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297. To determine the amount of £1, at compound interest, when the time and rate are known: Find the amount of £1 for the first “period”—i.e., for the first year, half-year, or quarter, according as the interest on principal falls due in yearly, half-yearly, or quarterly instalments; and raise this amount to the power indicated by the number of periods.'

We have just seen that the amount of £i for 3 years, at 5 per cent., compound interest, would be £1.053

year. 21.025 if the period were a half- -year. £1012519

quarter. It is obvious that—all other circumstances being the same when the amount of £ 1 is multiplied by

2
3 we obtain the amount of 73
4

£4
&c.

&c. 298. The principal, time, and rate being known, to determine the compound interest: Find the amount of £1 for the given time, and at the given rate (S 297); multiply this amount by the principal; and subtract the principal from the resulting product.

Example I.—What compound interest would £,1,000 have produced in 3 years, at 5 per cent.—interest on principal falling due in annual instalments ?

Amount of £I=£1053=£1.157625; amount of £1,000= £1157625 * 1,000= £1,157.625=£1,157 128. 6d. ; required compound interest=£157 125. 60. - the difference between £1,157 12s. 6d. and £1,000.

NOTE 1.-If interest on principal fell due in half-yearly instalments, we should have

Amount of £=£1025=£I'1597 (nearly); amount of £1,000=£1:1597 x 1,000= £1,1597=£1,159 148.; required compound interest £159 145.— the difference between £1,159 14s. and £ 1,000.

And if interest on principal fell due in quarterly instalments, we should have

Amount of £I=£14012512=£1:16075; amount of £1,000=

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£1'16075 X 1,000 = £1,16075=£1,160 158.; required com

= pound interest=£160 158. — the difference between £1,160 159. and £1,000.

NOTE 2.—The amount of £i for the first period being represented by a, and the number of periods by n, the amount of £i for n periods would be a" ($ 297); so that, for n periods, the amount of

£2)

(La" X 2 £3 would be £a" X3 £4

Za" X 4 &c.

&c. If, therefore, we put P for the principal, and A for the sum to which P would have amounted in n periods, we shall have

an XP=A. This easily-remembered formula enables us to deal with any exercise in Compound Interest.

299. The amount, time, and rate being known, to determine the principal: Find what the amount would be if the principal were for ($ 297); and divide the result into the given amount.

EXAMPLE II.--What sum would have amounted, in 3 years, at 5 per cent., to £1,157 128. 6d.—interest on principal falling due in annual instalments ?

Here we have a (the amount of £i for the first period)= £105; n (the number of periods)=3; a" (the amount of £I for n periods)=£1:053; A (the given amount)=£1,157-625; £1053XP=£1,157625; P (the required principal)=

(1,157.625= 61,157-625=£1,000.

1.053 1'157625 NOTE.-If interest on principal fell due in half-yearly instalments, we should have a= = £1'025; n=6; a"=£1'0256;

1,157-625 £1025*x P=£1,157-625; P=£1,157-625.

1'025

I'1597 £998 45. 3d. (nearly.)

And if interest on principal fell due in quarterly instalments, we should have a=£10125; n=12; an=£1012512 ; £10125" XP= £1,157.625; P=£

$1,157-625= $1,157-625_

. (

1'0125" I'16075 £997 68. 2d. (nearly.)

300. The principal, amount, and rate being known, to determine the time : Divide the amount by the principal, and find what power the resulting quo

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tient is of the amount of £ 1 for the first period; the index of this power will be the number of periods in the required time.

EXAMPLE III.-In what time would £1,000 have amounted to £1,157 12s. 6d., at 5 per cent.-interest on principal falling due in annual instalments?

Here we have a=£1:05; P= £1,000; A=£1,157.625; £1'05" X 1,000=£1,157.625; 1:05"= £1,157625 1,000= 1.157625; n=3—the number of times I'05 is contained, as factor, in 1'157625. So that, the period being a year, the required time is 3 years.

EXAMPLE IV.-In what time would £1,000 have amounted to £1,159 145., at 5 per cent.-interest on principal falling due in half-yearly instalments?

Here we have a= £1:025; P=£1,000; A=£1,15997; £1.025" X 1,000=£1,1597 ; 1'025"=1,1597–1,000=1°1597; n=6–the number of times 1'025 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 have amounted to £1,160 155., at 5 per cent.-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; 1'0125"=1,16075 + 1,000 =: 1:16075; n=12-the number of times 1'0125 is contained, as factor, in 1:16075. So that, the period being a quarter, the required time is 12 quarters, or 3 years.

301. 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 would £1,000 have produced £1,157 12s. 6d. in 3 years—interest on principal falling due in annual instalments?

Here we have n=3; P=£1,000; A=£1,157.625; aix 1,000 = 1,157:625; a 1,157.625-1,000 1.157625; a= ☺1•157625=1'05, the amount of £ 1 for one period—i.e., for a year; 1:05—1='05, the interest of £i for a year; '05 X 100 =5, the interest of £ 100 for a year—i.l., the required rate per cent.

EXAMPLE VII.-At what rate would £1,000 have amounted to £1,159 14s. in 3 years-interest on principal falling due in half-yearly instalments ?

Here we haven=6; P=£1,000; A=£1,15997; a® X 1,000= 1,597; a=I,I 59*7-1,000=II 597; a=.

= $ 1.1597 = 1.025, the amount of £i for one period-i.e., for a half-year; 1'025 – I='025, the interest of £ 1 for a half-year; `025 X 100=2-5, the interest of £100 for a half-year; 2-5X2=5, the interest of £100 for a year-i.e., the required rate per cent.

EXAMPLE VIII.—At what rate would £1,000 have amounted to £1,160 155. in 3 years—interest on principal falling due in quarterly instalments ?

Here we haven=12; P=£1,000; A=£1,160*75; a" X 1,000 =1,165-75; a"=1,160'75 1,000=1:16075; a=1'16075=

= 10125, the amount of £i for one period—i.e., for a quarter ; 10125 – I =0125, the interest of £i for a quarter; •0125X 100=1'25, the interest of £100 for a quarter; 1.25 X 455, the : interest of £100 for a year--i.e., the required rate per cent.

Note.To find the number of periods in which any principal would have doubled itself at compound interest, we divide the logarithm of 2 by the logarithm of the amount of £1 for the first period. Because, substituting 2 P for A in the formula

log 2 anxP=A, we have an XP=2P; a"=2; n=

So that,

log a if the rate were 5 per cent., money would, at compound interest, have (a) nearly doubled itself in 14 years, (b) very nearly doubled itself in 28 half-years, and (c) more than doubled itself in 56 quarters—according as interest on principal fell due in yearly, half-yearly, or quarterly instalments :

(a.)

log 2 *3010300
log 1'05 *0211893

=1492 (years)
log 2
(6.)

3010300=28.07 (half-years),
log 1'025 0107239

log 2 (c.)

*3010300—55.8 (quarters) log 1'0125 0053950

· It could be shown, in the same way, that the number of periods in which money would have trebled itself at compound interest is the quotient obtained when the logarithm of 3 is divided by the logarithm of the amount of Zi for the first period.

ANNUITIES.

cular perso

302. 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.

303. Annuities are of two kinds—CERTAIN and CONTINGENT. 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 partiperson, or upon

or upon the life of the survivor of two or more persons, an annuity is a “contingent”-or a LIFE-annuity.

304. A“ certain” annuity which is to continue for ever is called a PERPETUAL annuity, or a PERPETUITY.

305. 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 reversionaryannuity.

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

306. 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.

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