INTEREST. 



To the foregoing, a great number of ether cafes might 

 Ije added. But it will be fufficient to give only a few of 

 the mod curious and important ; fome of which having 

 occurred in praftice, it would perhaps be improper to omit 

 them. 



Theorem I — To determine the time in which i/. will be 

 doubled by compound intereft. 



5(?/«/;ort —Since i + r\" the amount of i/. at the end of 

 n years is in this cafe = 2 n, or the required time, will be 



= Jq ^ — ^ •; from which it appears that money doubles 



itfelf at 3/ per cent, in 2j§ years nearly ; at 47. per cent, in 

 17I years ; at 5/. per cent, in 14J- years ; at 6/. per cent, iu 

 1 1 ,% years ; at S/. per cent, m 9 years ; at j o/. per cent. 

 in 75 years ; and at 12/. per cent, in 6^ years nearly. 



Theorem 2. — Suppofing money to be doubled.by compaund 

 interell in n years, required r the rate of interelt at which 

 it has been improved. 

 ' Solution — The log. i + r, by the preceding theorem 



being given = ■ ' "' " , the value of r v.ill alfo be given. 

 Or, if • the hyperbolic logarithm of 2 be exprefTcd by h, it 



annuity to be received annually, and its intereft to be im. 

 proved half-yearly. 



So/ution.— The feries expreffing thefe accumulations is 



going exprefGon will then be 



67.1567 X 500 = 33>57S'j5- If the above annuity and 

 its intereft be improved yearly, the accumulation will be 

 equal to 33,219.4/. and if both principal and intereft be 

 improved half-yearly, they will amount to 33,998;^. 



Theorem 5. — To determine the fum ^ wliich at compound 

 intereft will amount in the time n to N, and in the time 

 m to M. 



will be found =1 — — — 



^ n 



Theorem^. — Suppofing 1/. to be doubled in n years at 

 the intereft r, and in m years at the intereft 2 /•, to iind the found = 

 ratio of m to n. 



'(Jut'wn.—la this cafe/i . i + ;■] = N, and/ . I + r\„ 

 M, from which two equations, the log. oi p may be 

 log. N — n . log. M 



Solution — Since 1 + r]' and I + 2 r\" are each of them 

 equal to 2, they will be equal to each other, and confe- 

 quently m will be to n as log. i + r to log. 1 -h 2 ?•, or 

 by fluxions and the binomial theorem, as i 4- ~ 



to one very nearly, or ftill more nearly, as | -(- 



Examples. — At 3 and 6 percent, mand n are rcfpeclively to 

 each other in the ratio of .50746 to one ; at 4 and 8 per cent. 

 of .5093410 one ; at 5 and 10/tr irtn/. of .51 197 to one ; at 

 6 and 12/fr cent, of 51417 to one ; at 8 aud 16 per cent, of 

 .5i889.to one ; at 10 and 20 />»r an/, of .52269 to one ; and 

 St 12 and 2^ per cent, of .52681 to one. From thefe examples 

 it appears that the higher the intereft the greater will be 

 tlie ratio of ra ton. It follows alfo from the expreffion 



J. 4. ^ ~ *" ' *"; &c. that money will not double itfelf at 

 " 4 



2 r intereft in h ilf the time that it doubles itfelf at r in- 

 tereft. The truth of this conclufion may perhaps be more 

 fatisfaAorily proved in the following manner. Since m : n 



Corollary 1 — If the fum p amounts to N in the time «, 

 the log. of the amount M in the time m will be = 



,n.lag.N->7^^i.log-/ 



Corollary 2. — If the fum p in « years amounts to K, 

 the time m in which it will amount to M, will be = 

 n . log. N — leg, p ^ 



log. M — log. /> 



Theorem 6. — Suppofing the fum a to become payable 

 annually for n years, at wliat time / might all the payments 

 (n a) be made at once, fo as to be an equivalent , to the 

 feveral annual payments. 



Solution. — The prefent value sf the annuity a for n years 



(fee AxsviTiEs) being = - — ■ „ , will in this 



cafe be = - , , from which equation / may be found 



_ log, nr + log. 



log. I 4- r 

 E.xampk. — 'L'tl a = 10, n = 15, and r =: .05, 

 other words, let an annuity of 10/. be payable a 

 for 15 years at 5/. per cent, at what period will th( 

 payment of 150/. be an equivalent to this annuity ? 

 cale / will be = 

 losf. 



log- 



log. I . 05 



iSlr_ 



inuallv 

 fmgl'e 

 la this 



7-5 



log. 1 4- r 

 than unity, therefore m will always be to n in a greater 

 ratio thai) as i to 2. 



Theorem 4. — To determine the fum to which an annuity 

 e will accumulate at the end of n year»> fuppoiing fuch 



nearly, or 7^ years. By the fame rule, if the annuity- 

 were for 30 or 50 years, a finglc payment made at the end 

 of 13.17 years, or at the end of 20. { years, will be an equiva- 

 lent to fuch annuities refpec'tively. 



Corollary. — If the payments, inftead of always being a, 



vary each year, fo that in the firft, fecond, and tliird years 



3 thejr 



