Exam. 7. A father leaves to his eldest child for 8 years a profit rent of $280, per annum, payable yearly, and the reversion of it for the 12 years succeeding, to his second child. What is the present value of the legacy of the second, at 4 per cent. per annum, compound interest? Here, by Prob. II., the value of an annuity of $1 for 20 years, at 4 per cent., is 13.590325, and for 8 years 6.732745; the difference of which is 6.85758, the present value of a reversion of $1 in the proposed circumstances. This being multiplied by $280, the product, $1920.1224, or $1920.121, nearly, is the value required. Ex. 5. What sum must be paid, to change into a perpetuity a lease for 16 years, which brings a profit rent of $286.65 per annum, payable yearly, compound interest being allowed at 4 per cent. per annum? Ans. $3465.34. 6. What sum must be paid, to add 25 years to a lease, which brings a profit rent of $562.50, and of which 14 years are unexpired, compound interest being allowed at 5 per cent. per annum? Ans. $4004.10.. 7. What is the present value of the reversion of a perpetuity of $300 per annum, payable yearly, but not to come into possession till the expiration of 100 years, compound interest being allowed at 6 per cent. per annum? Ans. $14.501. Annuities Contingent, or Life Annuities. 228. Life Annuities are those whose commencement or termination, or both, depend on the extinction of one or more lives. 229. When life annuities are in possession, they are often called simply annuities on lives; but when they are in reversion, they are generally called annuities on survivorships. 230. The value of a life is the present value of an annuity of $1 to continue during that life. 231. The expectation of a life of a given age, is the mean period during which persons of that age live. 232. The complement of a life is double the expectation of the same life. The calculation of life annuities depends on the joint ap plication of the rules of compound interest, and of the de trine of chances, to tables deduced from observations on the duration of human life. For the theory of life annuities, which is of a nature too complicated to be given in a work like the present, the wishers to be acquainted with this interesting and difficult subject, may have recourse to the writings of Simpson, De Moivre, and more particularly of Dr. Price and Morgan, where the subject will be found treated at great length, both in theory and practice. A se- . lection of the most useful rules, without the theory, is given in Professor Thomson's Arithmetic. The student is also referred to Joyce's Arithmetic. CHAPTER XVI. SHORT METHODS OF CALCULATION. 233. An Aliquot Part of a number is such a part as, when taken a certain number of times, will exactly make that number. Thus, 5 is an aliquot part of 20, 3 of 12, &c. 234. What is generally called Practice is only an abridged method of performing operations in the Rule of Proportion, by the use of aliquot parts; and is generally employed in calculating the prices of commodities. It may be also employed in calculating interest, discount, &c. These tables may be constructed by Prob. VII. page 132, or rather by dividing $1, £1, 1 acre, &c. by 2, 3, 4, &c. and selecting such of the quotients as are free from fractions. In the calculation of prices, the quantity of the commodity may be of one denomination, or of more than one; and accordingly the subject divides itself into two branches, with several varieties, as will appear from the following rules and illustrations. 235. RULE I. In finding the price of a commodity, when the price of each article, as well as the quantity, is of one denomination, the product of the given price and of the number of articles, will be the price required. Example 1. Required the price of 289 cwt. of beef, at $5 per civt. *This, in fact, is the weight that is now generally adopted by grocers and merchants throughout the United States. Here, the price of 289 cwt. at $5 per cwt. 289 cwt. at $1 per cwt. is evidently $289=price of 289 cwt. at $1 per cwt. $289; and the price of the same at $5 per 5 cwt. must obviously $1445-price of 289 cwt. at $5 per cwt, be 5 times that amount. Exercises.-1. Required the price of 25 cwt. of sugar, at 11 cents per cwt. Ans. 2.75. 2. Required the price of 125 barrels of flour, at $4 per barrel. Ans. $500. 3. Required the price of 756 yards of superfine black cloth, at £3 per yard. Ans. £2268. 4. Required the price of 26 pieces of linen, at £7 per piece. Ans. £182. 236. RULE II.-When the price is an aliquot part of a higher denomination, take a like part of the number of articles, and the result will be the price in the higher denomination. Exam. 2. What cost 96 lbs. of tea, at 50 cents per lb. ? 96 lbs. of tea, at 50 cents per lb. $96=price of 95 lbs. at $1 per lb. 50cts. of $1 $48=price of 96 lbs. at 50 cents per lb. Exam. 3. What cost 532 lbs. of tea, at 6s. 8d. per lb. ? 532 lbs. at 6s. 8d. per lb. 6s. 8d. of £1 £532 price of 532 lbs. at £1 each. £177 6s. 8d.=price at 6s. 8d. each. In this example, since 532 articles, at £1 each, would cost £532, it is evident, that at 6s. 8d. the same number of articles would cost one-third of that amount, 6s. 8d. being one-third of £1. We, therefore, divide 532 by 3, and the quotient £177 6s. 8d. is the required price. Ex. 5. What cost 253 lbs. coffee, at 20 cents per lb. ? Ans. $50.60. 6. What cost 2560 lbs. cotton, at 12 cents per lb. ? Ans. $320. 7. What cost 139 yards calico, at 25 cents per yard? Ans. $34.75. 8. What cost 335 lbs. vitriol, at 5 cents per lb. ? Ans. $16.75. 9. What cost 676 lbs. of bread, at 4 cents per lb. ? Ans. $27.04. 10. What cost 350 yards of linen, at 3s. 4d. per yard? Ans. £58 6s. 8d. 11. What cost 1200 yards of linen, at 2s. 6d. per yard? Ans. £150. 237. RULE III.-When the price of each article is not an aliquot part of a higher denomination, it is to be divided into such parts, that the price of the whole quantity at each of these prices, may be found by the first or second rule; and the sum of the prices thus obtained will be the whole price required. Exam. 4. Required the price of 479 cwt. of sugar, at $8.75 per cwt. 479 cwt. at $8.75 per cwt. at $8.75 per cwt. Ex. 12. What cost 35 casks of raisins, at $2.25 per cask? Ans. $78.75. 13. What cost 120 gallons of wine, at $2.37 per gal Ans. $285. lon ? 14. What cost 230 gallons of Madeira wine, at $2.50 per gallon? Ans. $575. 15. What cost 356 yards of cloth, at £2 17s. 9d. per yard? Ans. £1027 19s. Od. 238. RULE IV. When the quantity is not expressed by a whole number of one denomination, find the price of the integral quantity according to the method already illustrated, and then find the price of the fractional parts, or lower denominations, from the given rate, by Ꮓ |