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. 20lb. gross, tare 14lb. per cwt., tret 4lb. per 104? cwt. qrs. lb. 14=1)32 3 20 4 0 13 tare. Statements in the Rule of Three. 1st. For the tare. lb. lb. cwt. qrs. lb. 112 14:32 3 20: Ans. 4cwt. Oqr. 13lb. tare. 2d. For the tret. lb. lb. cwt. qrs. lb. 104: 4 :: 28 3 7: Ans. 1cwt. Oqr. 12lb. tret. When the remainder is less than 4lb. it is not reckoned. The reason of dividing by 26, is to shorten the work; 4lb. being part of 104, 4)84-26 2. What is the neat weight of 495cwt. 1qr. 2lb. gross, tare 281b. per cwt., and tret 4lb. for every 104lb.? Ans. 357cwt. Oqr. 181lb. 3. What is the neat weight of 4hhds. of sugar, weighing 33cwt. 1qr. 12lb. gross, tare 14lb. per cwt., tret 4lb. per 104, and cloff 2lb. per 3cwt.? Ans. 27cwt. 3qr. 16lb. 10oz. 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, 2lb., the cloff, being part of 3cwt. 2)33188. 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 them. 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 every 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 neat weight. When tare, tret, and cloff are allowed; how do you find the neat weight? A. First, find and deduct the tare 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 the 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. 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. Does the An Annuity is a sum of money payable at regular periods, generally every year, for a certain time or forever. 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 keeps 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 foreborne, 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 product 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 percent ? The interest of $100, at 7 per cent, for 1 $7. is year, for 2 years, for 3 years, 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 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 that must be 21 dollars, and the 100 dollars which becomes due at the end of the second year, runs on interest 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 time; 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, foreborne 5 years, simple interest, computed at 7 per ct.? Ans. $2280. 3. A man let 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 annuity 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 93,45,7 The present worth for ly. $ per cent? 107 114 121 82,64,4 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 à certain time, or forever. What are annuities said to be, when they depend on some contingency, as the life or death of a person? A. The annuity is then said to be contingent. How do you distinguish annuities, when they do not commence till some future period? A. They are then said to be in reversion. When is an annuity said to be in arrears? A. When the debtor keeps it beyond the time of payment. What is the present worth of an annuity? A. It is such a sum as being now put out at interest, would exactly pay the annuity as it becomes due. What is the amount of an annuity? A. The sum of the annuities for the time, with the interest due on each. How do you find the amount of an annuity at simple interest? A. First find the interest of the given annuity for one year, then for 2 years, three years, and so on, up to the given number of years, less one; then multiply the annuity by the given number of years, and add the product to the whole interest, and the sum will be the amount of the annuity. How do you find the present worth of an annuity at simple interest?A. First find the present worth of each yearly payment by itself, discounting from the time it becomes due; then the sum of all these will be the present worth. ALLIGATION, Teaches how to compound or mix together several simples of different qualities or prices, so that the composition may be of some intermediate quality or price. It is commonly distinguished into two kinds, Alligation Medial, and Alligation Alternate. ALLIGATION MEDIAL, Teaches to find the price or quality of the composition, from having the quantities and prices or qualities of the several simples given. CASE I.-To find the mean price or quality of any part of the composition, when the several quantities and their prices or qualities are given. RULE.-First, multiply the quantity of each ingredient, by its price or quality; then add all the products together, and add also all the quantities together, into another sum; Then say; As the whole composition, is to the sum of the products, so is any part of the composition, to its mean value or quality. NOTE.-Alligation means to mingle, tie or mix together two or more simples; and Medial means the middle or mean rate between the extremes. EXAMPLES. 1. A merchant mixes 20 gallons of brandy at 10 shillings per gallon, with 36 gallons of rum at 6 shillings per gallon, and 40 gallons of gin at 4 shillings per gallon ; what is a gallon of the mixture worth? R Ans. 6s. NOTE. The student will now perceive, that Alligation is nothing more than the Rule of Three applied to mixing different ingredients. 2. A grocer mixes 60lb. of sugar at 8d. per pound with 20lb., worth 12d. per pound; what is the value of 1 pound of the mixture? Ans. 9d. 3. A farmer mixes 10 bushels of wheat at 5s. a bushel, with 18 bushels of rye at 3 shillings a bushel, and 20 bushels of barley at 2 shillings per bushel; how much is a bushel of the mixture worth? Ans. 3s. 4. A refiner melted together 8oz. of gold, of 22 carats fine, 10oz. of 20 carats fine, 12oz. of 16 carats fine, 8oz. of 18 carats fine; will you make out the fineness of the composition? Ans. 1814 carats fine. 5. Of what fineness is that composition, which is made by mixing 4lb. of silver of 8oz. fine, with 2lb. 4oz. of 9oz fine, and 8oz. of alloy ? Ans. 74oz fine. NOTE.-An ounce of pure gold being reduced into 24 equal parts, these parts are called carats; but when gold is mixed with baser metal, the mixture is said to be so many carats fine; thus, if 22 carats of pure gold be mixed with 2 of alloy, it is said to be 22 carats fine; and if 20 carats of pure gold be mixed with four of alloy, it is said to be 20 carats fine. A pound of pure silver, losing nothing in trial, is said to be 12oz. fine, but if it lose 1oz. by the fire, or be mixed with loz. of alloy, it is said to be 11oz. fine, &c. ALLIGATION ALTERNATE, Teaches to find what quantity of any number of simples, whose rates are given, will compose a mixture of a given rate. So that it is the reverse of Alligation Medial, and may be proved by it. This is called Alligation Alternate, because the same question frequently admits of different answers. CASE I-When the prices of the several simples are given, |