Scale Questions:

When working with map scale questions, I suggest arranging your work as “two ratios with one unknown”.  It’s always “map over real world”.  Be sure to show units when appropriate!!  Then you simply cross-multiply & divide to get your answer.

 Example:

What is the real world distance between two features that are 5.4 cm apart on a 1:20,000 scale map?

		Rep. Fraction (map scale)		Measures
Map		1	=	5.4 cm
Real world		20000		? cm

Now cross-multiply (5.4 cm * 20000) and divide (by 1 in this case) and get 108,000cm or 1,080m.

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 What about the case of a map (photo) with an unknown scale?  Well, we can still use the same set-up (cross-multiply and divide) as before.  If we have a map with known scale and a map of the same area with unknown scale, then we would look for two features we can see on both maps (e.g. two road intersections).  Say we have the previous map at 1:20,000 and the distance between the two intersections is 5.4 cm (same as previous example).  Then we know the real world distance is 1,080 m.  On the unknown map the map measure is 3.3 cm.  What is the scale of this map?

 Step 1: is actually solving for the real world (ground) distance between the features using the map with known scale - this is shown above.

Step 2: is recognizing that this real world (ground) distance between the 2 features is constant - what has changed is the map measurement.  This is because the scale is different on the 2nd map.

		Known map		“Unknown map”
Map		5.4 cm		3.3 cm
Real world		108,000 cm		108,000 cm

 Step 3: As stated before, since we are measuring between the same two features (road intersections), the real world distance is the same (108,000 cm).  But, for the Unknown Map we have a different fraction (3.3 / 108,000).  Now all we have to do is simplify (i.e. divide the bottom and top by 3.3). 

		“Unknown Map”		Map Scale
Map		3.3 cm	=	1
Real world		108,000 cm		?

There are 2 ways to 'visualize this'

a) cross multiply & divide: cross-multiply (1 * 108000) and divide (by 3.3 in this case) and get 32,727 – which translates to an unconventional scale of 1 : 32,727.

b) divide the top & bottom values by the same value and reduce the ratio to 1/XX.  In this case we would divide the top by 3.3 to get a value of 1 ... hence we would divide the bottom value by the same number and get a value of 32,727.

Note  that we always work with ratios that are map / real world.  This helps keeps the idea of scale in focus.