Rule of Three

Date: 06/15/2002 at 16:30:49
From: Marjorie Wehr
Subject: Rule of three

Please explain to me briefly (and with an example) the concept of the 
rule of three. I have looked in algebra and under proportion, but I 
didn't find an exact explanation with examples.

Thank you.
Marjorie Wehr

Date: 06/17/2002 at 12:04:57
From: Doctor Peterson
Subject: Re: Rule of three

Hi, Marjorie.

The Rule of Three is an ancient mechanical method for solving 
proportions, which we can do fairly easily (and with more 
understanding) using algebra. Briefly, it says that if you know 
three numbers a, b, and c, and want to find d such that

    a/b = c/d  (that is, a:b::c:d)

then

    d = cb/a .

Algebraically, we would multiply the equation (proportion) by bd, 
giving

    ad = bc

("the product of the means equals the product of the extremes", 
according to another old rule), and then divide by a. But doing it 
mechanically, we just read the three numbers in reverse order, first 
multiplying and then dividing. This requires no understanding and 
gives little insight, but if it is needed often it can save a lot of 
thought.

Here are a couple of references I have found that explain it. The
first, in a discussion of old mathematical copybooks, quotes an 1821
text:

    Multiplication is vexation - John Hersee
    http://w4.ed.uiuc.edu/faculty/westbury/Paradigm/Hersee.html 

    As an example of style and method, the 'Single Rule of Three
    Inverse' illustrates many points. First we need the 'Single
    Rule of Three Direct', which:

        'Teacheth, by three numbers given, to find out a fourth,
        in such proportion to the third as the second is to the
        first.

        RULE. - First state the question; that is place the
        numbers in such order that the first and third be of one
        kind, and the second the same as the number required;...
        Multiply the second and third numbers together, and
        divide product by the first, the quotient will be the
        answer to the question...'

    But:

        'Inverse Proportion is, where more requires less, and
        less requires more...

        RULE. - Multiply the first and second terms together,
        and divide the product by the third; the quotient will
        bear such proportion to the second as the first does
        to the third.'

    Readers will recognize the style of questions that the Rule
    of Three solves: 'If 10 men can dig a trench in 4 days, how
    long will 7 men take to dig a similar trench?'

This shows both the Direct Rule of Three which I stated above, and 
the Inverse Rule of Three. The latter is stated with the four terms 
in the order, a, b, c, d where ab = cd, so that it gives d in terms 
of a, b, and c:

      d = ab/c

  This corresponds to the proportion

      a:c::d:b

  The sample problem at the end represents the proportion

      10 men * 4 days = 7 men * x days

  The inverse rule of three gives

      x = 10*4/7 = 5.7 days

Again, this page discusses the rule in ancient Sanskrit writings:

    2000 Years Transmission of Mathematical Ideas: Exchange and Influence 
    from Late Babylonian Mathematics to Early Renaissance Science
      - S. R. Sarma (Aligarh, India)
    http://www.iwr.uni-heidelberg.de/transmath/author20.html 

    In the history of transmission of mathematical ideas, the
    Rule of Three forms an interesting case. It was known in
    China as early as the first century AD. Indian texts dwell
    on it from the fifth century onwards. It was introduced
    into the Islamic world in about the eighth century.
    Renaissance Europe hailed it as the Golden Rule. The
    importance of the rule lies not so much in the subtlety of
    its theory as in the simple process of solving problems.
    This process consists of writing down the three given
    terms in a linear sequence (A -> B -> C) and then,
    proceeding in the reverse direction, multiplying the last
    term with the middle form and dividing their product by
    the first term (C x B : A). With this rule one can easily
    solve several types of problems even without a knowledge
    of the general theory of proportion. The writers in
    Sanskrit, however, were well aware of the theory.

This page comes from an old text, describing how the rule can be 
easily applied on a slide rule:

    Extracts from The Mechanic's Calculator, by William Grier
      (seventh edition 1839) - Peter Owen
    http://homepages.tesco.net/~pro/text/grier.html 

    40. To solve questions in the rule of three or simple
    proportion:

    Set the first term on the slider B to the second on A; then
    on the line A will be found the fourth term, standing
    against the third term on B.

    If 4 lbs. of brass cost 36 pence, what will 12 lbs. cost?
    Move the slider so, that 4 on B will stand against 12 on A;
    then against 36 on B will be found the fourth term 108 on A.

I found these sites by using Google.com to search for the phrase 
"rule of three" together with restricting words like "proportion" and "multiply".

I hope this answers your question. If you have any further questions, 
feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/