this by £1.192518 (the amount for four years) is £2.411708, or £28 22 nearly, the amount of £1 for 20 years. At simple interest, the amount would have been only £1 13.

Had the products here been found at full length, the labour would have been immense. In the last multiplication, one of the factors would have contained 49 figures, the other 18, and the product 61. It should be carefully remarked, however, that the decimal part of the amount found as above, will rarely be true in all its places. A trifling error in rejecting or over-estimating a figure at the end of a decimal may accumulate, and render the accuracy of the last figure, or the last two figures doubtful. Thus, in the preceding result, the two last figures should have been 14 instead of 08, which would occasion an error of rather more than a penny in the amount of £1000. When great accuracy is required, therefore, the amounts should be brought out to a greater number of places, and the last figure or two of the final amounts rejected, or not depended on. The larger the sum also whose interest is required, and the longer the time, this is the more necessary, as the effect of the error is the more perceptible.

Exam. 3. What is the amount, true to six places of decimals, of £1 for 6 years, payable half-yearly, at 5 compound interest?

cent.

annun

Here, the payments being half-yearly, the amount of £100 for half a year is £102 10, or £102.5; and, consequently, that of £1 for the same time is £1025: the square of this is 1:050625, or 10506250, the amount for two half years, or one year. Multiplying this by 1050625, by the contracted method, we obtain 11038129, the fourth power of £1.025, or the amount of £1 for two years; the third power of which is 1.3448888, the twelfth power of £1.025, or the amount of £1 for six years or if only six figures of decimals be retained, 1·344889.

Exercises. Find the amounts of £i in the following exercises, at the given rates cent. Pannum :

RULE II. To find the amount, or the interest, of any sum, at compound interest, for a given time, and at a given rate: Find the amount of £1 for the given time by rule I., and multiply it by the given sum; the product will be the amount required.

If the principal be subtracted from the amount, the remainder will be the interest.

Exam. 4. Required the amount of £760 14 4 for 12 years, at 5 cent. annum, compound interest.

By rule I., the amount of £1 is found to be £1795856.

would in that case have been found by multiplication alone.

Exercises. Find the amounts of the following sums at the given

15. If a boy 12 years old, have a legacy of £1396 16 8 left to him, how much will he have to receive at the age of 21, the legacy being improved by compound interest at 5 cent. Pannum.? Ans. £2166 18 114.

16. Required the amount of £589 10 5 from the third till the twenty-first year of a boy's life, at 5 Pcent. P annum, compound interest. Answ. £1418 15 0.

17. £648 from the 6th till the 21st year of a boy's life, at 43, &c. Answ. 1299 16 63.

18. If a merchant commence trade with a capital of £1200, and each year, after paying all expenses, increase the capital of the former year by a fifth part of itself; how much will he be worth at the end of 30 years? Answ. £284,851 11 64.

RULE III. To find the principal, which at a given rate, and in a given time, will amount to a given sum: Or, to find the present worth of a sum at compound interest for a given time, and at a given rate: Divide the given sum by the amount of £1, found by rule I., and the quotient will be the principal, or present worth required

The present worth of 1 may be found by dividing it by its amount for the given time.

Exam. 5. What sum must be lent at compound interest, at 5 cent. annum, at the birth of a child, so that the amount may be £3000 6 8 at the end of 21 years?

Here, the amount of £1 for 21 years being 2.785962, we have for answer £3000·3′÷2·785962 = £1076·9469 = £1076 18 114. Exercises. Required the present worths of the following sums, or the principals that would produce them, at compound interest, at the given rates cent. annum :

19. 324

18 6 for 9 years, at 5 cent., &c. 20. 264 11 8 12 21. 554 18 4 27

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www.

42

4

183 18 912

23. A brother is to pay his sister a portion of £4500 at the end of 11 years: how much will discharge the debt at the end of 4 years, compound interest being allowed at 4 cent annum on the sum he pays? Answ. £3306 14 63.

24. With what capita! must a merchant commence trade, to be worth £15000 at the end of 12 years, if he may be expected to clear annually an eighth of his capital? Answ. £3649 14 73.

25. Whether is it better to sell a farm for £1000 payable at present, £1000 payable at the end of years, and £1000 payable at the end of 10 years; or to sell it for £3000 payable at the end of 5 years, compound interest being allowed at 4 cent. Pannum? Answ. Better at three payments, by £31 14 2.

The reason of the first rule will appear from the following considerations. The amount of £1 for a year will evidently be a hundredth part of the amount of £100 and as £1 is to its amount for a year, so is any other principal to its amount for the same time. Hence, to take a particular instance, the amount of £1 for a year at 5 per cent., will be 1.05; and by the nature of compound interest, this will be the principal for the second year. Then, as the principal £1: £1·05, its amount :: the principal £1.05 : £1·052, which will be the amount at the end of the second year, and the principal for the third year. Again, as £1: £105, its amount:: the principal £1·052 : £1·053, the amount at the end of the third year, and the principal for the fourth year.. In this manner it will appear, that the amount of one pound for any number of years will be equal to £1.05 raised to the power denoted by the number of years. The amount of £1 being thus determined, it is evident, that the amount of any other principal will be had by multiplying the amount of £1 by that principal, since the amount will evidently be proportional to the principal; which proves the second rule.

The third rule is evidently the converse of the second, and hence its correctness is evident.*

It may be proper here to observe, that, if interest be payable yearly, the amount of £1 at the end of 6 months will be the square root of its mount for a year; its amount for 4 months, or one-third of a year, the cube root of the same; for three months the fourth root; for a day the 365th root, &c. Farther, also, the amount of £1 for 2 days will be the quare, for 3 days the cube, &c. of its amount for one day; its amount or 8 months will be the square of its amount for 4 months; its amount for 9 months the cube of its amount for 3 months: and, finally, its amount for a year and a quarter will be the product of its amounts for a year, and for 3 months; and its amount for 6 years and a half will be the product of its amounts for 6 years, and for half a year; and so in similar cases. All this will appear obvious from a due consideration of the nature of compound interest.

If the interest were payable half-yearly, however, at a given rate, per cent. per annum, as in the third example, the amount at the end of a year would be more than if it were payable yearly. Thus at 4 per cent. per annum, payable half-yearly, the amount at the end of half a year would be 102; at the end of a year, or two half years, 1022, or 1'0404; at the end of a year and a half, or 3 half years, 1023; at the end of 2 years, or 4 half years, 1'024, &c.-Had it been payable quarterly, the

a

* If r be put the amount of £1 for a year, p= the principal, t=the time, and a= the amount; the second rule expressed algebraically will become a pr1......(1); or, by taking the logarithms of both sides, log a = log p+t log r.........(2.) From equation 1 we have p, which is the algebraic expression for rule III.: also from equation 2 we have log plog at log r, which is a logarithmic formula for the same purpose. log a-log p

; and from equation 2, log r=

t

From equation 1, we have also r Each of these serves for the resolution of the problem in which it is required to find the rate at which a given principal will amount to a given sum in a given time. The first of them expressed in words gives this rule: Divide the amount by the principal, and that root of the quotient which is denoted by the number of years, will be the amount of £1 for 1 year, whence the rate will be known. Thus, if it be required to find at what rate £1000 would amount to £1360 17 8 in seven years, let the latter, with its shillings and pence reduced to a dccimal, be divided by the former: the quotient is 1-3608625, the seventh root of which is 1.045 nearly. Hence, the rate must be 44 per cent. per annum. The difficulty of the extraction of high roots, renders this mode of calculation inferior almost beyond comparison to that by the logarithmic formula above exhibited, unless, which is rarely the case, a degree of accuracy may be required which cannot be attained by logarithmic tables. In general, indeed, the great facility afforded by logarithms in calulation, will be as much felt in Compound Interest as in almost any other part of Mathematics. log a log P, We have also, from equatioh 2, t = -a formula which will serve to find 1. gr

This

the time in which a given principal at a given rate will amount to a given sum. may also be approximated without logarithms by Position: for if the given amount be divided by the principal, the quotient will be the amount of £1 for the required time: then this and the amount of £1 for a year, being known, the time will be found, either exactly or nearly so, without much difficulty.

amounts at the end of 1, 2, 3, &c. quarters would have been 101, 1'01o, 1013, &c. In such cases, to find the amount of £1 at the end of the proposed time, raise its amount at the end of the first payment, to the power denoted by the number of payments.

As calculations in Compound Interest are much facilitated by the use of interest tables, a table constructed by rule I., is given at the end of the book showing the amount of £1 at compound interest for any number of years not exceeding 50, at the most useful rates. The pupil, after hav ing wrought the preceding exercises by the rules already given, should be taught, instead of finding the amount of £1 by rule I, to take it, when he can, from the table. This table will also be useful in finding by inspection, in many instances, the time, from the principal, rate, and amount; and the rate, from the principal, time, and amount.

ANNUITIES.

An ANNUITY is a fixed sum of money payable at the ends ef equal periods of time, such as years, half-years, or quar

ters.

Annuities are of two kinds, Certain and Contingent.

ANNUITIES CERTAIN are those which commence at a fixed time, and continue for a determinate number of years. ANNUITIES CONTINGEN are those whose commencement, or continuance, or both, depend on some contingent event, usually the life or death of one or more individuals.

The PRESENT VALUE of an an annuity at compound interest, is such a sum as would, if lent at compound interest for the given time, amount to the same sum to which the annuity itself would amount, if forborn during the same time.

When an annuity does not come into possession till a given time has elapsed, or some particular event has taken place, it is said to be an ANNUITY IN REVERSION.

ANNUITIES CERTAIN

RULE I. To find the amount of an annuity, payable yearly, the payments of which are forborn for a given time, compound interest being charged on them as they become due: (1.) Subtract a unit from the amount of £1 for a year, and from its amount for the given time at compound interest: (2.) Di

* An annuity is commonly said to be worth as many years' purchase as there are pounds in the present value of an annuity of £1. Thus in the case of an annuity for 20 years at 5 per cent. per annum, because the present value of an annuity of £1 is £12·462 &© the annuity is said to be worth about 12 years' purchase.