{2109375000}{100^2} \text{m}^2 \\[10pt] = {} & 210937.5\,\, \text{m}^2 = \frac{210937.5}{1000^2}\,\,\text{km}^2 = 0.2109375 \,\,\text{km}^2 = 21.09375\,\,\text{hectares}. \end{align}