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the whole mixture, and 72-8 (=6+2, the whole mixture) 9 cents, the worth of 1 quart of the mixture. When the price and quantities of the simples are given, and it is required to find the price of a given quantity of the mixture, as in the preceding example, it is called

ALLIGATION MEDIAL.

RULE.

208. Multiply each quantity by its price, and divide the sum of the products by the sum of the quantities, the quotient will be the rate of the compound required.

QUESTIONS FOR PRACTICE.

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lons of wine at 4s. 10d. a gal-
lon, with 12 gallons at 5s. 6.
and 8 at 6s. 34d. a gallon;
what is a gallon of the mixture
worth?
Ans. 5s. 7d.

4. If 5lb of tea at 6s. per lb. 8lb. at 5s. and 4lb. at 4s. 6d. be mixed together, what is a pound of the mixture worth? Ans. 5s. 2d.

5. A goldsmith melted together 10 oz. of gold 20 carats fine, 8 oz. 22 carats fine, and 1 lb. 8 oz. 21 carats fine; what is the fineness of the mixture? Ans. 2018 carats fine.

ALLIGATION ALTERNATE.

209. When the prices of the simples, and also the price, or rate of the mixture, are given, the method of finding the proportion, or quantities of the several simples, is called Alligation Allernate.

I. A person has tea worth 40 cents a pound, which he wishes to mix with tea worth 60 cents a pound, in such manner that the mixture shall be worth 50 cents a pound; in what proportion must it be mixed? Ans. Equal quantities of each; for the price of one kind exceeds the mean just as much as the price of the other falls short of it, the difference between the given rate and the mean being 5 in each case.

2. In what proportion must I mix currants worth 9 cents a pound; with currants worth 12 cents a pound, in order that the mixture may be worth 10 cents a pound? Here a pound at 9 cents falls one cent short of the mean, and a pound at 12 cents exceeds the mean 2 cents; hence 2 lb. at 9 cts. will fall short of the mean by the same quantity that one lb. at 12 cents exceeds it; we must therefore take twice as many of the 9 cent currants as we do of those worth 12 cents, in order that the mixture may be worth 10 cents.

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2. A has 7 cwt. of sugar at 8d. per pound, for which B gave him 12 cwt. of flour; what was the flour per pound?

Ans. 4 d. 3. How much tea at 9s. 4d. per pound, must be given in barter for 156 gallons of wine, at 12s. 3 d. per gallon?

Ans. 201lb. 13-5 54 Oz.

4. B delivered 3 hhds. of brandy at 6s. 8d. per gallon to C for 126 yds. of cloth; what was the cloth per yard? Ans. 108.

5. A has coffee which he barters with B at 10d. per pound more than it cost him, against tea, which stands B in 10s. the pound, but puts it at 12s. 6d. I would know how much the coffee cost at first.

Ans. 3s. 4d.

6. A and B barter; A has 150 gallons of brandy at $1.20 per gal. ready money, but in barter, would have $1.40; B has linen at 60 cts. per yard, ready money; how ought the

linen to be rated in barter, and how many yards are equal to A's brandy?

Ans. barter price, 70cts. and B must give A 300 yds.

7. C has tea at 78cts. per lb. ready money, but in barter, would have 93cts.; D has shoes at 7s. 6d. per pair, ready money; how ought they to be rated in barter, in exchange for tea? Ans. $1.49.

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?

Ans. the cotton 9 d. per lb. in barter, and 7cwt. Oqrs. 161b. of cotton must be given for 100 doz. of candles.

NOTE. The exchange of one commodity for another, is called Barter.

9. If 6 men build a wall 20 feet long, 6 feet high, and 4 feet thick, in 32 days; in what time will 12 men build a wall 100 feet long, 4 feet high, and 3 feet thick? Ans. 40 days.

10. If a family of 8 persons in 24 months spend $480; how nuch would they spend in 8 months, if their number were doubled? Ans. $32.

11. Three men hire a pas

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

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 weighing 10 lb. and by immersing them in water, find the quantities displaced by each to be 8 by the copper, 7 by the mixture, and 5 by the gold; what part of the mixture is gold, and what part copper? 8-2

7

5

3:10 ::

1

And

S2: 63 copper,
1: S 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 to be assessed on the inventory 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. $ inv. $ 3000000 : 35000 :: 24600: 287 Ans.

2. A certain school, cousisting of 60 scholars, is supported 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. $125 on the scholar, and A pays $3.75 per quarter.

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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 be 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:

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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 preperty is 8276? By the table it appears that $200 pay $6, that $70 pay $2. 10, and that $6 pay 18 cents. Thus $200 is $6 00

70 2.10

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Proceed in the same way to find each individual'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 trife larger than the amount to be raised, to compensate for the loss of fractions.

REVIEW.

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

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

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

3. Does changing the places of the two middle terms affect the pro portion? Why not?

4. What is meant by inverse proportion?

5. What is meant by the Single Rule of Three? What is the general rule for stating questions in the Rule of Three? How is the answer then found? If the first and third terms be of different denominations, what is to be done? What, if there are different denominations in the second term? Of what denomination will the quotient be? What, if the quotient be not of the same denomination of the required answer? What is the method of proof in this rule?

6. What is compound proportion? By what other name is it called? What is the rule for stating questions in compound proportion?-for performing the operation?

7. What is Fellowship? What is meant by capital 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.(21) 2. Fractions are of two kinds, Vulgar and Decimal, which differ in the form of expression and the modes of operation.

3. A Vulgar Fraction is expressed by two numbers, called the numerator and denominator, written the foriner over the latter, with a line between. (21)

4. A Decimal Fraction, or a Decimal, is a fraction, which denotes parts of a unit which become ten times smaller by each successive division,(113) and is expressed by writing down the numerator only. (Sec Part II. Sect. III.) A decimal is read in the same manner as a vulgar fraction; thus, 0.5 is read 5 tenths, 0.25, 23 hundredths, and it is put into the form of a vulgar fraction by drawing a line under it, and writing as many ciphers under the line as there are figures in the decimal, with a 1 at the left hand; thus, 0.5 becomes 5, 0,25,2%, and 0.005, 100

VULGAR FRACTIONS.

215. 1. A proper fraction is one whose numerator is less than its denominator; as,,,, &c.(23)

2. An improper fraction is one whose numerator is greater than its denominator; as, f, g, V, &c.(24)

3 8

3. The numerator and denominator of a fraction are called its terms.(30)

4. A compound fraction is a fraction of a fraction; as, J of 1.

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