INTEREST, compound, is allowing inter est upon interest, or adding the interest as it becomes due to its principal, and considering the whole as a new principal, bearing interest at the name rate as be fore. Let r now represent the amount of one pound in one year, that is, principal and interest; let n = the number of years; p = the principal; a = the amount. Then 1 : r r r' the amount of 11. in 2 years, 1 : r : r' : r3 the amount of I/. in 3 years, 1 : r:: r3 : r4 the amount of 11. in 4 years, &c.
Whence it appears, that r raised to the power whose exponent is the given num ber of years, or rn, will be the amount of V. in those years; and as 11.:rn :p :a from which the following theorems are easily deduced, viz. if the principal, time, and rate of interest, are given, to find the amount.
Theo. 1. p = a If the amount, time, and rate, are given, to find the principal ? a Theo. 2.— = p.
rn If the principal, amount, and time, are given, to find the rate? n a Theo. 3. j If the principal, amount, and rate, are given, to find the time ? p a —=rn, therefore, — being Theo. 4.ivided by r till nothing re}-d mains, the number of divi sions will be n.
It seldom happens, however, that it is necessary to work questions relative to compound interest by these rules, as very extensive and accurate tables have been published by Mr. John Smart and others, which save much labour in such calcula tions, and are therefore generally resort ed to in practice. The principles on which such tables are formed will appear from what has been already said : thus, the numbers in a table shewing the amount of 11. in any given number of years, are merely the powers of 11. increased by its
interest for a year; that is, r, r', r3, r4, &c. and the numbers contained in a table of the present values of U. to be received at the end of a given number oryears, are 11. discounted for those years, or 11. divided by the powers of r, that is, 1 1 —,- 1 &c.
r r' r3 r4' Tables of this kind being usually con. fined to six or eight places of decimals, necessarily give the amount beyond the first three or four years somewhat less than the true amount, but the difference is so small as to be of no importance in the pur poses to which they are usually applied.
Ex. 1. What sum will 500/. increase to in 21 years, if improved at 5 per cent. compound interest ? 500 X 2.785962 1392/. 198. 74d.
Ex. 2. What sum, if improved at 5 per cent. compound interest, will accu mulate to a million in 50 years? 1000000 467399 = 872031. 14s. 8c?.
The increase of an annuity, if forborne for a given time, may be found by this table, in the same manner as the amount of a given sum ; for as each payment of the annuity will become due at an equal distance from the time in which it would have been due, the amount of the first payment must give that of each of the succeeding ones.
able thereon from testator's death; but if charged only on the personal estate, which cannot be immediately got in, it shall carry interest only from the end of the year after the death of the testator. Where lands are charged with payment of a sum in gross, they are also chargeable in equity with payment of interest for such sum.