What a scale says
A map is a drawing of real ground, made much smaller. Every distance on the ground is shrunk by the same amount, so the map keeps the right shape. The scale of the map tells you how much real distance one unit on the map stands for.
On a map with the scale 1 cm to 10 km, every centimeter on the map stands for 10 km on the ground. A road that is 5 cm long on the map is 5 lots of 10 km: 5 × 10 = 50 km long on the ground.
Each centimeter on the map is paired with 10 km on the ground, so 5 cm is paired with 50 km.
From the ground to the map
To go from the map to the ground, multiply by what one map unit stands for. To go from the ground back to the map, divide by it. A lake 35 km across is 35 ÷ 10 = 3.5 cm across on the same map.
Check the direction of your answer. Ground distances are much bigger than map distances, so a map-to-ground answer should be bigger, and a ground-to-map answer should be smaller.
A scale written as a ratio
Many maps write the scale as a ratio, such as 1 : 50,000. A ratio has no units, so it means that 1 of any unit on the map stands for 50,000 of the same unit on the ground. Usually the map is measured in centimeters, so 1 cm on the map stands for 50,000 cm on the ground.
50,000 cm is too large a number to picture, so change it into bigger units. There are 100 cm in a meter, so divide by 100: 50,000 cm is 500 m. There are 1,000 m in a kilometer, so divide by 1,000: 500 m is 0.5 km. On this map, 1 cm stands for half a kilometer, and a path 8 cm long on the map is 8 × 0.5 = 4 km long.
At 1 : 50,000, every 2 cm on the map is 1 km on the ground, so 8 cm is 4 km.
The usual mistakes
Adding instead of multiplying. At 1 cm to 10 km, 5 cm is not 5 + 10 = 15 km. Each of the 5 centimeters stands for 10 km, so the distance is 5 × 10 = 50 km.
Forgetting the units of a ratio scale. At 1 : 50,000, a path 8 cm long is 8 × 50,000 = 400,000 on the ground, but that number is in centimeters, the unit the map was measured in. It is 4,000 m, which is 4 km, not 400,000 m.
Worked example: A Hiking Trail on Two Maps with Different Scales
Question A hiking map is drawn to the scale 1 : 25000. On it, the trail from a parking lot to a waterfall measures 18.4 cm along its bends. (a) How long is the trail on the ground, in kilometers? (b) The same trail is drawn on a regional map on which 1 cm stands for 2 km. How long is the trail on the regional map, in centimeters?
1.On the hiking map, 1 cm stands for 25000 cm on the ground, which is 250 m.
On the hiking map, 1 cm stands for 25000 cm, which is 250 m on the ground. 2.The trail is 18.4 cm on the map, so on the ground it is 18.4 × 250 = 4600 m.
The trail is 18.4 cm on the map: 18.4 × 250 = 4600 m on the ground. 3.(a) 4600 m = 4.6 km, so the trail is 4.6 km long.
(a) 4600 m is 4.6 km. 4.(b) On the regional map, every 2 km of ground is 1 cm. The trail's 4.6 km is 4.6 ÷ 2 = 2.3 lots of 2 km.
(b) On the regional map, 1 cm stands for 2 km, and 4.6 ÷ 2 = 2.3. 5.So the trail is 2.3 cm long on the regional map. Check: 2.3 × 2 = 4.6 km, the same trail.
The same trail is 2.3 cm long on the regional map, which shows more ground in each centimeter.
Answer: (a) 4.6 km; (b) 2.3 cm
Common mistakes
- Writing 18.4 × 25000 = 460000 and calling it kilometers or meters. A scale written as a ratio has no unit, so the answer is in the map's unit, centimeters, and 100000 cm make one kilometer.
- Scaling the 18.4 cm up for the regional map. Each centimeter of the regional map stands for more ground than a centimeter of the hiking map, so the same trail must be shorter there; going through the ground distance, 4.6 km, keeps the direction right.