Sect. 3. The Prefent Worth of Annuities or Penfions, &c. computed at Simple Intereft. THE HE Business of purchasing Annuities, or taking of Leases, &c. for any affigned Time, depends upon the true equating of the Principal or Money laid out on the Purchase, with the Annuity or Yearly Rent, by allowing (or discompting) the fame Rate of Intereft to both Parties. Which may be eafily performed by duly applying the respective Theorems of the two laft Sections together; as will fully appear by the following Question. Question 1. What is 751. yearly Rent, to continue nine Years, worth in ready Money, at 5 per Cent. per Annum Simple Intereft ? 1. Per Theorem 1. of the laft Section, find what the propofed yearly Rent would amount to, if it were forborn 9 Years, at 5 per Cent. Thus & = 75, t = 9, and R= 0,05: ttu=6075 tu= Quære A. Then 2) 5400 (2700 Multiply R = 0,05 135, =810=A. +tu=675, S 2. Then by Theorem 2. Section 1. find what Principal, being put to Intereft for the fame Time, and at the fame Rate, will amount to 8101. A. Thus t R= 0,45 = 9× 0,05 ; 1 R + 1 = 1,45) 810 (558,6206= P : that is, P=5587. 125. 31⁄2 d. which is the Worth of 751. a Year, as was required. From the Work of these two Operations (duly confidered) it muft needs be eafy to conceive, how the two Theorems by which they were performed, may be combined in one. For 1. tt Ru- -1 Ru+2tu 2 Confequently PtR+P= =A; and 2. PtR+P= A. tt Ru-t Ru+2tu 2 this Equation may be deduced the following Theorems. tt Ru-tRu+21a Theorem I. 21R +2 And from = P, or ttR-R+2t xu-P By this Theorem all Queftions of the fame Kind with the last (viz. that above) may be easily and readily answered at one Operation. 1 ttu-tu 2 P + ** Ru 4 2 P Ru 2 = t. By the fecond and fourth Theorems, two very ufeful Questions may be eafily answered. 1. As for Inftance: If it be required to find what Annuity, or yearly Rent, &c. may be purchased, for any propofed Sum, to continue any affigned Time, allowing any Rate of Intereft? This Queftion may be answered by Theorem 2. 2. Again: If it be required to find how long any yearly Rent, Penfion, or Annuity, &c. may be purchafed (or enjoyed) for any propofed Sum, at any given Rate of Intereft? All Questions of this Kind are easily answered by Theorem 4. In these Questions it is fuppofed, that the Purchase or yearly Rent, is to commence or be immediately entered upon. But if it be required to find the Value or Purchase of an Annuity or yearly Rent, &c. in Reverfion; that is, when it is not to be entered upon until after some Time, or Number of Years are paft; then you must first find what the Sum propofed to be laid out in the Purchase, would amount to, if it were put to Intereft, during the Time the Annuity, &c. is not to be put in present Poffeffion; and make that Amount the Sum for the Purchase, proceeding with it as in either of the two laft Queftions, &c. Note, From the first Question of this Section it will be easy to conceive how to perform the Equation of Payments, between Debtor er Creditor, at any Rate of Intereft, without doing any Damage ro either Party. That is, when feveral Sums of Money are to be paid, at several different Times, to find the Time when all the Payments may be truly difcharged at once: as if one Sum were to be paid at the End of two Months, another at fix Months, and perhaps a third Sum at cight Months End, &c. And if it were required to find the Time when all thofe Sums may be truly discharged at one Payment without Lofs, &c. CHAP. CHA P. XII. Of Compound Interest, and Annuities, &ċ. COMPOUND Interest is that which arises from any Princi pal and it's Interest put together, as the Intereft fo becomes. due; fo that at every Payment, or at the Time when the Payments became due, there is created a new Principal; and for that Reafon it is called Intereft upon Intereft, or Compound Intereft. As for Inftance; Suppofe 100l. were lent out for two Years, at 6 per Cent, per Annum, Compound Intereft: then at the End of the first Year, it will only amount to 106 l. as in Simple Intereft. But for the fecond Year this 106 7. becomes Principal, which will amount to 112. 7s. 2 d. at the fecond Year's End, whereas by Simple Intereft it would have amounted to but 112 7. And altho' it be not lawful to let out Money at Compound Intereft; yet in purchafing of Annuities or Penfions, &c. and taking Leafes in Reverfion, it is very ufual to allow Compound Intereft to the Purchaser for his ready Money; and therefore it very requifite to understand it. is Let Sect. 1. Of Compound Interest. P = the Principal put to Intereft. A R= the Amount of the Principal and Intereft. {the Amount of 11, and it's Interest for 1 Year, at any given Rate, which may be thus found. Viz. 100 106: 1 : 1,06 the Amount of 1. at 6 per Cent. Or 100 105 :: I: 1,05= the Amount of 17. at 5 per Cent. and fo on for any other affigned Rate of Intereft. R4 Rs = the Amount of 1 l. for four Years. the Amount of 17. for five Years. Here t=5 For 1: R:: R: RR :: RR: RRR:: RRR: R4:: R+: R5: &c. in. As one Pound is to the Amount of one Pound at one That is Year's End :: so is that Amount: to the Amount of one Pound at two Year's End, &c. { Whence Whence it is plain, that Compound Intereft is grounded upon a Series of Terms, increafing in Geometrical Proportion continued; wherein t (viz. the Number of Years) does always affign the Index of the last and highest Term: Viz. the Power of R, which is R. Again, As 1: R' : : P : P R' — A the Amount of P for the Time, that R= the Amount of 1 l. As one Pound is to the Amount of one Pound for any That is given Time :: fo is any proposed Principal (or Sum): to it's Amount for the fame Time. From the Premifes (I prefume) the Reafon of the following Theorems, may be very eafily understood. Theorem 1. PRA, as above. From hence the two following Theorems are easily deduced, A Theorem 2. R A P. Theorem 3. = Rt. P By there three Theorems, all Questions about Compound Intereft may be truly refolved by the Pen only, viz. without Tables; tho' not fo readily as by the Help of Tables, calculated on Purpofe; as will appear farther on. Question 1. What will 2561. 10s. amount to in feven Years, at 6 per Cent. per Annum, Compound Intereft? Here is given P = 256,5; t = 7; and R = 1,06 which being involved until it's Index t (viz. 7.) will become R1 1,50363. Then 1,50363 x 256,5=385,6811=A= 385 ↳ 13. 7d. which is the Anfwer required. L Question 2. What Principal or Sum of Money must be put (or det) out to raise a Stock of 3851. 13s. 7 d. in feven Years, at 6 per Cent. per Annum, Compound Intereft? 2 Here is given 385,6811; R= 1,06; and 1 = 7; to find P. by Theorem 2. Thus R 1,50363) 385,6811=A (256,5=P. That is, P2561. 10s. which is the Principal or Sum, as was required. 2 Question Question 3. In what Time will 2561. 10s. raife a Stack of (or amount to) 3851. 13s. 7 d. allowing 6 per Cent per Annum, Compound Intereft? Here is given P = 256,5; A= 385,6811; R = 1,06; to A 385,6811 find by the third Theorem. R' = =3 P 256,5 Thus = 1,50363, which being continually divided by R= 1,06 until nothing remain, the Number of thofe Divifions will be 7=t. 1,06) 1,50363 (1,41852. And 1,06) 1,41852 (1,338225. Again 1,06) 1,338225 (1,262477. And fo on until it become 1,06) 1,06 (1. which will be at the feventh Divifion. Therefore it will be t7 the Number of Years required by the Question. Question 4. If 2561. 10s. will amount to (or raife a Stock of) 3851. 138. 7 d. in feven Years Time; what must the Rate of Intereft be, per Cent. per Annum? Here is given P=256,5; A= 385,6811, and t = 7, Quære A R. By Theorem 3. = R = 1,50363; as before in the last Queftion. And if R P R 1,50363, then R=7/1,50363, which may be thus extracted. Put I G7 2- 77 re=R, then 2r7+7r6e + 21 r5 e e = R7 = 1,50363=G 37r6e + 21 r5 e e = G — r7 37rs 4re+3ee= Gr7 = D 7r5 Then I 0,06 :: 100: 6 the Rate per Cent. required. The first three Queftions may be much more eafily performed by the following Table, which is only the Amounts of one Pound for thirty-nine Years. That |