132 A £ I T H 
A LIIG AT I ON. 
When corn, wine, fpices, metals, &c. are to be mixed 
together, the method of proportioning fuch mixture is 
called the rule of Alligation ; and is of two kinds, medial 
atid alternate. 
AfucATiox Medial is the method of finding the 
rate or quality of the compofition, from having given the 
rates-and quantities of the fimples or ingredients. The 
rule of operation is this: Multiply each quantity by its 
rate, and add all the products together; then divide the 
film of the products by the fum of the quantities, or whole 
Compound, arid the quotient will be the rate fought. For 
example, Suppofe it were required to mix together 6 gal¬ 
lons of wine, worth 53. a gallon; 8 gallons, worth 6 s. the 
gallon ; and 4 gallons, worth 8s. the gallon ; and to find 
the worth or value, per gallon, of the whole mixture. 
Gall. s. Products. 
Here 6 mult, by 5 gives 30 
8 by 6 48 
_4 by 8 
Whole comp. 18 11 o fum of products 
Then iS) 110(6-^-or 6*s. is the rate fought. 
108 
2 
Alligation Alternate, is the method of finding 
the quantities of ingredients or fimples, neceffary to form 
a compound of a given rate. The rule of operation is 
this : 1 ft, Place the given rates of the fimples in a column, 
under each other; noting which rates are lefs, and which 
are greater, than the propofed compound. 2d, Connect 
or link, with a crooked line, each rate which is lefs than 
the propofed compound rate, with one or any number of 
thofe which are greater than the fame ; and every greater 
rate with one or any number of the lefs ones. 3d, Take 
the difference between the given compound rate and that 
of each fimple rate, and fet this difference oppofite every 
rate with which that one is linked. 4th, Then, if only 
one difference ftand oppofite any rate, it will be the quan¬ 
tity belonging to that rate; but, when there are fevera'l 
differences to any one, take their fum for its quantity. 
For example, Suppofe it were required to mix together 
gold of various degrees of finenefs, viz. of 19, of 21, and 
of 23, caradts fine, l'o that the mixture fhall be of 20 ca- 
radts fine. Hence, 
Comp, rate 20 
Rates 
DiiFs. Sums of Differences 
1 f 1 of 2i caradts fine 
i-4-3< 4 of 19 caradts fine 
1 ^ 1 of 23 caracts fine 
That is, there mud be an equal quantity of 21 and 23 ca¬ 
radts fine, and 4 times as much of 19 caradts fine. Various 
limitations, both of the compound and the ingredients, 
may be conceived ; and, in fuch cafes, the differences are 
to be altered proportionally. Queftions of this fort are 
however commonly belt and ealieff refolved by common 
Algebra, of which they form a fpecies of indeterminate 
problems, as they admit of many or an indefinite number 
of anl'wers. 
VULGAR FRACTIONS. 
A fradt ion is fome part or parts of an unit, or any other 
number or quantity whatever, and confifls of two numbers 
or terms, called the numerator, and the denominator, placed 
one above another with a line of feparation between them, 
as / 3 ’y imerat01 "1 which is called three-fourths, the 
\4 denominator J 
numerator being firlt named, then the denominator. The 
denominator fhews how many parts the integer is divided 
into; and the numerator expreffes how many of thofe parts 
the fradtion confifls of. Therefore with a given denomi¬ 
nator, if you increafe the numerator, you increafe the va¬ 
lue of the fradtion; but, with a given numerator, if you 
M £ T I C. 
increafe the denominator, you diminifh its value; for the 
value of every fradtion is diredtly as its numerator, and 
inverfely as its denominator. 
A proper fraction is that vvhofe numerator is lefs than 
its denominator, as orf, or-*-, orf, Sec. 
An improper fraction, is that whofe denominator is lefs 
than or equal to the numerator, as 4 -, or Z-, Sec. 
A Jingle fraction, is that which confifls but of one nu¬ 
merator and one denominator, as f, or-Z, Sec. 
A compound fraction, or fraction of a fraction, is that 
whofe parts are vulgar fradtions connedled by the word of. 
as i of £ of f of 7-t. 
A complex fradtion is that whofe numerator or denomi¬ 
nator, or both, is a vulgar fradtion, or mixed number, as 
5 ? 
or 
8 ” sl '“ 9 f 
A mixed number is a whole number with a fradtion 
ne.xed, as 9 |, 17R, &c. 
A traction is laid to be inverted, when the numerator is 
fet in the place of the denominator, and the denominator- 
in the place of the numerator, as inverted becomes f, 
Scholium. In any fradtion, as £, the numerator may 
he confidered as a dividend, the denominator as a divifor^ 
and A reprefents the quote. Hence fradtions whofe nume¬ 
rator are lefs than, equal to, or greater, than their denomi¬ 
nator, are refpedtively lefs than, equal to, or greater than, 
unity or 1. 
Lemma I. To multiply a fradtion by a whole number 
is to multiply the numerator, but to divide a fradtion by a 
whole number is to multiply the denominator, by the faid 
number, as for example: 
Let | both be multiplied and divided by 9. 
-± 42 = lf theprodua. ' 
Thus ■> 7 7 
7 
Sec. 
an- 
4 4 
the quotient. 
-7X9 , 6 3 
Lemma II. To multiply one fradtion by another, is to 
multiply the numerators together for the numerator, and 
the denominators together for the denominator of the pro. 
dudt; but to divide one fradtion by another, is to multi¬ 
ply the dividend into the divifor inverted; as for example, 
Let £ be multiplied and divided by £. 
Thus^I=— the produdi, and — the quotient. 
6X9 54 6x7 42 
REDUCTION of VULGAR FRACTIONS. 
By Redudtion, fradtions are converted out of one form, 
or denomination, into another, for the greater eafe in work¬ 
ing, but flill always retaining the fame value. 
Problem I. To reduce a fraBion into another of equal 
value. Multiply or divide both terms of the fradtion by 
one and the fame number, and you will have a new frac¬ 
tion equivalent to that given. 
Ex. Reduce A- to two other fradtions, each of the fame 
value. 
— The new fradtions each of the 
96 ^ fame value with that is, 
_3 ’ 
^ 24 -r 3 ~ 8 
Problem II. To reduce a whole number to the form of a 
fraBion. Place 1 under it for a denominator 
Ex. Reduce 3, 6, 7, 8, and u, each to the form of a 
fradtion. Thus L, z., - 2 ., <J. 
Problem III. To reduce a whole number to a fraBion , 
with a given denominator. Multiply the whole number by 
the given denominator, and under the produdt write the 
fame denominator. 
Ex. Reduce 3, 4, and 5, each to a fradtion whofe deno¬ 
minator may be 9 . 
J 9X4 
24 X 4 
I 9~r—3 _ 3 _ 
2*V—ff=f. This Ihould be 
well remembered. 
Thus 
3X9 2 7 4 X 9 36 
and 
5 X 9 
9 . 9 
fradtions required. 
Problem IV. 
. 4 5 
' 9 
the 
To reduce a mixed number to an improper 
fraBion „ 
