The scale of a map

The scale of a map is the ratio of a length on the map to the length it stands for in the real world, both lengths in the same unit.

• On a hiking map at a scale of $\displaystyle 1:50{,}000$, every centimetre on the map is $\displaystyle 50{,}000$ cm on the ground – half a kilometre.
• The first number is the length on the map, the second the length in the real world.
• The map is a reduction of the land: every distance on it is smaller by the same factor, so the shapes on it – roads, coasts, borders – look as they really are.

On a hiking map at a scale of $\displaystyle 1:50{,}000$, a trail is 8.4 cm long.
How long is the trail on the ground, in kilometres?

The solution below multiplies the length on the map by $\displaystyle 50{,}000$, and then changes centimetres into kilometres.

• Every centimetre on the map is $\displaystyle 50{,}000$ cm on the ground: $\displaystyle {8.4\cdot50{,}000}={420{,}000}$ cm.
• A kilometre is $\displaystyle 100{,}000$ cm: $\displaystyle {420{,}000\div100{,}000}={4.2}$ km.

At a scale of $\displaystyle 1:n$, the real length $\displaystyle L$ is the length on the map $\displaystyle l$ times $\displaystyle n$, in the same unit:

On a map at a scale of $\displaystyle 1:50{,}000$, two springs are 6.2 cm apart.
How far apart are they on the ground, in kilometres?

Two towns are 36 km apart as the crow flies.
How many centimetres is that on a map at a scale of $\displaystyle 1:200{,}000$?

• 36 km is $\displaystyle {36\cdot100{,}000}={3{,}600{,}000}$ cm.
• Every $\displaystyle 200{,}000$ cm in the real world is one centimetre on the map: $\displaystyle {3{,}600{,}000\div200{,}000}={18}$ cm.

From the map to the real world we multiply by $\displaystyle n$, and from the real world to the map we divide by $\displaystyle n$: the real length is always the larger.

Tel Aviv and Jerusalem are about 54 km apart as the crow flies.
How far apart are they on a map at a scale of $\displaystyle 1:250{,}000$, in centimetres?

On a map, 3 cm stand for 1.5 km in the real world.
What is the map’s scale?

• We write both lengths in the same unit: 1.5 km is $\displaystyle 150{,}000$ cm.
• The ratio: $\displaystyle 3:150{,}000$.
• We divide both numbers by 3, so that the first number is 1: $\displaystyle 1:50{,}000$.

Find the scale, with 1 as its first number.

In the corner of many maps there is a bar like this one, with a note that its numbers are kilometres:

• The part between 0 and 1 stands for one kilometre in the real world. On the printed map it is 2 cm long.
• So the scale is $\displaystyle 2:100{,}000$, that is $\displaystyle 1:50{,}000$.
• When the map is enlarged on a screen or by a photocopier, the bar grows with it: it stays right, while the written scale no longer is.

The map has been enlarged on a screen, and the bar’s one-kilometre part is now 4 cm long.
What is the scale of the map as the screen shows it?

On a map at a scale of $\displaystyle 1:200{,}000$, a cycling route is 12.5 cm long.
A cyclist rides at 20 km/h.
How long will the ride take, in hours?