KBAT Problems
KBAT Problem: Scale, Ratio and a Real Map
A KBAT problem using a map scale to move between drawing distance and real distance, proportional reasoning in a real setting.
Set the ratio the right way
A map scale such as 1 : 50000 means one unit on the map is 50000 real units. The KBAT care is keeping the units consistent, convert centimetres to kilometres at the end, not partway through, to avoid a common slip.
Understand the problem
An original map problem. Two towns are 8 cm apart on a map with scale 1 : 250 000.
A follow-up asks how long a car travelling at 60 km/h would take between them. The scale is a ratio of the same units, so 1 cm on the map stands for 250 000 cm on the ground, the first job is to turn that into a friendly unit before doing anything else.
Plan and solve
- 1 cm represents 250 000 cm = 2 500 m = 2.5 km.
- Real distance = 8 × 2.5 = 20 km.
- Time = distance ÷ speed = 20 ÷ 60 h = ⅓ h = 20 minutes.
Check and a variant
Check backwards: 20 km ÷ 2.5 km per cm = 8 cm, matching the map reading, and 20 km at 60 km/h is a third of an hour, which is 20 minutes, both sensible. A twist: on a 1 : 500 000 map the same 8 cm would be 40 km, because 1 cm now represents 5 km.
Reversing the question 'a real distance of 15 km is how many cm on the first map?' gives 15 ÷ 2.5 = 6 cm.
Frequently asked questions
How do I read a scale like 1 : 250 000?
It means one unit on the map represents 250 000 of the same units in real life. So 1 cm on paper is 250 000 cm on the ground.
Convert that big number into metres or kilometres early, 250 000 cm is 2.5 km, so the rest of the arithmetic stays simple and the answer reads sensibly.
Which way do I multiply, is it easy to flip?
Going from map to real life you multiply by the scale; going from real life to map you divide. A quick check: real distances should come out much larger than map ones, so if your 'real' answer is smaller than the centimetres you started with, you have divided when you should have multiplied.
Reverse it.
Do I have to convert centimetres to kilometres?
You must give the answer in whatever unit the question asks, and mixing units mid-calculation is a common slip. It is usually cleanest to turn the scale into 'so many km per cm' first, then everything downstream is in kilometres.
Just keep the conversion visible so a marker can follow it.