A great example of the application of simple math to real-world problems is provided in a recent Numberphile video on YouTube by the British mathematician James Grime.

It is taken from a real story from World War II where the British were trying to estimate the number of tanks the Germans were producing each month.  The spies came up with an estimate of about 1500 tanks per month, whereas the mathematicians estimated the number to be closer to 250 tanks per month.  How the mathematicians did this is explained by Grime.  (I added an explanation of one step that whizzed by in Grime’s presentation and that I didn’t understand at first.)

See Counting Tanks.