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54. If the transportation of 12 cwt. 2qr. 6 lb, 275 miles cost $27.78, how far, at that rate, may 3 tons Ocwt. 3gr. be carried, for $234.78?

55. A cistern 17] feet in length, 10 feet in breadth, and 13 feet deep, holds 546 barrels of water. Then how many barrels will fill a cistern, that is 16 feet long, 7 feet broad, and 15 feet deep?

56. If 25 pears can be bought for 10 lemons, and 28 lemons for 18 pomegranates, and 1 pomegranate for 48 almonds, and 50 almonds for 70 chestnuts, and 108 chestnuts for 2, cents, how many pears can I buy for $1.35?

57. In how many days, working 9 hours a day, will 24 men dig a trench 420 yards long, 5 yards wide, and 3 yards deep; if 248 men, working 11 hours a day, in 5 days dug a trench 230 yards long, 3 yards wide, and 2 yards deep?

58. If the interest on 347 dollars for 3 years be 72 dollars 87 cents, what will be the interest, at the same rate, on 537 dollars for 2 years ?

59. What must be paid for the carriage of 4cwt., 32 miles, if the carriage of 8cwt., 123 miles, cost 12 dollars 80 cents ?

60. By working 9 hours a day, 5 men hoed 18 acres of corn in 4 days. How many acres will 9 men hoe, at that rate, in 3 days, working 10 hours a day?

61. One pound of thread makes 2 yards of linen cloth, 5 quarters wide. Then how many pounds of thread will be required to make 50 yards of linen yd. wide ?

62. If 6 men, working 7 hours a day, mowed 28 acres of

grass in 4 days, how many men, at that rate, will mow 16 acres in 8 days, working 6 hours a day?

63. If 5 men can make 300 pair of boots in 40 days, how many men must be employed to make 900 pair in 60 days?

64. If 3 compositors set 151 pages in 2 days, how many will be required to set 692 pages in 6 days?

65. If 36 yards of cloth, 7 quarters wide be worth $ 98, what is the value of 120 yd. of cloth of equal texture, but only 5 quarters wide ?"

XXVI.

CONJOINED PROPORTION.

CONJOINED PROPORTION--called by merchants, The Chain Rule-consists of a series of terms bearing a certain proportion to each other, and so connected, that a comparison is instituted between two of the terms, through the medium of all the others.

The principles of this rule are included in proportion. The rule is chiefly employed in the higher operations of exchange, arbitrations of bullion, specie and merchandise. For the purpose of elucidation, however, we propose the following familiar example.

If 3 lb. of tea be worth 4 lb. of coffee, and 6 lb. of coffee worth 201b. of sugar, how many pounds of sugar may be had for 9 lb. of tea?

This question, we know, may be solved by a statement in compound proportion; but the following is the solution by conjoined proportion.

Distinguish the several terms into antecedents and consequents, and connect them by the sign of equality in the way of equations, as follows.

First, enter on the right the given sum or term on which the operation is to be performed, (which in the foregoing question is 9 lb. of tea) and call this the term of demand

Secondly, on the left of this term, and a line lower, enter the first antecedent, which must be of the same kind or name with the term of demand, and equal in value to the annexed consequent.

Thirdly, in the same manner, let the second antecedent be of the saine name with the second consequent, and equal in value to the third consequent: and so on, for any number of terms.

Fourthly, the terms being thus arranged, divide the product of the consequents by the product of the antecedents, and the quotient will be the answer in the denomination of the last consequent, or odd term.

20X409

720
18

–40 lb. sugar,

9 lb. of tea, term of demand 3lb. tea

4 lb. of coffee.
6 lb. coffee =20lb. sugar, the odd term.
Hence

the answer. 6X3 By the above example it will be seen, that in the arrangement of antecedents and consequents, each sort is entered twice, except that in which the answer is required, and which is called the odd term.

It should also be observed, that no two entries of the same denomination are in the same column; and, as they ire placed in the way of equations, it is evident that the juantities on each side, which are equal in value to one another, are cancelled in the operation; and, therefore ine quotient or answer will obviously be in the denomination of the last consequent, which is the odd term.

This rule may be proved by reversing the operation; taking the answer as the term of demand, and making the first antecedent the last consequent or odd term, as follows

40lb. of sugar.
20 lb.
sugar

6 lb. coffee.
4 lb. coffee 3 lb. tea.
Ther

9 lb. of tea, the proof. The operation may be abridged by omitting such num. bers as are the same in both columns, whenever such instances exist.

When fractions occur, the most convenient method is to convert them into whole numbers. Thus, an antecedent of 12 and a consequent of 9 may be changed (by multiplying both by 12) into 7 and 108, and the ratio will not be altered. So 5 and 11 have the same ratio with 20 and 47.

The rule may be exemplified by a question in reduction; thus, — It is required to reduce 2 tons to ounces.

2 tons, term of deinand. 1 ton

= 20cwt.
1 cwt. Zo

28 lb.
1 lb. 16 oz. the odd term.
Then

=71690 ounces, the answer.

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1 If 17 lb. of raisins are worth 20 lb. of almonds, and 5lb. of almonds worth 8 lb. of figs, and 37 lb. of figs worth 30 lb. of tamarinds, how many pounds of tamarinds are equal in value to 42} lb. of raisins ?

2. Suppose 100 lb. of Venice weight is equal to 70 lb. of Lyons, and 60 lb. of Lyons to 50lb. of Rouen, and 20!b. of Rouen to 25 lb. of Toulouse, and 50 lb. of Toulouse to 37 lb. of Geneva; then how many pounds of Geneva are equal to 25 lb. of Venice?

3. If i French crown is equal in value to 80 pence of Holland, and 83 pence of Holland to 48 pence English, and 40 pence English to 70 pence of Hamburgh, and 64 pence of Hamburgh to 1 form of Frankfort, how many florins of Frankfort are equal to 166 French crowns ?

4. If A can do as much work in 3 days as B can do in 4 days, and B as much in 9 days as C in 12 days, and C as much in 10 days as D in 8 days, how many days' work of D are equal to 5 days' work of A?

5. If 70 braces at Venice are equal to 75 braces at Leghorn, and 7 braces at Leghorn are equal to 4 yards in the United States, how many braces at Venice are equal to 64 yards in the United States ?

6. A merchant in St. Petersburg owes 1000 ducats in Berlin, which he wishes to pay in rubles by the way of Holland; and he has for the data of his operation, the following information, viz. That 1 ruble gives 47 stivers; that 20 stivers make 1 florin; 2. florins 1. rix dollar of Holland; that 100 rix dollars of Holland fetch 142 rix dollars of Prussia; and that 1 ducat in Berlin is worth 3 rix dollars Prussian. How many rubles will pay

the debt? 7. If 94 piasters at Leghorn are equal to 100 ducats at Venice, and 1 ducat is equal to 320 maravedis at Cadis, and 272 maravedis are equal to 630 reas at Lisbon, and 400 reas are equal to 50 d. at Amsterdam, and 56 d. are equal to 3 francs at Paris; and 9 francs are equal to 94 d. at London, and 54 d. are equal to 1 dollar in the United States, how many dollars are equal to 800 piasters ?

XXVII.

DUODECIMALS.

Duodecimals are compound numbers, the value ol whose denominations diminish in a uniform ratio of 12 They are applied to square and cubic measure.

The denominations of duodecimals are the foot, (f.), the prime or inch, (), the second, ("), the third, ("), the fourth, (""'), the fifth, ("*"'), and so on. Accordingly, the expression 3 1'7" 9" 6''" denotes 3 feet 1 prime 7 seconds 9 thirds 6 fourths.

The accents, used to distinguish the denominations below feet, are called indices.

The foot being viewed as the unit, duodeciinals present the following relations. l'=ts of 1 foot. 12 of of 1 foot.

144 of 1 foot. I"" I of la of Is of 1 foot. .

1728 of 1 foot. 1"=of 1 of 1 of 1 of 1 foot. 20736 of 1 foot.

&c. Addition and subtraction of duodecimals are performed as addition and subtraction of other compound numbers; 12 of a lower denomination making 1 of a higher. Mul tiplication, however, when both the factors are duodecimals, is peculiar, and will now be considered.

When feet are multiplied by feet, the product is in feet. For instance, if required to ascertain the superficial feet in a board 6 feet long and 2 feet wide, we multiply the length by the breadth, and thus find its superficial, or square feet to be 12. But when feet are multiplied by any number of inches (primes], the effect is the same as that of multiplying by so many twelfths of a foot, and therefore the product is in twelfths of a foot, or inches: thus a board 6 feet long and 6 inches wide contains 36 inches, because the length being multiplied by the breadth, that is, 6 feet by ta of a foot, the product is já of a fool

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