Question

What is the distance between two cities, $4.5$ cm apart, on a map drawn to scale $1$ cm $=45$ km?

Answer 1

Let $D=$ the actual distance in kilometers (km) between any two locations, and let $d=$ the corresponding measured distance on a map in centimeters (cm) between the same two locations.

Since the scale on a map is $1$ cm $=45$ km, then the ratio of the distance on the map to the actual distance is equal to $\frac{1cm}{45km}$; $\therefore$ the quotient of any measured distance $d$ on the map divided by the corresponding much longer actual distance $D$ would have to maintain (equal) this same ratio; In other words, we can set up the following proportion to solve for the actual distance $D$ between two cities given the measured map distance $d=4.5$ cm between them as follows:

$\frac{1cm}{45km}=\frac{d}{D}$

Substituting, we get:

$\frac{1cm}{45km}=4.5\frac{cm}{D}$

Cross-multiplying, we get:

$D\left(1cm\right)=\left(4.5cm\right)\left(45km\right)$

$\frac{D\left(1cm\right)}{\left(1cm\right)}=\frac{\left[\left(4.5cm\right)\left(45km\right)\right]}{\left(1cm\right)}$

$D=202.5$km is the actual distance between the two cities.