Wool, bales, Smyrna 10 lbs. per bale

do. do. Hamburg 3 per cent.

actual.

do.

do do. South America 15 lbs. per bale do.

EXAMPLES.

1. Find the net weight of a hogshead of sugar, weighing gross 1228 lbs. Ans. 1075 lbs.

OPERATION.

1228 lbs. gross weight.
7 lbs. draft.

1221

12 per cent. of 1221 lbs. 146 lbs. tare.

1075 net weight.

NOTE. For draft and tare, let the pupil examine the preceding tables.

2. Required the net weight of 6 boxes of sugar, weighing gross, as follows-No. 1, 450 lbs.

OPERATION.

2914 lbs. gross.

No. 2, 470 do.

No. 3. 510 do.

No. 4, 496 do.

6X4 24 draft.

3. What is the net weight of 4 chests of tea, which weigh as

Ans. 384 lbs.

follows.

NOTE. — In making allowances, if there be a fraction of more than half a pound, 1 pound is added to the tare.

AMERICAN DUTIES.

The duties on merchandise imported into the United States are either specific or ad valorem duties.

Specific duty is a certain sum paid on a ton, hundred weight, pound, square yard, gallon, &c.; but when the duty is a certain per cent. on the actual cost of the goods in the country, from which they are imported, it is called an ad valorem duty.

6. What is the duty on 6 hogsheads of sugar, weighing gross, as follows; No. 1, 1276 lbs., No. 2, 1280 lbs., No. 3, 1178 lbs., No. 4, 1378 lbs., No. 5, 1570 lbs., No. 6, 1338 lbs., duty 2 cents. per lb. ? Ans. $175.52.5 7. What is the duty on an invoice of woolen goods, which cost in London 986£ sterling, at 44 per cent. ad valorem ? Ans. $1928.17.+ 8. Required the duty on 5 pipes of Port wine, gross gauge as follows; No. 1. 176 gallons, No. 2, 145 gallons, No. 3, 128 gallons, No. 4, 148 gallons, No, 5, 150 gallons; wants of each pipe, 4 gallons; duty 15 cents per gallon. Ans. $106.80. weighing 270 tons, Ans. 8100.

9. Required the duty on a cargo of iron,

at $30 per ton ?

10. Compute the duty on 7890 pounds of tarred cordage, at 4 cents per pound.

Ans. $310.08.

11. What duty should be paid on 10 casks of nails, weighing each 450 lbs. gross at 4 cents per lb ? Ans. $164.12.

15

SECTION XLIII.

RATIO.

RATIO is the mutual relation, which one magnitude or number, has to another of the same kind, without the intervention of a third. The ratio which the first of any two numbers has to the second, may be expressed by dividing the first number by the second. Thus the ratio of 10 to 5 = 10 = 2; and the ratio of 7 to 21==}. Quantities of different kinds have no ratio to each other; as no one would inquire how often 5 dollars were contained in 15 miles, or what part of 8 bushels were 2 minutes. A ratio shows how many times one number or term, is contained in another.

NOTE.The French mathematicians express the ratio, which one number has to another by dividing the consequent by the antecedent. The ratio of 10 to 5 they would write thus, and the ratio of 7 to 21, thus, 21—3. It is doubtful whether this innovation is productive of any good.

SECTION XLIV.

PROPORTION.

PROPORTION is the likeness or equalities of ratios. Thus, because 5 has the same relation or ratio to 10 that 8 has to 16, we say such numbers are in proportion to each other, and are therefore called proportionals.

If any four numbers whatever be taken, the first is said to have the same ratio or relation to the second, that the third has to the fourth, when the first number or term contains the second, as many times as the third contains the fourth; or when the second contains the first, as many times as the fourth does the third. Thus, 8 has the same ratio to 4 that 12 has to 6; because 8 contains 4, as many times as 12 does 6. And 3 has the same relation to 9, that 4 has to 12; because 9 contains 3, as many times as 12 does 4. Ratios are represented by colons; and the equalities of ratios by double colons.

398 24 is read thus ; 3 has the same ratio or relation to 9, as 8 to 24. The first and third numbers of a proportion are called antecedents, and the second and fourth are called consequents; also, the first and fourth are called extremes, and the second and third are called means.

Whatever four numbers are proportionals, if their antecedents or consequents be multiplied or divided by the same numbers, they are still proportionals; and if the terms of one proportion be multiplied or divided by the corresponding term of another proportion, their products and quotients are still proportionals.

This will appear evident, from the various changes, that the following example admits.

3:6 By division.

4X4: 8X8:33:6X6 By compound ratios.

By division.

That the product of the extremes is equal to that of the means, is evident from the following consideration. Let the following proportionals be taken. 12:38:2. From the definition of proportion, the first term contains the second, as many times as the third does the fourth; therefore, 12—, but

, and; and if 24, the numerator of the first fraction, which is a substitute for the first term, be multiplied by 6, the denominator of the second fraction, and a substitute for the fourth term, the product will be the same as if 6, the denominator of the first fraction, and a substitute for the second term, be multiplied by 24, the numerator of the second fraction, and a substitute for the third term. Thus 24 X 66 X 24. Therefore the product of the extremes iş, in all cases, equal to that of the means.

If, then, one of the extremes be wanting, divide the product of the means by one of the extremes; or, if one of the means be wanting, divide the product of the extremes by one of the

means.

To apply this, we will take the following question. If 5 yards of cloth cost $15, what will 7 yards cost? It is evident, that twice the quantity of cloth would cost twice the sum, and that three times the quantity, three times the sum, &c.— That is, the price will be in proportion to the quantity purchased. We then have three terms of a proportion given; one of the extremes and the two means, to find the other extreme.

Thus, 57:: 15. Therefore to find the other extreme by the rule above stated, we multiply the two means, 7 and 15, and divide their product by the extreme given, and the quotient is the extreme required. 7 × 15=105. 105-521 dollars, the answer required.

To perform this question by analysis, we reason thus. If 5 yards cost 15 dollars, 1 yard will cost one-fifth as much, which is 3 dollars; and if one yard cost 3 dollars, 7 yards will cost 7 times as much, which is 21 dollars.

From the illustrations above given, we deduce the following

RULE. *

State the question by making that number, which is of the same name or quality of the answer required the third term; then, if the answer required is to be greater than the third term, make the second term greater than the first; but if the answer is to be less than the third term, make the second less than the first.

Reduce the first and second terms to the lowest denomination, mentioned in either, and the third term to the lowest denomination mentioned in it.

Multiply the second and third terms together, and divide their product by the first, and the quotient is the answer in the same denomination to which the third is reduced.

If any thing remains after division, reduce it to the next lowest denomination, and divide as before.

If either of the terms consists of fractions, state the question, as in whole numbers, and reduce the mixed numbers to improper fractions, compound fractions to simple ones, and invert the first term, and then multiply the three terms continually together, and the product is the answer to the question. Or the fractions may be reduced to a common denominator; and their numerators may be used as whole numbers. For when frac

tions are reduced to a common denominator, their value is as their numerators.

*This rule was formerly divided into the Rule of Three Direct, and Rule of Three Inverse. The Rule of Three Direct, included those questions, where more required more and less required less, thus; If 5 lbs. of coffee cost 60 cents, what would be the value of 10 lbs., would be a question in the Rule of Three Direct, because the more coffee there was, the more money it would take to purchase it.

But if the question were thus. If 4 men can mow a certain field in 12 days, how long would it take 8 men; it would be in inverse, because the more men, the less would be the time to perform the labor, that is, more would require less. The method for stating questions was this; To make that number which is the demand of the question, the third term, that which is of the same name the first, and that which is of the same name as the answer required, the second

term.

If the question was direct, the first and second terms must be multiplied together, and their product divided by the first; but if it was inverse, the first and second terms must be multiplied together, and their product divided by the third.