570. To find the amount of a given sum in any time at simple interest.

Let P be the principal in pounds.

n the number of years for which interest is taken.

r the interest of one pound for one year.

M the amount.

Sincer is the interest of one pound for one year, Pr is the interest of P pounds for one year, and therefore nPr the interest of P pounds for n years;

therefore

M=P+Pnr.

From this equation if any three of the four quantities M, P, n, r are given, the fourth can be found; thus

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

Let R denote the amount of one pound in one year, so that R=1+r, then PR is the amount of P in one year; the amount of PR in one year is PRR or PR, which is therefore the amount of P in two years at compound interest. Similarly the amount of PR3 in one year is PR3, which is therefore the amount of P in three years. Proceeding thus we find that the amount of P in n years is PR";

The interest gained in n years is M - P or P (R” — 1).

572. Next suppose interest is due more frequently than once a year; for example, suppose interest to be due every quarter,

and let 4

be the interest of one pound for one quarter. Then, at

compound interest, the amount of P in n years is P (1 + 1)" ;

for the amount is obviously the same as if the number of years

were 4n, and the interest of one pound for one year. Simi

4

larly, at compound interest, if interest be due q times a year, and for each interval, the amount of

the interest of one pound be

At simple interest the amount will be the same in the cases supposed as if the interest were payable yearly, r being the interest of one pound for one year.

573. The formula of the preceding articles have been obtained on the supposition that n is an integer; we may therefore ask whether they are true when n is not an integer. Suppose

1

n = m + −,

μ

1

where m is an integer and

a proper fraction. At

simple interest the interest of P for m years is Pmr; and if the borrower has agreed to pay for any fraction of a year the same

fraction of the annual interest, then

Pr
μ

is the interest of P for

th

Pr м

of a year; hence the whole interest is Pmr + that is,

Pnr, and the formula for the amount holds when n is not an integer. Next consider the case of compound interest; the amount of P in m years will be PR"; if for the fraction of a year interest is due in the same way as before, the interest of PRTM for PR and the whole amount is PR (1+ μ

th

of a year is

(1 + 2).

On this supposition then the formula is not true when n is not an integer. To make the formula true the agreement must be of a year

that the amount of one pound at the end of

th

shall be (1+r)2, and therefore the interest for

1

th

of a year

(1+r)μ − 1. This supposition though not made in practice is

often made in theory, in order that the formulæ may hold uni

versally.

Similarly if interest is payable q times a year the amount of P in n years is P (1+

(1+2)", by Art. 572, if n be an integer; and

it is assumed in theory that this result holds if n be not an integer.

574. The amount of P in n years when the interest is paid

g times a year is P (1+ increase without limit, this becomes Pe" (Art. 552), which will therefore be the amount when the interest is due every moment.

(1+2)", by Art. 572; if we suppose q to

575. The Present value of an amount due at the end of a given time is that sum which with its interest for the given time will be equal to the amount. That is, (Art. 567), the Principal is the present value of the amount.

576. Discount is an allowance made for the payment of a sum of money before it is due.

From the definition of present value, it follows that a debt due at some future period is equitably discharged by paying the present value at once; hence the discount will be equal to the amount due diminished by its present value.

577. To find the present value of a sum due at the end of a given time and the discount.

Let P be the present value, M the amount, D the discount, r the interest of one pound for one year, n the number of R the amount of one pound in one year.

years,

At simple interest:

M= P(1+nr), (Art. 570);

578. In practice it is very common to allow the interest of a sum of money paid before it is due, instead of the discount as here the payer

defined. Thus at simple interest, instead of

would be allowed Mnr for immediate payment.

Mnr 1 + nr

EXAMPLES OF INTEREST.

1. Shew that the discount is half the harmonic mean between the sum due and the interest on it.

2. The interest on a certain sum of money is £180, and the discount on the same sum for the same time and at the same rate is £150; find the sum.

3. If the interest on £A for a year be equal to the discount on £B for the same time, find the rate of interest.

4. If a sum of money doubles itself in 40 years at simple interest, what is the rate of interest?

5. A tradesman marks his goods with two prices, one for ready money, and the other for a credit of 6 months; what ratio ought the two prices to bear to each other, allowing 5 per cent. simple interest?

6. Find in how many years £100 will become £1050 at 5 per cent. compound interest; having given

log 141-14613, log 15 1.17609, log 16 = 1.20412.

=

7. Find how many years will elapse before a sum of money trebles itself at 3 per cent. compound interest; having given

8. If a sum of money at a given rate of compound interest accumulate to p times its original value in m years, and to q times its original value in n years, prove that

n = m log, q.

XLII. EQUATION OF PAYMENTS.

579. When different sums of money are due from one person to another at different times, we may be required to find the time at which they may all be paid together, so that neither lender nor borrower may lose. The time so found is called the equated time.

580. To find the equated time of payment of two sums due at different times supposing simple interest.

2

Let P1, P, be the two sums due at the end of times t1, t ̧ respectively; suppose t, greater than t1; let r be the interest of one pound for one year, x the equated time.

2

The condition of fairness to both parties may be secured by supposing that the discount allowed for the sum paid before it is due is equal to the interest charged on the sum not paid until after it is due.

The discount on P, for t2-x years

therefore

2

2

the interest on P, for x-t1 years is P, (x−t)r;