Proportion problems and the Rule of Three

Here is a problem from an old textbook :

If 750 men require 22,500 rations of bread for a month; how many rations will a garrison of 1200 men require?  (Charles Hutton, A course of mathematics for the use of academies as well as private tuition, 4th American ed., vol. 1, p. 46, New York, 1825.)

 Such problems were set up as proportions, i.e.

750 : 22,500 :: 1200 : ?

Each of the four quantities (three known and one to be calculated) is called a term. The expression can be read

'750 men is to 22,500 rations of bread as 1200 men is to how many rations of bread?'

To calculate the last term, Hutton has one multiply the two middle terms and then divide the result by the first term:

`\qquad\qquad\qquad? = \frac{"22,500 " \times " 1,200"}{750} = 36,000.`

This method of solving proportion problems was common in Hutton's day, when it was known as the Rule of Three.  As Abraham Lincoln says of his own education,

Of course when I came of age I did not know much. Still somehow, I could read, write, and cipher to the Rule of Three; but that was all. I have not been to school since. The little advance I now have upon this store of education, I have picked up from time to time under the pressure of necessity.  (Letter, accompanying his Autobiography, to Jesse Fell, Dec. 20, 1859.)

To 'cipher' is do arithmetic, using the familiar Hindu-Arabic number system.

In more recent times, this problem might be set up

`\qquad\qquad\qquad\frac{750" men"}{"22,500 rations of bread"}=\frac{1200 " men"}{x}.`

where we have used quantities convention used in public by physicists, engineers, chemists and other technical people, rather than the numbers(-only) convention used mathematics and scribbled calculation. 

One isolates `x` through a series of steps.  One can multiply both sides by `x`, giving

`\qquad\qquad\qquad\frac{"(750 men) "  x}{"22,500 rations of bread"}=1200 " men".`

Dividing both sides by `"750 men"` gives

`\qquad\qquad\qquad\frac{x}{"22,500 rations of bread"}=\frac{1200 " men"}{"750 men"}.`

Multiplying both sides by `"22,500 rations of bread"` then gives

`\qquad\qquad\qquad x  = \frac{1200 " men " \times " 22,500 rations of bread"}{"750 men"},`

`\qquad\qquad\qquad    = "36,000 rations of bread".`

So far, this is bookwork---useful, but perhaps a little boring.  It is essential background, however, to dealing with proportion tables, our next object of concern.