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401.

SIMPLE INTEREST.

To find the Amount of a given sum, in any time, at

simple interest.

Let P be the principal, in pounds,

n the No. of years for which the interest is to be calculated*. r the interest of £1 for one year†.

M the amount.

Then, since the interest of a given sum, at a given rate, must be proportional to the time, 1 (year) : n (years) :: r: nr, the interest of £1 for n years; and the interest of P£ must be P times as great, or Pnr; therefore the amount M= P+Pnr.

402.

From this simple equation, any three of the quantities P, n, r, M being given, the fourth may be found; thus

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Ex.

What sum must be paid down to receive 600£, at the end of nine months, allowing 5 per cent. abatement? Or, which is the same thing, what principal P will in nine months amount to 600£, allowing interest at the rate of 5 per cent. per annum?

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403. To find the amount of a given sum in any time at compound interest.

Let R=1£ together with its interest for a year; then at the end of the first year, R becomes the principal, or sum due;

When days, weeks, or months, not making an exact number of years, enter the calculation, n is fractional. ED.

+ It must always be borne in mind that r is not the rate per cent. but only the hundredth part of it. Thus for 4 per cent. r=0·04£, for 5 per cent. r=0·05£; and so on. ED.

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and so on; so that R" is the amount of 1£ in n years. And if P£ be the principal, the amount must be P times as great, or

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COR. 2.

The interest = M-P=PR"-P=P{R^— 1}.

Ex. 1. What must be paid down to receive 600£ at the end of 3 years, allowing 5 per cent. per annum compound interest? In this case R=1'05, n=3, M=600;

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Ex. 2. Find the amount of 5£ in 2 years at 3 per cent., compound interest.

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.. PR"=5×1·0767=5·3835=5£. 7s. Sd., the amount required.

404. When compound interest is named, it is usually meant that interest is payable only at the end of each year; but there may be cases in which the interest is due half-yearly, quarterly, &c.; and then the amount found in the last Article will be altered. Thus, if r be the interest of 1£ paid at the end of the year, it has been shewn that the amount of P£ at the end of n years=P(1+r)".

But if be the interest of 1£ paid at the end of each half-year, then

2

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Similarly, if be the interest of 1£ paid at the end of each quarter,

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And, generally, if interest be considered due q times a year, at equal intervals, each payment for 1£ being,

q

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Ex. Find the amount of 100£ in 1 year at 5 per cent. per annum, when the interest is due, and converted into principal, at the end of each half-year.

Here P=100, r=0·05, n=1,

.. Amount required=100×(1+0·025)2=105·0625£,

=105£. 1s. 3d.

405. Required the amount of a given sum at compound interest, the interest being supposed due every instant.

The interest being paid q times per annum, by last Art., the amount

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Let q be indefinitely great, that is, the intervals between the payments

indefinitely small, then, neglecting and its powers,

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406. To determine the advantage, when compound interest is reckoned, of having interest paid half-yearly, quarterly, &c. instead of yearly.

It appears from Art. 404 that the advantage per 1£ for a year, when interest is paid half-yearly, and the half-yearly payment is half the yearly

one,

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In the case of quarterly payments of interest, the advantage

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And generally, when the interest is paid q times a year, the advantage

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Hence it appears, that, for a single year, the advantage of having interest paid frequently is very small. But it increases as the number of years increases, and is expressed in n years, for every 1£, (when interest is paid q times a year at equal intervals, being the payment per 1£,) by

r

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407. It must be observed always, when interest is paid q times per annum, each payment being for

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every 1£, that the true annual rate of

interest is not r, but (1+2)-1, since this

rest for 1£ for a year.

expresses the value of the inte

PROB. In what time will any sum of money double itself, at any given rate of interest simple or compound?

I. In the case of simple interest M=P+Pur,

... here 2P=P+Pnr,

or n=

1, the number of years required.

II. In the case of compound interest paid yearly M=P(1+r)",

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The following Table, calculated from these two results, and shewing the several times in which any sum will double itself at the rates of interest there given, is taken from Baily's Doctrine of Interest and Annuities, and will furnish good practice to the learner, who will verify it by means of the Logarithmic Tables.

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In general practice, compound interest is only reckoned for an integral number of years, so that if there be any fractional part of a year remaining, for this simple interest is taken.

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