sign. Otherwise, if we have 10 equations, we could for example
isolate a on the left of each equation and get:
a=B
a=C
a=D
a=E
a=F
a=G
a=H
a=J
a=K
a=M
We then know that
a=B=C=D=E=F=G=H=J=K=M
I'm not going to try to write these as individual equations. There
would be 10*9*8*7*6*5*4*3*2 or 3628800 equations.
My incorrect method is sometimes more powerful and requires less
thinking.
Regards,
Brian
--- In Math4u@yahoogroups.com, Opher Liba <opherel@...> wrote:
>
>
> an equation has only one equal sign.
>
> if there are two equal signs, it is (in fact) a shortening for a
system of two equations.
>
> opher
>
> Opher LibaMathematics Educator, Researcher and
AuthorMathematically (R), Center for Educational EnrichmentJerusalem,
Israelwww.Mathematically-JLM.com
>
>
> To: Math4u@...: brianejensen@...: Fri, 21 Dec 2007 05:19:50
+0000Subject: [Math4u] Re: 4 situations
>
>
>
>
> My thanks to Buck for clarifying this issue. My apologies to
Vinaire who was correct.Books are perhaps even better reference, but
websites are more accessible. (What few books I have are hard to
locate.)One rule I will not personally follow is that a ratio can
only be two numbers. I solve many problems where I write a:b:c=d:e:f.
I also write equations which have more than 2 sides and more than one
equal sign. If a puzzle is given with 6 equations and these equations
can be combined into one equation with 7 sides and 6 equal signs,
then it is easier to solve in the rearranged form.I understand from
the definition that it is not proper to say "the ratio of apples to
oranges" but rather "the rate of apples to oranges." Everyone uses
the word "ratio" in this case however.I cringe that this definition
makes it look like the ratio sign is the same as the division sign.
So let's say we pitch our aqueduct at 1 inch per 100 feet. This is a
rate. Can we use a colon? It is safer to use the word "per." So the
ratio would be 1:1200 or 0.000833. The workers would need to hire a
mathematician to figure it out.Regards,Brian--- In
Math4u@yahoogroups.com, Buck Dick <kcubar34@> wrote:>> Brian and
Vinaire,> > From "The Universal Encyclopedia of Mathematics," Simon
and Schuster, 1964:> > "The RATIO of two quantities a and b is their
quotient a:b = a/b. We read "a to b" or "a over b". If a:b = m, m is
referred to as the VALUE of the RATIO. Two quantities, in order to
posess a RATIO must be either PURE NUMBERS or MEASURES OF THE SAME
TYPE. e.g. 20:5 = 20/5 = 4; 15 lb: 3 lb = 5. The VALUE of the RATIO
(PROPORTIONALITY FACTOR) is always a pure number."> > So, please (as
Vinaire as indicated) don't confuse a RATIO with a RATE (different
measures) or a RATIONAL NUMBER.> > I have no idea whose websites
those are, but I will bet they are not edited by mathematicians if
they allow a RATIO to have different units, as in 2000 miles: 40
hours.> > Buck Dick> > Brian Edward Jensen <brianejensen@> wrote:> As
usual I disagree with you. > > First website I found > >
http://www.mathleague.com/help/ratio/ratio.htm > > gives ratios with
different units, defines proportion, and says rate > involves time. >
> Second website I found > >
http://www.math.com/school/subject1/lessons/S1U2L1GL.html > > gives
ratios with different units > > Third website I found > >
http://www.purplemath.com/modules/ratio.htm > > gives ratios with
different units > > The "Understanding" of others is often made up
nonsense. What is > taught in school is not reliable. Let's get some
websites to back up > information. > > Regards, > > Brian> > --- In
Math4u@yahoogroups.com, Vinaire <vinaire@> wrote:> >> > I think that
by "Rationnel" you mean a rational number. A number > and units are
two different things. A rational number is a number > that can be
expressed as a ratio of two integers. > > > > When you compare two
quantities of the same unit you have a RATIO. > When you compare two
quantities of different units you have a RATE. > That is my
understanding. Others may clarify it further.> > > > PROPORTION
simply means two equal ratios.> > > > I did not understand what you
meant by "et." You seem to be mixing > some other language with
English.> > > > I did not understand what you meant by "rapport."> >
> > DIVISION simply means how many times you can take one quantity
out > of another. It may be looked upon as repeated subtraction. When
you > divide a smaller quantity (number with a unit) by a larger
number you > get fractions.> > > > I do not understand the situation
where you would divide a number > by a quantity. Can you give me a
practical example of that?> > > > Vinaire> > > > > > > > > > > > -----
Original Message ----> > From: testimprove <bonnerentree2003@>> >
To: Math4u@yahoogroups.com> > Sent: Sunday, December 16, 2007
12:20:31 PM> > Subject: [Math4u] Re: 4 situations> > > > Hi,> > "I am
not familiar with some of the symbols and terminology you are >
using"> > -Which symbols and terminology I use and you are not
familiar with?> > your question :> > "Were you taught that way in
school?"suprised me.> > I'd like before answering your question to
let me know what you > thing about this way.> > Badriya> > > > > > >
> > > > > --- In Math4u@yahoogroups. com, Vinaire <vinaire@> wrote:>
> >> > > I asked that question because I am not familiar with some of
the > symbols and terminology you are using. Were you taught that way
in > school?> > > > > > V.> > > > > > ----- Original Message ----> >
> From: testimprove bonnerentree2003@ ...> > > To:
Math4u@yahoogroups. com> > > Sent: Sunday, December 16, 2007 6:54:25
AM> > > Subject: [Math4u] Re: 4 situations> > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >
Hi> > > > > > "What country are you from?"> > > > > > even I don't
know what is the relation between this question and > the
mathematic,I give an answer to your question:> > > > > > I am from
France> > > > > > I wonder if my style in writing English is the
purpose of your > inquiry.> > > > > > I have studied Math in
French ,and I really will be happy if you > point out the errors
I 'll probably make while typing questions in > this wonderful forum>
> > > > > Badriya > > > > > > --- In Math4u@yahoogroups. com, Vinaire
vinaire@ wrote:> > > >> > > > What country are you from?> > > > > > >
> Vinaire> > > > > > > > ----- Original Message ----> > > > From:
testimprove bonnerentree2003@ ...> > > > To: Math4u@yahoogroups. com>
> > > Sent: Saturday, December 15, 2007 5:17:43 PM> > > > Subject:
[Math4u] 4 situations> > > > > > > > > > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >
> > > > > Hello,> > > > > > > > I'd like to know if it is exact that
any Rationnel has 4 > meanings.> > > > > > > > · example the
rationnel 17/6> > > > > > > > · a) the numbers 17 and 6 are different
units for> > > > > > > > example 17 liters of milk for making 6 kg of
cheese> > > > > > > > so liters and kg are two diferent units.Here
the > > > > > > > > > > > > > > > > fraction17/6 means a proportion >
> > > > > > > > > > > > > > > · b) 17 et 6 are the same units , for
example the > > > > > > > > heigth of a triangle is 17mm and the side
is 6 mm, > > > > > > > > here the fraction 17/6 means a rapport > > >
> > > > > · c) the number 17 is a unit and the number 6 > > > > > > >
> > > > > > > > > deosn't have a unit . For example to cut 17meters
in> > > > > > > > > > > > > > > > 6 equal pieces.Here 17/6 means a
divison > > > > > > > > > > > > > > > > · d) The number 17 doesn't
have a unit and 1/6 > > > > > > > > mm,we have 17x1/6 so here 17/6 is
a fraction of the 1/6x17> > > > > > > > > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >
> > > > > > <!--> > > > > > > > #ygrp-mkp{> > > > border:1px solid
#d8d8d8;font- family:Arial; margin:14px > 0px;padding: 0px 14px;}> >
> > #ygrp-mkp hr{> > > > border:1px solid #d8d8d8;}> > > > #ygrp-mkp
#hd{> > > > color:#628c2a; font-size: 85%;font- weight:bold; line-
height: > 122%;margin: 10px 0px;}> > > > #ygrp-mkp #ads{> > > >
margin-bottom: 10px;}> > > > #ygrp-mkp .ad{> > > > padding:0 0;}> > >
> #ygrp-mkp .ad a{> > > > color:#0000ff; text-decoration: none;}> > >
> -->> > > > > > > > > > > > > > > > <!--> > > > > > > > #ygrp-
sponsor #ygrp-lc{> > > > font-family: Arial;}> > > > #ygrp-sponsor
#ygrp-lc #hd{> > > > margin:10px 0px;font-weight: bold;font-
size:78%; line-height: > 122%;}> > > > #ygrp-sponsor #ygrp-lc .ad{> >
> > margin-bottom: 10px;padding: 0 0;}> > > > -->> > > > > > > > > >
> > > > > > <!--> > > > > > > > #ygrp-mlmsg {font-size:13px; font-
family: arial, helvetica, > clean, sans-serif;}> > > > #ygrp-mlmsg
table {font-size:inherit; font:100% ;}> > > > #ygrp-mlmsg select,
input, textarea {font:99% arial, helvetica, > clean, sans-serif;}> >
> > #ygrp-mlmsg pre, code {font:115% monospace;}> > > > #ygrp-mlmsg *
{line-height: 1.22em;}> > > > #ygrp-text{> > > > font-family:
Georgia;> > > > }> > > > #ygrp-text p{> > > > margin:0 0 1em 0;}> > >
> #ygrp-tpmsgs{> > > > font-family: Arial;> > > > clear:both;}> > > >
#ygrp-vitnav{> > > > padding-top: 10px;font- family:Verdana; font-
size: 77%;margin: > 0;}> > > > #ygrp-vitnav a{> > > > padding:0 1px;}
> > > > #ygrp-actbar{> > > > clear:both;margin: 25px 0;white-space:
nowrap;color: #666;text- > align:right; }> > > > #ygrp-actbar .left{>
> > > float:left;white- space:nowrap; }> > > > .bld{font-weight:
bold;}> > > > #ygrp-grft{> > > > font-family: Verdana;font- size:77%;
padding:15px 0;}> > > > #ygrp-ft{> > > > font-family: verdana;font-
size:77%; border-top: 1px solid #666;> > > > padding:5px 0;> > > > }>
> > > #ygrp-mlmsg #logo{> > > > padding-bottom: 10px;}> > > > > > > >
#ygrp-vital{> > > > background-color: #e0ecee;margin- bottom:20px;
padding:2px 0 > 8px 8px;}> > > > #ygrp-vital #vithd{> > > > font-
size:77% ;font-family: Verdana;font- weight:bold; > color:#333; text-
transform: uppercase; }> > > > #ygrp-vital ul{> > > >
padding:0;margin: 2px 0;}> > > > #ygrp-vital ul li{> > > > list-style-
type: none;clear: both;border: 1px solid #e0ecee;> > > > }> > > >
#ygrp-vital ul li .ct{> > > > font-weight: bold;color: #ff7900;float:
right;width: 2em;text-> align:right; padding-right: .5em;}> > > >
#ygrp-vital ul li .cat{> > > > font-weight: bold;}> > > > #ygrp-vital
a{> > > > text-decoration: none;}> > > > > > > > #ygrp-vital a:hover
{> > > > text-decoration: underline; }> > > > > > > > #ygrp-sponsor
#hd{> > > > color:#999;font- size:77%; }> > > > #ygrp-sponsor #ov{> >
> > padding:6px 13px;background- color:#e0ecee; margin-bottom: >
20px;}> > > > #ygrp-sponsor #ov ul{> > > > padding:0 0 0
8px;margin:0; }> > > > #ygrp-sponsor #ov li{> > > > list-style-type:
square;padding: 6px 0;font-size: 77%;}> > > > #ygrp-sponsor #ov li a
{> > > > text-decoration: none;font- size:130% ;}> > > > #ygrp-
sponsor #nc{> > > > background-color: #eee;margin- bottom:20px;
padding:0 8px;}> > > > #ygrp-sponsor .ad{> > > > padding:8px 0;}> > >
> #ygrp-sponsor .ad #hd1{> > > > font-family: Arial;font-
weight:bold; color:#628c2a; font-size: > 100%;line- height:122% ;}> >
> > #ygrp-sponsor .ad a{> > > > text-decoration: none;}> > > > #ygrp-
sponsor .ad a:hover{> > > > text-decoration: underline; }> > > >
#ygrp-sponsor .ad p{> > > > margin:0;}> > > > o{font-size: 0;}> > >
> .MsoNormal{> > > > margin:0 0 0 0;}> > > > #ygrp-text tt{> > > >
font-size:120% ;}> > > > blockquote{margin: 0 0 0 4px;}> > > > .replbq
{margin: 4;}> > > > -->> > > > > > > > > > > > > > > > > > > > > > >
> > > > > > > > > > > > > ____________ _________ _________ _________
_________ _________ > _________ _________ _________> > > > Looking
for last minute shopping deals? > > > > Find them fast with Yahoo!
Search. http://tools. search.yahoo. > com/newsearch/ category. php?
category= shopping> > > >> > > > > > > > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > > <!--> > > > > > #ygrp-mkp{> > >
border:1px solid #d8d8d8;font- family:Arial; margin:14px >
0px;padding: 0px 14px;}> > > #ygrp-mkp hr{> > > border:1px solid
#d8d8d8;}> > > #ygrp-mkp #hd{> > > color:#628c2a; font-size: 85%;font-
weight:bold; line-height: > 122%;margin: 10px 0px;}> > > #ygrp-mkp
#ads{> > > margin-bottom: 10px;}> > > #ygrp-mkp .ad{> > > padding:0
0;}> > > #ygrp-mkp .ad a{> > > color:#0000ff; text-decoration: none;}
> > > -->> > > > > > > > > > > > <!--> > > > > > #ygrp-sponsor #ygrp-
lc{> > > font-family: Arial;}> > > #ygrp-sponsor #ygrp-lc #hd{> > >
margin:10px 0px;font-weight: bold;font- size:78%; line-height: >
122%;}> > > #ygrp-sponsor #ygrp-lc .ad{> > > margin-bottom:
10px;padding: 0 0;}> > > -->> > > > > > > > > > > > <!--> > > > > >
#ygrp-mlmsg {font-size:13px; font-family: arial, helvetica, > clean,
sans-serif;}> > > #ygrp-mlmsg table {font-size:inherit; font:100% ;}>
> > #ygrp-mlmsg select, input, textarea {font:99% arial, helvetica, >
clean, sans-serif;}> > > #ygrp-mlmsg pre, code {font:115% monospace;}
> > > #ygrp-mlmsg * {line-height: 1.22em;}> > > #ygrp-text{> > > font-
family: Georgia;> > > }> > > #ygrp-text p{> > > margin:0 0 1em 0;}> >
> #ygrp-tpmsgs{> > > font-family: Arial;> > > clear:both;}> > > #ygrp-
vitnav{> > > padding-top: 10px;font- family:Verdana; font-size:
77%;margin: 0;}> > > #ygrp-vitnav a{> > > padding:0 1px;}> > > #ygrp-
actbar{> > > clear:both;margin: 25px 0;white-space: nowrap;color:
#666;text- > align:right; }> > > #ygrp-actbar .left{> > >
float:left;white- space:nowrap; }> > > .bld{font-weight: bold;}> > >
#ygrp-grft{> > > font-family: Verdana;font- size:77%; padding:15px 0;}
> > > #ygrp-ft{> > > font-family: verdana;font- size:77%; border-top:
1px solid #666;> > > padding:5px 0;> > > }> > > #ygrp-mlmsg #logo{> >
> padding-bottom: 10px;}> > > > > > #ygrp-vital{> > > background-
color: #e0ecee;margin- bottom:20px; padding:2px 0 8px > 8px;}> > >
#ygrp-vital #vithd{> > > font-size:77% ;font-family: Verdana;font-
weight:bold; > color:#333; text-transform: uppercase; }> > > #ygrp-
vital ul{> > > padding:0;margin: 2px 0;}> > > #ygrp-vital ul li{> > >
list-style-type: none;clear: both;border: 1px solid #e0ecee;> > > }>
> > #ygrp-vital ul li .ct{> > > font-weight: bold;color:
#ff7900;float: right;width: 2em;text- > align:right; padding-
right: .5em;}> > > #ygrp-vital ul li .cat{> > > font-weight: bold;}>
> > #ygrp-vital a{> > > text-decoration: none;}> > > > > > #ygrp-
vital a:hover{> > > text-decoration: underline; }> > > > > > #ygrp-
sponsor #hd{> > > color:#999;font- size:77%; }> > > #ygrp-sponsor #ov
{> > > padding:6px 13px;background- color:#e0ecee; margin-bottom:
20px;}> > > #ygrp-sponsor #ov ul{> > > padding:0 0 0 8px;margin:0; }>
> > #ygrp-sponsor #ov li{> > > list-style-type: square;padding: 6px
0;font-size: 77%;}> > > #ygrp-sponsor #ov li a{> > > text-decoration:
none;font- size:130% ;}> > > #ygrp-sponsor #nc{> > > background-
color: #eee;margin- bottom:20px; padding:0 8px;}> > > #ygrp-
sponsor .ad{> > > padding:8px 0;}> > > #ygrp-sponsor .ad #hd1{> > >
font-family: Arial;font- weight:bold; color:#628c2a; font-size: >
100%;line- height:122% ;}> > > #ygrp-sponsor .ad a{> > > text-
decoration: none;}> > > #ygrp-sponsor .ad a:hover{> > > text-
decoration: underline; }> > > #ygrp-sponsor .ad p{> > > margin:0;}> >
> o{font-size: 0;}> > > .MsoNormal{> > > margin:0 0 0 0;}> > > #ygrp-
text tt{> > > font-size:120% ;}> > > blockquote{margin: 0 0 0 4px;}>
> > .replbq{margin: 4;}> > > -->> > > > > > > > > > > > > > > > > > >
> > > > > > > > ____________ _________ _________ _________ _________
_________ > _________ _________ _________> > > Looking for last
minute shopping deals? > > > Find them fast with Yahoo! Search.
http://tools. search.yahoo. > com/newsearch/ category. php?category=
shopping> > >> > > > > > > > > > >
__________________________________________________________>
______________> > Never miss a thing. Make Yahoo your home page. > >
http://www.yahoo.com/r/hs> >> > > > > > > ----------------------------
-----> Looking for last minute shopping deals? Find them fast with
Yahoo! Search.>
>
>
>
>
>
>
> _________________________________________________________________
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