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### 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

("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

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

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

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/
```
Associated Topics:
Middle School Algebra
Middle School Fractions
Middle School History/Biography
Middle School Ratio and Proportion

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