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/
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