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AMERICAN DUTIES.

Art. 164. DUTIES are taxes imposed by the government of a country upon most articles of merchandise imported from foreign countries. They are of two kinds, ad valorem and specific.

An ad valorem duty is a specified per cent. on the actual cost of the goods in the country from which they were imported.

A specific duty is a specified sum on a square yard, pound, gallon, &c.

Duties are to be paid only on the articles of merchandise, and not on the boxes, casks, bags, &c., which contain them; hence, certain deductions are to be made from their gross weight or measure, called allowances. These allowances are draft, tare, leakage, and breakage.

Gross weight is the whole weight of the goods, including the box, bag, &c., which contains them.

Neat weight is the weight of the goods after all allowances have been deducted.

Draft is an allowance made for waste, which is to be deducted from the gross weight, and is as follows:

On

112 lbs.

1 lb.

Above 112 lbs. and not exceeding 224 lbs. 2 lbs.

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Tare is an allowance for the actual or supposed weight of the cask, box, chest or bag, which contains the article of merchandise, and is to be deducted after the draft has been deducted.

Leakage is an allowance of 2 per cent., which is to be deducted from the gauge or measure of all liquids imported

in casks.

Breakage is an allowance of 10 per cent. on porter, ale, and beer; and of 5 per cent. on all other liquors, when imported in bottles. A dozen bottles of the common size are estimated to contain 2 gallons.

In deducting allowances, a fraction is disregarded, unless it exceeds one half, when it is considered a unit.

1. What is the amount of duty on an invoice of sheetings, imported from Liverpool, which cost £125. 10s. sterling, at 20 per cent. ad valorem? Ans. $121.484.

ALLIGATION.

Art. 165. ALLIGATION is the method of finding the mean value of a mixture which is composed of several ingredients, when the price and quantity of each of the ingredients are given; also, of finding the quantity of each of several ingredients, whose prices are given, that will be required to compose a mixture of a given mean value.

Art. 166. TO FIND THE MEAN VALUE OF A MIXTURE COMPOSED OF SEVERAL INGREDIENTS, WHEN THE PRICE AND QUANTITY OF EACH OF THE INGREDIENTS ARE GIVEN.

Suppose a farmer should mix 20 bushels of corn, worth 60 cents a bushel, with 20 bushels of oats, worth 30 cents a bushel; what would be the mean value of the mixture? $12.00, the value of 20 bushels of corn. $ 6.00, the value of 20 bushels of oats.

$.60 X 20
$.30 X 20

Bu. 40

=

$18.00, the total value of the mixture.

$18.00 ÷ 40=45 cents, the mean value required. Hence, the following rule.

RULE. Find the value of the several ingredients by multiplying the price of each by its quantity; then divide the sum of the several products by the sum of the several quantities, the quotient will be the mean value of the mixture.

1. A grocer mixed 2 pounds of tea, worth 50 cents a pound, with 3 pounds, worth 40 cents a pound; what was the mean value of the mixture?

2. If 4 pounds of sugar, worth 8 cents a pound, be mixed with 8 pounds, worth 10 cents a pound, what will be the value of each pound of the mixture?

3. A grocer mixed 125 pounds of sugar, worth 9 cents a pound, and 75 pounds, worth 8 cents a pound, with 50 pounds, worth 7 cents a pound; what was the mean value of each pound of the mixture? Ans. 8 cents.

4. A goldsmith melted together 4 ounces of gold of 15 carats fine, 5 ounces of 18 carats fine, and 6 ounces of 20 carats fine; what was the fineness of the mixture?

Ans. 18 carats fine.

NOTE. A carat is the twenty-fourth part of any quantity of gold, and the fineness of gold depends upon the number of carats of pure gold it contains. Thus, gold that is 18 carats fine, is composed of 18 parts of pure gold, and 6 parts of some other metal. The fineness of silver depends upon the number of ounces of pure silver in a pound.

Art. 167. TO FIND THE QUANTITY OF EACH OF SEVERAL INGREDIENTS, WHOSE PRICES ARE GIVEN, THAT WILL BE REQUIRED TO COMPOSE A MIXTURE OF A GIVEN MEAN VALUE.

Suppose we wish to mix tea that is worth 60 cents a pound, with tea that is worth only 40 cents a pound, in such proportions that the mixture shall be worth 50 cents a pound; it is plain that we must take equal quantities of each, because the price of one kind is as much greater than the mean price, as the price of the other is less.

Again; suppose we wish to mix sugar that is worth 7 cents a pound, with sugar that is worth 10 cents a pound, in such proportions that the mixture shall be worth 9 cents a pound.

The sugar that is worth 10 cents a pound is worth 1 cent more than the mean price, and the sugar that is worth 7 cents a pound is worth 2 cents less than the mean price; hence, it will be necessary to use twice as much sugar at 10 cents a pound as we use at 7 cents a pound.

A goldsmith has gold of 12, 15, 20, and 22 carats fine, and he wishes to mix the four kinds in such proportions that the mixture shall be 18 carats fine.

The proportion of each kind of gold may be found by writing the fineness of the several kinds under one another, and then connecting one that is less than the mean fineness with one that. is greater, and writing the difference between the fineness of each kind and the required mean fineness of the mixture, opposite to that with which it is connected; these differences will express the proportion or quantity of each kind.

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By connecting a less rate of fineness with a greater, and placing the differences between them and the mean rate alternately, there is precisely as much gained by one quantity as is lost by the other; therefore the gain and the loss upon the whole must be equal.

From the preceding illustrations we obtain the following

RULE. Write the rates or prices of the ingredients in a column under one another, and place the mean rate or price at the left of the column. Connect with a continued line, the price of each ingredient which is less than the mean rate

or price, with one or more that is greater; then write the difference between the price of each ingredient and the mean rate, opposite the price of the ingredient with which it is connected, and the difference, or the sum of the differences, if there is more than one opposite the price of each ingredient, will be the required quantity of that ingredient.

5. A grocer would mix tea at 35 cents, 40 cents, 50 cents, and 55 cents a pound, in such proportions that the mixture shall be worth 45 cents a pound; how many pounds of each kind must he use?

Ans. 10 lbs. at 35 cents, 5 lbs. at 40 cents, 5 lbs. at 50 cents, and 10 lbs. at 55 cents.

Questions of this kind admit of as many different answers as there are various ways of connecting the rates or prices of the ingredients, and all of them equally correct, which may be proved by Art. 165. Art. 168. WHEN ONE OF THE INGREDIENTS IS LIMITED TO

A GIVEN QUANTITY.

RULE. Find the proportion of each ingredient, as in Art. 166; then make the number opposite that ingredient whose quantity is given, a common denominator, and the number opposite each of the other ingredients, the numerator of a fraction; multiply the given quantity by each of these fractions, the several products will express the required quantities of the other ingredients.

6. A trader has 90 pounds of sugar worth 7 cents a pound, which he would mix with some at 8 cents a pound, some at 10 cents, and some at 12 cents a pound. How much of each kind must he mix with the 90 pounds to make a mixture worth 9 cents a pound?

Ans. 30 lbs. at 8 cts., 30 lbs. at 10 cts., and 60 lbs. at 12 cents.

Art. 169. WHEN THE WHOLE COMPOUND IS LIMITED TO A GIVEN QUANTITY.

RULE. Find the proportional part of each ingredient as in Art. 166; then make the sum of these proportional parts a common denominator, and the proportional part of each ingredient the numerator of a fraction; multiply the given quantity by each of these fractions; the several products will express the required quantity of each ingredient.

7. A grocer has three kinds of sugar, at 7, 8, and 10 cents a pound; what quantity of each kind must he take to fill a cask holding 230 pounds, so that the mixture shall be worth 9 cents a pound?

INVOLUTION.

Art. 170. INVOLUTION is multiplying a number by itself.

The number itself is called the first power, or root, and the product obtained by multiplying any number by itself is called the second power, or square of that number. If the second power or square be multiplied by the first power, the product is called the third power, or cube of that number.

The power of a number is sometimes indicated by a small figure placed at the right and a little above the number, called the exponent or index, and it shows how many times the number is to be used as a factor to produce the required power.

Thus, 52 = 5 X 5 53 5 × 5 × 5

=

545 × 5 × 5 × 5: Any required power of a

25, the second power of 5. 125, the third power of 5. 625, the fourth power of 5. number may be found by con

tinually multiplying each succeeding higher power by the given number until the required power is obtained.

Any required power of a fraction may be found in a similar manner; thus, the second power of 2 is 2x=; the third power is 1. The second power of the decimal fraction .3 is .3 .3.09; the third power is .09 X.3=.027.

If the power of a mixed number be required, reduce it to an improper fraction, or reduce the fractional part to a decimal, and then involve it to the required power.

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