Ratio
The
ratio of width to height of standard-definition television.
In mathematics, a ratio is a relationship between two
numbers of the same kind[1] (i.e., objects, persons,
students, spoonfuls, units of whatever identical dimension), usually expressed
as "a to b" or a:b, sometimes expressed
arithmetically as a dimensionless quotient of the two[2] which explicitly indicates how
many times the first number contains the second (not necessarily an integer).[3]
Notation and terminology
The ratio of numbers A and B can be expressed as:[4]
§ the ratio of A to B
§ A is to B
§ A:B
§ A rational number which is the quotient of A divided by B
The numbers A and B are sometimes called terms with A being the antecedent and B being the consequent.
The proportion expressing the equality of the ratios A:B and C:D is written A:B=C:D or A:B::C:D. this latter form, when spoken or written in the English
language, is often expressed as
A is to B as C is to D.
Again, A, B, C, D are called the terms of the
proportion. A and D are called the extremes, and B and C are called the means. The equality of three or more proportions is
called a continued proportion.[5]
History and etymology
It is impossible to trace the origin of the concept of ratio, since the ideas from
which it developed would have been familiar to preliterate cultures. For
example, the idea of one village being twice as large as another is so basic
that it would have been understood in prehistoric society.[6] However, it is possible to
trace the origin of the 'number' "ratio" to the Ancient Greek λόγος (logos). Early translators rendered this into Latin as ratio ("reason"; as in the
word "rational"). (A rational number may be expressed as the quotient
of two integers.) A more modern interpretation of Euclid's meaning is more akin
to computation or reckoning.[7] Medieval writers used the word proportio ("proportion") to
indicate ratio and proportionalitas("proportionality")
for the equality of ratios.[8]
Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers.[9] The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus
. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.[10]The existence of multiple theories seems unnecessarily
complex to modern sensibility since ratios are, to a large extent, identified
with quotients. This is a comparatively recent development however, as can be
seem from the fact that modern geometry textbooks still use distinct
terminology and notation for ratios and quotients. The reasons for this are twofold.
First, there was the previously mentioned reluctance to accept irrational
numbers as true numbers. Second, the lack of a widely used symbolism to replace
the already established terminology of ratios delayed the full acceptance of
fractions as alternative until the 16th century.[11]
Euclid's definitions
Book V of Euclid's Elements has 18 definitions, all of which relate to
ratios.[12] In addition, Euclid uses ideas
that were in such common usage that he did not include definitions for them.
The first two definitions say that a part of a quantity is another quantity which
"measures" it and conversely, a multiple of a quantity is another quantity which it
measures. In modern terminology this means that a multiple of a quantity is
that quantity multiplied by an integer greater than one and a part of a
quantity (meaning aliquot part) is that which, when multiplied by an integer greater than
one, gives the quantity. Euclid does not define the term "measure" as
used here but one may infer that if a quantity is taken as a unit of
measurement, and a second quantity is given as an integral number of these
units, then the first quantity measures the second. Note that these
definitions are repeated, nearly word for word, as definitions 3 and 5 in book
VII.
Definition 3 describes what a ratio is in a general way. It
is not rigorous in a mathematical sense and some have ascribed it to Euclid's
editors rather than Euclid himself.[13] Euclid defines a ratio to be
between two quantities of the
same type, so by
this definition the ratios of two lengths or of two areas are defined, but not
the ratio of a length and an area. Definition 4 makes this more rigorous. It
states that a ratio of two quantities exists when there is a multiple of each
which exceeds the other. In modern notation, a ratio exists between quantities p and q if there exist integers m and n so that mp>q and nq>m. This condition is
known as the Archimedean property.
Definition 5 is the most complex and difficult; it defines
what it means for two ratios to be equal. Today, this can be done by simply stating
that ratios are equal when the quotients of the terms are equal, but Euclid did
not accept the existence of the quotients of incommensurables, so such a
definition would have been meaningless to him. Thus, a more subtle definition
is needed where quantities involved are not measured directly to one another.
Though it may not be possible to assign a rational value to a ratio, it is
possible to compare a ratio with a rational number. Specifically, given two
quantities, p and q, and a rational number m/n we can say that the ratio of p to q is less than, equal to, or
greater than m/n when np is less than, equal to, or
greater than mq respectively. Euclid's definition of equality can
be stated as that two ratios are equal when they
behave identically with respect to being less than, equal to, or greater than
any rational number. In modern notation this says that given quantities p, q, r and s, then p:q::r:s if for any positive integers m and n, np<mq, np=mq, np>mq according asnr<ms, nr=ms, nr>ms respectively. There is a
remarkable similarity between this definition and the theory of Dedekind cuts used in the modern definition
of irrational numbers.[14]
Definition 6 says that quantities that have the same ratio
are proportional or in proportion. Euclid uses the Greek ἀναλόγον (analogon), this has the same root
as λόγος and is related to
the English word "analog".
Definition 7 defines what it means for one ratio to be less
than or greater than another and is based on the ideas present in definition 5.
In modern notation it says that given quantities p,q, r and s, then p:q>r:s if there are positive integers m and n so that np>mq and nr≤ms.
As with definition 3, definition 8 is regarded by some as
being a later insertion by Euclid's editors. It defines three terms p, q and r to be in proportion when p:q::q:r. This is extended
to 4 terms p, q, r and s as p:q::q:r::r:s,
and so on. Sequences which have the property that the ratios of consecutive
terms are equal are called Geometric progressions. Definitions 9 and 10 apply this, saying that if p, q and r are in proportion then p:r is the duplicate ratio of p:q and p, q, r and s are in proportion then p:s is the triplicate ratio of p:q. If p, q and r are in proportion then q is called a mean proportional to (or the geometric mean of) p and r. Similarly, if p, q, r and s are in proportion then q and r are called two mean proportionals to p and s.
Examples
The quantities being compared in a ratio might be physical
quantities such as speed, or may simply refer to amounts of particular objects.
A common example of the latter case is the weight ratio of water to cement used in concrete, which is commonly stated as
1:4. This means that the weight of cement used is four times the weight of
water used. It does not say anything about the total amounts of cement and
water used, nor the amount of concrete being made. Equivalently it could be
said that the ratio of cement to water is 4:1, that there is 4 times as much
cement as water, or that there is a quarter (1/4) as much water as cement..
Older televisions have a
4:3 "aspect ratio", which means that the width is 4/3 of the height;
modern widescreen TVs have a 16:9 aspect ratio.
Fraction
If there are 2 oranges and 3 apples, the ratio of oranges to
apples is 2:3, whereas the fraction of oranges to total fruit is 2/5.
If orange juice concentrate is to be diluted with water in
the ratio 1:4, then one part of concentrate is mixed with four parts of water,
giving five parts total; the fraction of concentrate is 1/5 and the fraction of
water is 4/5.
Number of terms
In general, when comparing the quantities of a two-quantity
ratio, this can be expressed as a fraction derived from the ratio. For example,
in a ratio of 2:3, the amount/size/volume/number of the first quantity will be 
that of the second quantity.
This pattern also works with ratios with more than two terms. However, a ratio
with more than two terms cannot be completely converted into a single fraction;
a single fraction represents only one part of the ratio since a fraction can
only compare two numbers. If the ratio deals with objects or amounts of
objects, this is often expressed as "for every two parts of the first
quantity there are three parts of the second quantity".
Percentage ratio
If we multiply all quantities involved in a ratio by the same
number, the ratio remains valid. For example, a ratio of 3:2 is the same as
12:8. It is usual either to reduce terms to the lowest common denominator, or to express them in parts per hundred (percent).
If a mixture contains substances A, B, C & D in the ratio
5:9:4:2 then there are 5 parts of A for every 9 parts of B, 4 parts of C and 2
parts of D. As 5+9+4+2=20, the total mixture contains 5/20 of A (5 parts out of
20), 9/20 of B, 4/20 of C, and 2/20 of D. If we divide all numbers by the total
and multiply by 100, this is converted to percentages: 25% A, 45% B, 20% C, and
10% D (equivalent to writing the ratio as 25:45:20:10).
Proportions
If the two or more ratio quantities encompass all of the
quantities in a particular situation, for example two apples and three oranges
in a fruit basket containing no other types of fruit, it could be said that
"the whole" contains five parts, made up of two parts apples and
three parts oranges. In this case, 
,
or 40% of the whole are apples and 
, or 60% of the whole are
oranges. This comparison of a specific quantity to "the whole" is
sometimes called a proportion. Proportions are sometimes expressed as percentages as demonstrated above.
Reduction
Note that ratios can be reduced (as
fractions are) by dividing each quantity by the common factors of all the
quantities. This is often called "cancelling." As for fractions, the
simplest form is considered to be that in which the numbers in the ratio are
the smallest possible integers.
Thus the ratio 
may be considered
equivalent in meaning to the ratio 
within contexts
concerned only with relative quantities.
Mathematically, we write: "" 
"" (dividing both quantities by 20).
Grammatically, we would say,
"40 to 60 equals 2 to 3."
An alternative representation is: "40:60::2:3"
Grammatically, we would say,
"40 is to 60 as 2 is to 3."
A ratio that has integers for both quantities and that cannot
be reduced any further (using integers) is said to be in simplest form or lowest terms.
Sometimes it is useful to write a ratio in the form 
or 
to enable comparisons of
different ratios.
For example, the ratio 
can be written as 
(dividing both sides by
4)
Alternatively, can be written as 
(dividing both sides by
5)
Where the context makes the meaning clear, a ratio in this
form is sometimes written without the 1 and the colon, though, mathematically,
this makes it a factor or multiplier.
Dilution ratio
Ratios are often used for simple dilutions applied in
chemistry and biology. A simple dilution is one in which a unit volume of a
liquid material of interest is combined with an appropriate volume of a solvent
liquid to achieve the desired concentration. The dilution factor is the total
number of unit volumes in which your material will be dissolved. The diluted
material must then be thoroughly mixed to achieve the true dilution. For
example, a 1:5 dilution (verbalize as "1 to 5" dilution) entails
combining 1 unit volume of solute (the material to be diluted) + 4 unit volumes (approximately) of the solvent to give 5 units
of the total volume. (Some solutions and mixtures take up slightly less
volume than their components.)
The dilution factor is frequently expressed using exponents:
1:5 would be 5e−1 (5−1 i.e. one-fifth:one); 1:100 would be 10e−2 (10−2 i.e. one hundredth:one), and so on.
There is often confusion between dilution ratio (1:n meaning 1 part solute to n parts solvent) and dilution
factor (1:n+1) where the second number (n+1) represents the total volume of solute + solvent. In
scientific and serial dilutions, the given ratio (or factor) often means the
ratio to the final volume, not to just the solvent. The factors then can easily
be multiplied to give an overall dilution factor.
In other areas of science such as pharmacy, and in
non-scientific usage, a dilution is normally given as a plain ratio of solvent
to solute.
Odds
Odds (as in gambling) are expressed as a ratio. For
example, odds of "7 to 3 against" (7:3) mean that there are
seven chances that the event will not happen to every three chances that it
will happen.
Different units
Ratios are unit-less when
they relate quantities which have units of the same dimension.
For example, the ratio 1 minute : 40
seconds can be
reduced by changing the first value to 60 seconds. Once the units are the same,
they can be omitted, and the ratio can be reduced to 3:2.
In chemistry, Mass concentration "ratios" are usually
expressed as w/v percentages, and are really
proportions.
For example, a concentration of 3% w/v usually means 3g of substance in every 100mL of solution. This cannot easily be converted to a pure ratio because of density considerations, and the second figure is the total amount, not the volume of solvent.
http://www.khanacademy.org/video/multiplying--mixed-numbers?playlist=Developmental%20Math
http://www.khanacademy.org/video/multiplying-fractions-word-problem?playlist=Developmental%20Math
http://www.khanacademy.org/video/dividing-mixed-numbers?playlist=Developmental%20Math
http://www.khanacademy.org/video/dividing-mixed-numbers-and-fractions?playlist=Developmental%20Math
http://www.khanacademy.org/video/dividing-fractions-word-problem?playlist=Developmental%20Math