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2. What is the neat weight of 85cwt. 2qrs. 18lb. gross; tare 2cwt. 3qrs. 12lb. ?

Ans. 82cwt. 3qrs. 6lb. 3. What is the neat weight of 4 hogsheads of sugar, the gross weight and tare, as follows ? Ans. 39cwt. Qer. 161b cwt. qrs. lb.

16. Ist hhd. 9 3 18

Tare

1047 2d 10 1 12

98 3d 11 3 9

90 4th 10 2 11

106 Gross weight 42 2 22

Whole tare 398lb. 398lb.=3 2 6

Ans. 39 0 16 neat weight. 4. What is the neat weight of 3 casks of indigo, each weighing 4cwt. 2qrs. 14lb. gross; tare on each cask lcwt. Oqr. 1216. ?

Ans. 10cwt. 2qrs. 6lb. CASE II.-- When the tare is so much per hundred weight:

RULE.-Find, by Practice, the tare, by taking aliquot parts of a cwt.; then deduct the tare from the gross, the remainder will be the neat weight. Or, by the Rule of Three; as 112 is to the tare per cwt., so is the given weight to the tare. Then deduct the tare from the gross weight, and the remainder will be the neat weight.

EXAMPLES 1. What is the neat weight of 20 hogsheads of tobacco, weighing in all 246cwt. 3qrs. 7lb. gross; tare 16 lb. per ewt. ?

Ans. 21 lcwt. 2qrs. 6lb. 1b. cwt. qrs. lb. 16=+)246 37

35 1 1 tare deducted. Ans. 211 2 6 neat weight.

By the Rule of Three.

Ib. 16. cwt. qr, lb. 112: 16 :: 246 3 7

4 987

28 7903 1974 27643

16

Carried up.

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165858

Brought up. 27643

28) 4) 112)442288 ( 3949 ( 141 336 28

35cwt. lqr. llb. Tarè.
1062 114
1008 112

cwt. qr. 16.
548 29

246 3 7
448
23

35 1 1
1008 ilb. Ans. 211 2 6 neat weight.

1:008 Dem.- It is plain, that the tare, in the fourth term, bears the same proportion to the gross weight, in the 3d term, as the tare, in the 20 term, bears to the gross weight, in the first term; and this tare subtracted must leave the neat weight.

2. What is the neat weight of 123cwt. lqr. 20lb. gross; tare 141b. per cwt. ?

Ans. 108cwt. Note.--If the tare be 1416 per cwt., deduct of the gross weight for the tare; the remainder will be the neat weight, 141b. being of a cwt. or ' 1121b., which is shown thus, 14)1=reduced to its lowest terms. If the tare be 1616. per cwt., deduct 7 of the gross weight for the tare; the remainder will be the neat weight, 16lb. being of a cwt. or 1121b, which is shown, thus, 16) 16= If the tare be any other quantity than 1415. or 1616. per cwt., first obtain the fare for 141b., or 161b. per cwt. and then take aliquot parts of the quotient.

3. What is the neat weight of 18 hogsheads of sugár weighing 167cwt. 2qrs. gross, tare 20lb. per cwt. ?

16. cüt. qrs. lb.
16=5)167 2 0
4=+) 23 3 20
20lb.

5 3 26

29 3 18 Tare. Ans. 137 2 10 Neat weight. 4. In 115cwt. Iqr. 20lb. gross, 'tare 21lb. per cwt.; Whát is The neat weight?

Ans. 93cwt. 3qrs. 41b. 5. In 115cwt. Iqr. 2016. gross, tare 7lb. per cwt., what is the neat weight ?

Ans. 108cwt. Oqr. 24bb. CASE III.- When Tret is allowed with Tare.

RULE. Find the tare as in the preceding case, which subtract from the gross, and call the remainder suttle. Divide the suttle by 26, and the quotient will be the tret, which subtract from the suttle, and the remainder will be the neat weight.

EXAMPLES. 1. What is the neat weight of 32cwt. 3qrs. 2016. gross, tare 14lb. per cwt., tret 4lb..per 104?

cut. qrs. 16. Statements in the Rule of Three. 14=5)32 3 20

1st. For the tare. 4 0 13 tare. 16. lb. cwt. qrs. lb. 4=+)28 3 7 suttle.

112: 14 :: 32 3 20 : 1 0 12 tret.

Ans. 4cwt. Oqr. 131b. tare.

2d. For the tret. Ans. 27 2 23 neat.

16. lb. cwt. grs. lb.
104:4:: 28 3 :

Ans. lcwt. Oqr. 12lb. tret. When the remainder is less than 2lb. it is not reckoned.

The reason of dividing by 26, is to shorten the work; 41b, being to part of 104, 4)101=ztó.

2. What is the neat weight of 495€ wt. Iqr. 21b. gross, tare 281b. per cwt, and tret 4lb. for every 10415. ?

Ans. 357cwt. Oqr. 183lb. 3. What is the neat weight of 4hhds. of sugar, weighing 33cwt. lær. 121b. gross, tare 14lb. per cwt., tret 41b. per 104, and cloff 2lb. per 3cwt. ? Ans. 27cwt. 3qr. 16lb. 100%.

NOTE.-When cloff is allowed, deduct the tare and tret as usual, then divide what remains by 168, and subtract the quotient, the remainder will be the neat weight, Alb., the cloff, being Too part of 3cwt. 2)337=168.

QUESTIONS ON TARE AND TRET. What is Tare and Tret? A. An allowance made to the buyer, on the weight of goods. What is gross weight? A. It is the weight of the goods together with the box, bag, or whatever contains thein: What is tare? A. An allowance made to the buyer for the weight of the box, cask, bag, or whatever contains the goods. What is tret ? A. An allowance of 4 pounds on every 104 pounds for waste, dust, &c. What is cloff? A. An allowance of 2 pounds upon cvery 3cwt. What is suttle ? A. It is what remains after one or two allowances have been deducted. When the tare, on the whole quantity, is given, how do you find the neat weight? A. By subtracting the tare from

the gross weight. When the tare is so much per cwt., how do you find the neat weight? A. By taking aliquot parts of a cwt. for the tare, or find the tare by the Rule of Three, and deduct the tare from the gross weight. When tare and tret are allowed, how do you find the neat weight? A. First deduct the tare as usual, then divide the suttle by 26, for the tret, which subtract, the remainder will be the heat weight. When tare, tret, and cloff are allowed ; how do you find the neat weight ? A. First, find and deduct the tåre and tret as usual; then divide what remains by 168, and the quotient will be the cloff, · which subtract; and the remainder will be the neat weight. What is

he use of finding the neat weight? A. The use is, that the buyer only pays for the neat weight, at the price agreed on per cwt. Does the buyer pay for the freight of his goods, by the gross or neat-weight? A. Gross weight is reckoned, except where otherwise provided for by statute.

ANNUITIES. An Annuity is a sum of -money payable at regular periods, generally every year, for a certain time or for ever. The annuity sometimes depends on some contingency, as the life or death of a person and it is then said to be contingent.' When the annuity does not commence until a certain number of years has elapsed, it is then said to be in reversion. The annuity is said to be in arrear, when the debtor ieeps it beyond the time of payment. The present worth of an annuity is such a sum as being now put out to interest, would exactly pay the annuity when it becomes due, and such a sum as must be given for the annuity, if it be paid at the commencement. The amount is the sum of the annuities for the time it has been forborne, with the interest due on each payment or annuity.

The rules in annuity are only particular applications of the Rule of Three.

Case I.-To find the amount of an annuity at Simple Interest.

RULE. First, find the interest of the given annuity for one year; and then for 2, 3, and so on, up to the given number of years, less 1. Then multiply the annuity by the given number of years, and add the prodact to the whole interest, and the sum will be the amount sought.

EXAMPLES. 1. What is the amount of an annuity of $100 for four years, computing interest at 7

per

cent? The interest of $100, at 7 per cent, for 1 year, is.. $7.

..14.

for 3 years,.-------21. Four years' annuity, at $100 per year is $100X4=400

Ans. 442. DEM. It is plain, as 100 dollars is forborne 4 years, which is due at the end of each year, that the 100 dollars must be multiplied by 4

for 2 years,"

for the annuity, without the interest, and the 100 dollars, which becomes due at the end of the first year, runs on interest three years, consequently the interest on thạt must be 21 dollars, and the 100 dol lars which becomes due at the end of the second year, runs on inter: est two years, which gives 14 dollars interest, and the $100, which becomes due at the end of the third year, runs on interest one year, which gives 7 dollars interest, and the 100 dollars which becomes due at the end of the fourth year, does not run on interest, because it is paid or reckoned when it becomes due, consequently we must always reckon interest for one year less than the given tine; and it is plain, if these several annuities be added to the interest due on them, the sum must be the amount of the annuity,

2. What is the amount of an annuity of $400, forborne 5 years, simple interest, computed at 7 per ct:? Ans. $2280.

3. A man let a house upon a lease of 6 years, at $300 per annum, and the rent being in arrear for the whole term; what sum must he demand at the end of the term, simple interest being allowed at 7 per cent ?

Ans. $2115. Note.- Pensions in arrears are reckoned the same as in the preceding examples.

Case II.- To find the present worth of an annuity at Simple Interest.

RULE. First, find the present worth of each year by itself, discounting from the time it becomes due; then, the sum of all these will be the present worth.

NOTE.-This rule depends on the principles of discount. The an. muity for each year, may be considered as a debt due 1, 2, 3, 4, or 5 years hence, and so on, of which the present worth is to be found; hence the sum of all these will be the present worth.

EXAMPLES 1. What is the present worth of 100 dollars per annum, to continue 4 years, at 7 per cent ?

* cts. m. 107

93,45,7 The present worth for 1 year. 114

87,71,9 100 : : 100 : 121

82,64,4. 128 78,12,5.

4 years. Ans. $341,94,5 present worth required. 2. What is the present worth of an annuity of 400 dollars, continued five years, at 6 per cent?

Ans. $1703, 75cts. 5m. QUESTIONS ON ANNUITIES. What is an annuity? A. A sum of money payable at regular periods, for a certain time, or for ever. What are annuities said to bą,

2 years. 3 years.

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