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1.2

0.30 10309=14.2066, that is 14 1.1.05 0.0211893 years and 75 days nearly,

SCHOLIUM,

Many other fuppofitions might be made with regard to the improvement of money by compound interest. The interest might be fupposed to be joined to the capital, and along with it to bear interest at the end of every month, at the end of every day, or even at the end of every instant, and suitable calculations might be formed; but these suppositions, being seldom used in practice, are omitted.

III. Of. Annuities.

An annuity is a payment made annually for a certain term of years, and the chief problem, with regard to it is, to determine

its

* its present worth.' The supposition on which the solution proceeds is, that the money received by the seller, being improved by him in a certain manner during the continuance of the annuity, amounts to the same sum as the several payments received by the purchaser, improved in the same manner.

The suppositions with regard to the improvement may be various.

be various. What is called the method of fimple interest, in which simple interest only is reckoned upon the purchase-money, and simple interest on each annuity from the time of payment, is so manifestly unequitable, as to be unive:sally rejected; and the supposition which is now generally admitted in practice is the highest improvement possible on both sides, viz. by compound intereit. As the taking compound interest is prohibited by law, the realizing of this supposed improvement requires punctual payment of interest, and therefore the interest in such calculations is usually made low. Even with this advantage, it can hardly be rendered effectual in its full extent; it is however universally

acquiesced

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acquiesced in, as the most proper foundation of general rules ; and when peculiar circumstances require any different hypothesis, a suitable calculation

may

be made. Let then the annuity be called a, and let p be the present worth of it or purchasemoney, t the time of its continuance, and let the other letters denote as formerly.

The seller, by improving the price received p, at compound interest, at the time the annuity ceases, has PR'.

The purchaser is supposed to receive the first annuity a at the end of the first year, which is improved by him for t—1 years; it becomes therefore (Th. 2.) aR-.

He receives the ad annuity at the end of the 2d year, and when improved t-2, it becomes aRi-?.

The 3d annuity becomçs aR-, &c.

The last annuity is simply a, therefore the whole amount of the improved annuities is the geometrical series a taRTaR, &c. ... aR-'. The sum of the series by

R-I R-I. Chap. 6. Sect. 2. is a x =ax

RI

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But,

But, from the nature of the problem,
R-I

R-I PR' =a x and hence p=ax

TR*

r r

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The same conclusion results from calculating the present worth of the several annuities, considered as fums payable în reversion.

Cor. 1.

Of these four p, a, R, t, any three being given, the fourth may be found, by the solution of equations ; t is found easily by logarithms, R or r can be found only by resolving an adfected equation of the t order.

Cor. 2.

If an annuity has been unpaid for the term t, the arrear, reckoning com

RI pound interest, will be ax

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3. The present worth of an annuity in reversion, that is to commence after a certain time (n), and then to continue t

Cor.

years,

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years, is found by subtracting the present
worth for n years from the present worth
for ntt years, and then,

I

I

R-I

R p=ax Fax

th
TR

ron
Also of R, t, n, a, b, any four being gi-
ven, the fifth may be found.

Cor.

4. If the annuity is to continue for-
ever, then R-I and R

*
may

be considered

RI as the fame; and p=ax

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Cor. 5. A perpetuity in reversion (by
Cor. 3.) fince R-I=R', is p=

a

r

Prob. When 12 years of a lease of 2 1 were expired, a renewal for the same term was granted for 1000 l. ; 8 gears are now expired, and for what sum muft a correfponding renewal be made, reckoning 5 per cent. compound interest?

From the first transaction the yearly profit rent must be deduced ; and from this the proper fine in the second may be computed.

In

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