Ru + u By the fecond and fourth Theorems, two very ufeful Queftions may be easily 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 anfwered 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 Queftions of this Kind are eafily anfwered 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 fome Time, or Number of Years are paft; then you must firft find what the Sum proposed 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 prefent Poffeffion; and make that Amount the Sum for the Purchase, proceeding with it as in either of the two laft Questions, &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 to ither 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 discharged 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 eight Months End, &t. And if it were required to find the Time when all thofe Sums may be truly difcharged at one Payment without Lofs, &c. CHAP. CHAP. XII. Of Compound Interest, and Annuities, &c. COMPOUND Intereft is that which arifes from any Principal and it's Intereft 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 Reason 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 firft Year, it will only amount to 1067. as in Simple Intereft. But for the fecond Year this 106% becomes Principal, which will amount to 1121. 7s. 24 d. at the fecond Year's End, whereas by Simple Intereft it would have amounted to but 1127. 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 is very requifite to understand it. Let Sect. 1. Of Compound Interest. Pthe Principal put to Intereft. 4the Amount of the Principal and Intereft. the Amount of 1 7. and it's Intereft for 1 Year, at R={any given Rate, which may be thus found. Viz. 100 106 :: 1: 1,06 the Amount of 17. at 6 per Cent. Or 100 105, :: 1: 1,05= the Amount of 1 l. at 5 per Cent. and fo on for any other affigned Rate of Interest. Then if Rthe Amount of 1 l. for one Year, at any Rate. Rthe Amount of 1 1. for two Years. R3 the Amount of 1 l. for three Years. the Amount of rl. for four Years. Rthe Amount of 1 l. for five Years. Here t=5 For 1:R:: R:RR:: RR: RRR:: RRR: R4:: RR:&c. in. : As one Pound is to the Amount of one Pound at one That is Year's End :: fo 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 laft and higheft Term: Viz. the Power of R, which is Rt. Again, As 1: R' :: P: PR=A the Amount of P for the Time, that R= the Amount of 1 1. 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 I. PRA, as above. From hence the two following Theorems are easily deduced. Theorem 2. A A =P. Theorem 3.=R. By thefe three Theorems, all Queftions 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 Purpose; 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 R' 1,50363. Then 1,50363 x.256,5 = 385,6811=A=385 %. 13. 7 d. which is the Answer required. Question 2. What Principal or Sum of Money must be put (or let) out to raise a Stock of 3851. 13 s. 7 d. in feven Years, at 6 per Cent. per Annum, Compound Intereft? Here is given A 385,6811; R=1,06; and t=7; to find P. by Theorem 2. Thus R1,50363) 385,6811=A (256,5=P. That is, P 256. 10s. which is the Principal or Sum, as was required. Question Question 3. In what Time will 2561. 10s..raife a Stock 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 = 1,50363, 256,5 find t by the third Theorem Rt=+= which being continually divided by R 1,06 until nothing remain, the Number of thofe Divifions will be 7=t. Thus 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. 13s. 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. = R1,50363; as before in the laft P Queftion. And if RR 1,50363, then R=1,50363, which may be thus extracted. Put re=R, then 167 2-7 2r+7re + 21 r3 e e = R2 = 1,50363=G 37 re+ 21rs ee Gr ·3÷7r 4 re+3ee= G— =D 725 Then: 0,06 1,00 r 0,06=e 1,18) 0,0719 (1,000 1,06=r+e=R II to be rejected." 100: 6 the Rate per Cent. required. The firft 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 That is, of R. RR. RRR. R. R3. and fo on to R39 11.06 R 14 2.2609039557 27 4.8223459407 II The Title of this Table fhews it's Conftruction, and it's Ufe will eafily appear by an Example or two. What will 3751. 10s. amount to in nine Years, at 6 per Cent. per Annum, &c ? The tabular Number against 9 Years is 1,689479 which being multiplied with the Principal 375,5 will produce 634,3993 &c. viz. 634 1. 8 s. ferè, being the Amount or Answer required. EXAMPLE 2.. What Principal (or Sum) must be put to Intereft to raise a Stock of 6341. 8 s. in nine Years Time, at 6 per Cent. per Annum, &c. If the propofed Stock (viz. 634,4) be divided by the tabular Number that is against the given Number of Years (viz. 9.) the Quotient will be the Principal (or Sum) required. Viz. against 9 is 1,689479. Then 1,689479) 634,4 (375,5 = 375 % 10s. the Principal (or Sum) required. EXAMPLE 3. In what Time will 3751, 101, raife a Stock of (ar amount to) 634 1. 8 s. at 6 per Cent. &? I Divide |