EXAMPLE IV.-In what time would £1,000 amount to 1,159 14s. at 5 per cent. per annum-interest on principal falling due in half-yearly instalments?

=

Here we have a=1025; P=£1,000; A=£1,159'7; £1.025" × 1,000 £1,159°7; 1025"=1,159′7÷1,000 = 11597; n=6, the number of times 1025 is contained, as factor, in 1.1597. So that, the period being a half-year, the required time is 6 half-years, or 3 years.

EXAMPLE V.-In what time would £1,000 amount to 1,160 158. at 5 per cent. per annum-interest on principal falling due in quarterly instalments?

Here we have a=10125; P=£1,000; A=£1,160.75; £10125" X 1,000 £1,160.75; 10125"-1,160 75÷1,000 =116075; n=12, the number of times 10125 is contained, as factor, in 116075. So that, the period being a quarter, the required time is 12 quarters, or 3 years.

302. The principal, amount, and time being known, to determine the rate: Divide the amount by the principal; from the resulting quotient evolve the root indicated by the number of periods in the given time; subtract I from this root; and multiply the remainder by 100. The product so obtained will be the rate, or half the rate, or a fourth of the rateaccording as the period is a year, a half-year, or a quarter.

EXAMPLE VI.-At what rate per cent. per annum would £1,000 produce £1,157 128. 6d. in 3 yearsinterest on principal falling due in yearly instalments?

Here we have n=3; P= £1,000; A= £1,157.625; a3× 1,000=1,157·625; a3=1,157·625÷1,000=1•157625; a=√1157625=1'05, the amount of £1 for one periodi.e., for a year; 105-105, the interest of 1 for a year; 05 × 100=5, the interest of £100 for a year—i.e., the required rate per cent. per annum.

EXAMPLE VII.-At what rate per cent. per annum would £1,000 amount to £1,159 148. in 3 years-interest on principal falling due in half-yearly instalments?

Here we have n = 6; P=1,000; A = £1,159'7 ; a6 × 1,000 = 1,159°7; a6 = 1,159°7 ÷ 1,000 = 1*1597;

a=√11597=1025, the amount of £1 for one period— i.e., for a half-year; 1025-1='025, the interest of £1 for a half-year; 025 × 100=2*5, the interest of £100 for a half-year; 2.5×2=5, the interest of £100 for a year-i.e., the required rate per cent. per annum.

EXAMPLE VIII.-At what rate per cent. per annum would £1,000 amount to £1,160 158. in 3 years-interest on principal falling due in quarterly instalments?

Here we have n=12; P=£1,000; A=£1,160.75; a12 × 1,000=1,160°75; a12=1,160·75÷1,000=1*16075; a= 1.16075 = 10125, the amount of £1 for one period-i.e., for a quarter; 10125-10125, the interest of 1 for a quarter; 0125 × 100=125, the interest of £100 for a quarter; 125 × 45, the interest of £100 for a year—i.e., the required rate per cent. per annum.

NOTE. To find the number of periods in which any principal would double itself at compound interest, we divide the logarithm of 2 by the logarithm of the amount of 1 for the first period. Because, substituting 2P for A in the formula a" × P=A, we have a"× P=2P; log 2 a"=2; n=; If, therefore, the rate were 5 per cent. log a per annum, money would, at compound interest, (a) nearly double itself in 14 years, (b) very nearly double itself in 28 half-years, and (c) more than double itself in 56 quarters according as interest on principal fell due in yearly, half-yearly, or quarterly instalments:

It could be shown, in the same way, that the number of periods in which money would treble itself at compound interest is the quotient obtained when the logarithm is divided by the logarithm of the amount of £1 for the first period.

of 3

303. Property of any description, when given as an equivalent for a fixed annual income,-the income being payable in yearly, half-yearly, or quarterly instalments, according to agreement,-is said to be converted into an ANNUITY.

Although, however, usually employed in this restricted sense, the term "annuity" may be applied to any fixed annual income-such as a salary, a pension, &c.

CONTINGENT.

304. Annuities are of two kinds-CERTAIN and An annuity is "certain" when it is to continue for a definite number of years (10, 20, 50, &c.—as the case may be), or for ever; but when the length of time depends upon the life of some particular person, or upon the life of the survivor of two or more persons, an annuity is a "contingent "or a LIFE—annuity.

305. A "certain" annuity which is to continue for ever is called a PERPETUAL annuity, or a PERPETUITY.

306. When an annuity-either certain or contingent becomes payable at once, it is said to be immediate, or in possession: when, on the other hand, it is not to be available until a certain period of time shall have elapsed, or until a future event (somebody's death, for instance) shall have occurred, an annuity is known as a deferred-or a reversionary-annuity.

Annuities "in possession" are sometimes spoken of as annuities on lives; and "reversionary" annuities, as annuities on survivorship.

307. When an annuity is allowed to remain unpaid for a certain length of time, the sum to which, for arrears and compound interest chargeable upon them, the annuitant becomes entitled is called the AMOUNT of the annuity.

308. By the PRESENT VALUE of an annuity is meant the ready-money which, if improved at compound interest, would exactly pay the annuitant's claim in full.

If an annuity, falling due in annual instalments, remained unpaid for (say) 10 years, the annuitant would be entitled, not only to the arrears, but also to compound interest for

and to a year's (simple) interest on the 9th instalment.*

If, therefore, the instalments were I each, and the rate of interest 3 per cent., the amount of the annuitant's claim for the 10 years would be (§ 298)—

£ £ £ £

£

£ £ £ £ £ 1+1·03+1·032+1·033+1·031+1·035+1·036+1°032+1·038+1°039

* Upon the 10th instalment, due at the end of the 10th year, no interest would be chargeable.

It will be seen that the numbers 1, 103, 1032, 1·033, &c., form a geometrical progression, which has 103 for common ratio; so that (§ 267) the sum of the terms is 1039 × 103-1 10310 — I

103-I

=

*03

If the instalments were half-yearly ones of £1 each, the amount would be-

amount would be

66

66

18th

+2 half-years' COMPOUND interest

17th

"

+3

39

39

Ist

The sum of the terms of this progression, which has

If the instalments were quarterly ones of £1 cach, the

66

+19

66

+ 101519