| XXX | (318-(Annuities)) | XXX |
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AN Annuity is a fum of money, payable yearly, halfyearly, or quarterly, to continue a certain number of years, for ever, or for life. An annuity ia faid to be in arrear, when it continues unpaid' after it falls due. And an annuity is laid to be in reverfion, when the purcHafer, upon paying the price, does not immediately enter upon poffeffion; tire annuity not commencing till fome time after. Intereft on annuities may be computed either in the way of limple or compound interell. But compound intereft,, being found molt equitable* both for buyer and feller, the computation by fimple intereft is univerfally difufed. I. Annuities for a certain Time. Problem i. Annuity, rate, and time, given, to find the amount, or fum of yearly payments, and intereft. Rule. Make i the firft term of a geometrical feries, and.the amount of il. for a year the common ratio; continue this feries to as many terms as there are years in the queftion; and the fum of this feries is the amount of 11. annuity for the given years; which, multiplied by the given annuity, will produce the amount fought. Example. An annuity of 40I. payable yearly, is forborn and unpaid till the end of 5 years: What will then be due, reckoning compound intereft at ^ger cent. on all the payments then in arrear ? 123 : 4 5 1 : 1.05 : 1.1025 1.157625 : 1.21550625; whole fum is 5,525631251.; and 5.52563125 X 40 — 221.02525 = 2211. os. 6 d. the amount fought. The amount may alfo be found thus : Multiply the given annuity by the amount of 1 1. for a year; to the produifc add the given annuity, and the fum is the amount in 2 years; which multiply by the amount of 1 h for a year; to the product add the given annuity, and the fum is the amount in 3'years, <fcc. The former queftion wrought in this manner follows. 40 am. in 1 year. 126.1 am. in 3 years. 1.05 1.05 42.00 132.405 AO 40 . 82 am. in 2 years, 172.405 am. in 4 years. 1.05 1.05 86.10 181.02525 40 40 126 1 am. in 3-years. 221.02525 am.in yyears. If the given time be years and quarters, find the amount for the whole years, as above; then find the-amOunt of 11. for the given quarters; by which multiply the amount for the whole years; and to the produdt add
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fuch a part of the annuity as the given quarters are of 3 year. If the given, annuity be payable half-yearly, or quarterly, find the amount of 11. for half a year or a quarter; by which find the amount for the feveral half-years or quarters, in the fame manner as the amount for the feveral years is found above. Prob. 2. Annuity, rate, and time given, to find the prefent worth, or fum of money that will purchafe the annuity. Rule. Find the amount of the given annuity by the former problem; and then, by compound intereft, find the prefent worth of this amount, as a fum due at the end of the given time. Examp. What is the prefent worth of an annuity of 401. to continue 5 years, difcounting at 5 per cent, compound intereft ? By the former problem, the amount of the given annuity for 5 years, at 5 per cent, is 221.02525; and by compound intereft, the amount of if. for five years, at 5I. per cent, is 1.2762815625 And, 1.2762815625)221.02525000(173.179 = 173 1. 38. yd. the prelent worth fought. The prefent worth may alfo be found thus : By compound intereft, find the prefent worth of each year by itfelf, and the fum of thefe is the prefent worth fought. The former example done in this way follows. 1.2762815625)40.000000000(31.3410 1.21550625)40.0000000 (32.908a 1.157625)40.00000 (34-5535 1.1025)40.000 (36.2811 1.05)40.0 (38.0952 Prefent worth, 173.1788 If the annuity to be purchafed be in reverfion, find firft the prefent worth of, the annuity, as commencing immediately, by any of the methods taught above; and then, by compound- intere’ft, find the prefent worth of that prefent worth, rebating for the time in reverfion; and this laft prefent worth is the anfwer. Examp. What is the prefent worth of a yearly penfion or rent of 75 1. to continue 4 years, but not to commence till 3 years hence, difcounting at 5 percent,? .05 : 1 " 75 : I5°° 1.05 X 1.05 X 1.05 X 1.05 = 1.21550625 1.21 5 50625) I 5QO.OOOOO( 1234.05 371 1500 1234.05371 265.94629, prefent worth of the annuity, if it was. to commence immediately. 1.05^ r.oy X 1.05 = 1.157625. L. s. d 1.157625)265.94629(229.7344 = 229 14 8£ Prob. 3. Prefent u'cyth, rate and time given, to find the annuity. Rule,
