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Alligation Alternate is the reverse of Alligation Medial, and may be proved by it Questions under this rule admit of as many different answers as there are different ways of linking.

211. When the whole composition is limited to a certain quantity. RULE. Find the differences by linking as before; then say, As the sum of the quantities or differences, thus determined is to the given quantity :: so is each of the differences: to the required quantity of that rate.

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6. How much water at 0 cts. per gallon, must be mixed with brandy at $1.25 per gallon, so as to fill a vessel of 80 gallons, and that a gallon of the mixture may be worth $1?

25

100 (1.25-100

gal. gal.

125:80::

1.25

Ans.

QUESTIONS FOR PRACTICE.

oz.

gal. gal.

Given quantity 80

7. How much silver of 15, of 17, of 18, and 22 carats fine, must be melted together to form a composition of 40 oz. 20 carats fine?

5 of 151

5 of 17 5 of 18 25 of 22

8. A grocer would mix teas at 3s., 4s., and 4s. 6d. per lb.; and would have 30 lb. of the mixture worth 3s. 6d. per lb. how much of each must he take?

car. fine.

Ans.

16 water.

9. How of

{100: 64 brandy. water worth 03. per gallon,

lb.

18 at 3s.

at 4s.

6 at 4s. 6d.

must be mixed with wine worth
3s. per gallon, so as to fill a
cask of 100 gallons, and that
a gallon of the mixture may
be afforded at 2s. 6d. ?

gall.
Ans. 831 wine.

water.

212. When one of the simples is limited to a certain quantity RULE. Find the differences as before; then, As the difference standing against the given quantity is to the given quantity :: so are the other differences, severally,: to the several quantities required.

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8. C. has candles at 6s. per dozen, ready money; but in barter he will have 6s. 6d. per dozen; D has cotton at 9d. per lb. ready money; what price must the cotton be at in barter, and how much cotton must be bartered for 100 dozen of candles?

ture for $48; A puts in 80
sheep for 4 months, B 60 sheep
for 2 months, and C 72 sheep
for 5 months; what share of
the rent ought each to pay?
A $19.20

B 7.20 Ans.
C 21.60

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12. If I have a mass of pure gold, a mass of pure copper,

and a mass, which is a mixture
of gold and copper, each weigh-
ing 10 lb., and by immersing
them in water, find the quanti-
ties displaced by each to be 8
by the copper, 7 by the mix-
ture, and 5 by the gold; what
part of the mixture is gold, and
what part copper?
8-2

7

{i

And

3:10: :

S2: 63 copper
{1:31 gold.

This is the celebrated problem of Archimedes, by which he detected the fraud of the artist employed by Hiero, king of Syracuse, to make him a crown of pure gold (211).

ASSESSMENT OF TAXES.

1. Supposing the Legislature should grant a tax of $35000

of all the rateable property in
the State, which amounts to
$3000000, what part of it
must a town pay, the inventory
of which is $24600?
$ inv. $ tax.
8 inv. $.
3000000: 35000 :: 24600:287

2. A certain school, consisting of 60 scholars, is supportto be assessed on the inventoryed on the polls of the scholars, and the quarterly expense of the whole school is $75; what is that on the scholar, and what does A pay per quarter, who has 3 scholars?

Ans.

Ans. $1.25 on the scholar, and A pays $3.75 per quarter.

3. If a town, the inventory of which is $24600, pay $287, what will A's tax be, the inventory of whose estate is $525.75?

24600.00 287 :: 525.75 :
$6.133 Ans.

4. The inventory of a certain school district is $4325, and the sum to be raised on this inventory for the support of schools, is $86.50; what is

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$1 pays .03

"L .06

"

"

"

"L

7 ""

8 26 .24

9 " 10 "L

.09

.12

.15

.18

.21

213. In assessing taxes, it is generally best, first to find what each dollar pays, and the product of each man's inventory, multiplied by this sum, will be the amount of his tax. In this case, the sum on the dollar, which is to be employed as a multiplier, must be expressed as a proper decimal of a dollar, and the product must he pointed according to the rule for the multiplication of decimals (122); thus 2 cents must be written .02, 3 cents, .03, 4 cents, .04, &c. It is sometimes the practice to make a table by multiplying the value on the dollar by 1, 2, 3, 4, &c. as follows:

.27

.30

Thus $200 is 6.00

70 is 2,10
6 is 0.18

276 $8,28

$100 pays 3.00

200 66

6.00 300 66 9.00 12.00

400

66 "L

500

15,00

600 "L

18.00

2.10

700 "L

2.40

800 66

2.70

"L

900 1000 66

3.00

This table is constructed on the supposition that the tax amounts to three cents on the dollar, as in example 5th. USE.-What is B's tax, whose rateable property is $276? By the table, it appears that $200 pay $6, that $70 pay $2.10, and that $6 pay 18 cents.

Proceed in the same way to find each indi vidual's tax, then add all the taxes together, and if their amount agree with the whole sum proposed to be raised, the work is right. It is sometimes best to assess the tax a trifle larger than the amount to be raised, to compensate for the loss of fractions.

B's tax.

$10 pays

TABLE. .30 20 46 .60

30

"

.90

40 ་

1.20

1.50

1.80

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60 70

280

90

100

that on the dollar, and what is
C's tax, whose property inven-
tories at $76.44?
$4325 86.50

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& 76.44X.02 $1.528, C's tax. 5. If a town, the inventory of which is $16436, pay a tax of $493.08, what is that on the dollar?

$16436: $493.08 :: 1:.03 cts. Ans.

How

1. What is meant by ratio? is ratio expressed? What is the first term called? the second term?

1.02 cts. Ans.

REVIEW,

2. What is proportion? What general truth is stated respecting the

21.00

24.00

27.00

30.00

four terms of a proportion? How is this truth shown?

Does changing the place of the two middle terms affect the proportion? Why not?

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7. What is Fellowship? What is meant by canital or stock? What by dividend? What is the rule when the times are equal? What, when they are unequal? What is the method of proof?

8. What is Alligation? What is Alligation Medial?-Alligation Alternate? What is the rule for finding the proportional quantities to form a mixture of a given rate? Explain by analysis of an example. When the whole composition is limited to a certain quantity, how would you proceed? How, when one of the simples is limited to a certain quantity? How is Alligation proved?

9. What is Barter? What is meant by a tax? What is the common method of making out taxes?

SECTION VII.

Fractions.

DEFINITIONS.

214. 1. Fractions are parts of a unit, or of a whole of any kind.

If any number, or particular thing, be divided into two equal parts, those parts are called halves; if into 3 equal parts, they are called thirds; if inte 4 equal parts, they are called fourths, or quarters (11); and, generally, the parts are named from the number of parts into which the thing, or whole, is divided. If any thing be divided into 5 equal parts, the parts are called fifths; if into 6, they are called sixths; if into 7, they are called sevenths; and so on. These broken, or divided quantities are called fructions. Now if an apple be divided into five equal parts, the value of one of those parts would be one fifth of the apple, and the value of two parts two fifths of the apple, and so on. Thus we see that the name of the fraction shows, at the same time, the number of parts into which the thing, or whole, is divided, and how many of those parts are taken, or signified by the fraction. Suppose I wished to give a person two fifths of a dollar; I must first divide the dollar into five equal parts, and then give the person two of these parts. A dollar is 100 cents-100 cents divided into 5 equal parts, each of those parts would be 20 cents. Hence, one fifth of 100 cents, or of a dollar, is 20 cents, and two fifths, twice 20, or 40 cents.

The tediousness and inconvenience of writing fractions in words has led to the invention of an abridged method of expressing them by figures. One hof is written, one third, }, two thirds,, &c. The figure below the Lue shows the number of parts into which the thing, or whole, is divided, and the figure above the line shows how many of those parts are signified by the fraction. The number below the line gives name to the fraction, and is therefore called the denominator; thus, if the number below the line be 3, the parts signified are thirds, if 4, fourths, if 5, fifths, and so on. The number written above the line is called the numerator, because it enume

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