ments, work with 8 times the amount, 8 times the firft product, a fourth of the annuity, and 4 times the number of years. If the annuity in the question had been payable half yearly, the amount would have been 2021. 251. and each payment 1251. number of payments in 7 years 14. •.* 125 × 14—1750 the first product whence per the rule and N. B. 2091.25 X 4 — 1759 × 4. .06 the ratio of the rate as 1750 X 14 — I before. For quarterly payments, the amount per cafe ift. is 2104.3751. a fourth of the annuity 621. 10s. and 28 quarterly payments. 62.5X281750 the first product, whence per N. B. and rule. 2104.375X8-1750X 8 1750 X 28-1 .06 the ratio of the rate. Cafe 4th. The annuity, amount, and rate given to find the time. See in the use of the Square root. The prefent worth of annuities &c. Cafe 1ft. When the annuity, time and rate are given, to find the prefent worth. R ULE. ft. Multiply the time by the ratio of the rate, which call the firft product. Then to the firft product add 2, which fum multiply by the time, and from that product, fubtract the first product for a dividend: which divide by the first product + 1, multiplied by 2, and the quotient multiplied by the annuity is the prefent worth. What is 250l. per annum to continue 7 years, worth in ready money, allowing the purchafer 61. per cent fimple intereft? ift. 7X .06.42 the first product. •42 + 2 x 7.42X 250=1454.225=14541.4 6 .42+1 X 2 the answer. For For half yearly and quarterly payments, obferve the N. B. in Cafe ift. of annuities, penfions &c. in arrears. The number of payments 14. each payment 1251. and half the ratio .03.14 X.03.42 the first product. and 42 + 2 × 14—.42 × 125 = £1472 14 21 + .42 + IX 2 anfwer. If the annuity in the example had been paid quarterly the number of payments 28, each payment 621. Ios. and one fourth of the ratio .0151. ·.· 21 × .015 .42 first product. And •42 + 2 X 28—.42 × 62.5= 14811. 1s. 5d. zq、 .42+ IX 2 Hence appears the advantage of having an annuity paid quarterly, or half yearly, rather than yearly. Cafe 2d. When the prefent worth, time, and rate are given, to find the annuity. I ULE. To the product of the ratio and time, add 1 for a dividend, then multiply the faid product by the time-1, and add twice the time to the product for a divifor, the quotient arifing multiplied into twice the prefent worth, gives the annuity fought. What annuity to continue 7 years, can be purchased or 14541. 4s. 6d. ready money, allowing 6 per cent per annum fimple interest ? Ift. 7 X .06 .42 the product of the ratio and time. T n .42 + I +42X7—1+7 X2 × 1454.225×2 = 2501. N. B. For half yearly payments, work with half the ratio, number of payments, and multiply by 4 times the prefent worth. For quarte: ly payments take a 4th of the ratio, number of payments, and 8 times the present worth. The The prefent worth of the annuity in the laft queftion payable half yearly is 14721. 14s. 2 d. whence to find that annuity. Here half the ratio is .03 number of payments 14. .3 × 14.42 the product of half ratio and number of payments, 1472.71125 X 4 = 5890.845 .42+1 *.42 X-1-3* + 13 x 2 If the prefent worth of an annuity payable quarterly, for seven years be 14811. 1s. 5d. 2q. at 61. per cent, query that annuity? Here a fourth of the ratio .0151. number of payments 28 whence, 1481.0739375 × 8=11848.5915 And .015 x 28.42 the product of a fourth of ratio and number of payments. Cafe 3d. When the prefent worth, the annuity or penfion, and time are given, to find the rate of intereft per cent. ULE. Firft multiply the annuity by the time Rwhich call the firft product. 2d. from that firft product fubtract the prefent worth for which double a dividend; then to twice the prefent worth, add the annuity, from that fum fubtract the firft product, and multiply the remainder by the time for a divisor, fo will the quotient, be the ratio of the rate. + At what rate per cent, will 14541. 4s. 6d. ready money, purchase an annuity of 2501. for 7 years? First 250 × 71750 the first product. Second 1750-1454.225 X 2 1454.225 X 2 + 250—1750 × 7 whence 6 per cent is the answr. K3 For For half yearly payments, work with half the annuity, and number of payments inftead of time in years, and the quotient will be half the ratio of the rate per cent. For quarterly payments, work with a fourth of the annuity, or a quarterly payment, number of payments and the quotient will be a fourth of the ratio of the rate. If the present worth in the laft queftion be changed to 1472.711251. and the payments to be made half yearly, query the rate per cent? Here 125 X 14 1750 the first product as before. . Second 1750-1472.71125 X 2 1472.71125 X 2 + 125-1750 X 14 .03, which doubled is .06 the ratio as before. If the present worth had been 14811. is. 5d. 2q. — 1481.0739375 and the payments quarterly. Then one fourth of the annuity, or quarterly payment is 62.51. the number of payments 28. Whence 62.5 X 28 1750 the first product as before. Then 1750-1481.0739375 X 2 2 X 1481.0739375+ 62.5-1750 X 28 =.0151. which X 4.06 the ratio as before. Cafe 4th. When the annuity, rate per cent, and prefent worth are given, to find the time. See the use of the Square Root. Annuities, Penfions &c. taken in Reverfion. Cafe ift. To find the prefent worth of an annuity in reverfion. RULE R ULE. Find the prefent worth of the annuity at the given rate, for the time of its continuence per cafe firft of prefent worths. Second. call the present worth, the amount, and find what principal put to intereft, will come to that amount at the fame rate, for the time to come before the annuity commences per cafe fecond of interest accounts. Old Timon left his nephew Jack, who was 14 years old, a legacy of 50l. per annum, to continue 10 years to commence when Jack is 21 years old, but Jack contracts with Arifto for ready money, allowing 5 per cent to the purchaser; query how much Jack may receive? Firft per cafe first of prefent worths. Then .05 X 10.5 the firft product. And per cafe fecond of intereft accounts 490 7x.05+1 =3621. 962 = 3621. 195. 3d. the answer. Cafe zd. To find the yearly income of any annuity or penfion, &c. in reverfion. R ULE. First find the amount of the present worth, at the given rate and time before the annuity commences, per cafe first of interest accounts. Second confider the amount as prefent worth, and find per cafe fecond of prefent worths, what annuity it will purchase at the fame rate, and for the time of its continuance. Jack who is 14 years old, has an annuity left him for 10 years; not to commence until he is 21 years old, but wanting cash to carry on his extravagancies, has fold it for 3621. 19s. 3d. .111, allowing 5 per cent to the buyer; what was the annuity left him? First |