1.055-1055-ratio. Hence the rate is 5 per cent. per annum, Answer.

DISCOUNT BY COMPOUND INTEREST.*

The sum, or debt to be discounted, the time and rate, given, to find the present worth.

RULE. Divide the debt by that power of the amount of 11. or D. for 1 year, denoted by the time, and the quotient will be the present worth, which, subtracted from the debt, will leave the discount.

EXAMPLES.

1. What is the present worth, and discount, of 6001. due 3 years hence, at 61. per cent. per annum, compound interest?

3

Divide by 106=119101)600-00000(503-7741 = £.503 15s. 5jd. present worth, and .600-£.503 15 54.96 4s. 6d. = discount.

In this Table, corresponding to the time and rate, we have ⚫839619-present worth of 11. for the time and rate.

Multiply by 600 debt, or principal.

503-771400-present worth of the debt.

2. What is the present worth of 3121. 10s. due 2 years hence, at 44 per cent. per annum, compound interest?

Answer, .286 3s. 3d. 2.97qrs. 3. What

* Let mfum, or debt, to be discounted, and the other letters as before: Then the following Theorems will show all the cafes in Discount by Compound Interest.

I. ==p. II. pr=m. III. —=r which being continually divided by r, tiil Р

nothing remain, the number of these divifions will be equal to t.

IV. —=r which being extracted, (the time, given in the question, showing the

1

Note. Cafe 2d. may be wrought by Table 1, thus: Find that power of 11. for year, denoted by the time: multiply the prefent worth by it, and the product will be the answer.

Or, by Table 2d. thus: Find the present worth of 11. for the given time, by which divide the prefent worth, and the quotient will be the debt o principal. Cafe Sd. thus: Divide the debt by its prefent worth, and feek the quotient in Table 1, under the given rate, and in the line with it you will (ee the time.

Cafe 4th. is wrought in the fame manner, only, feek the quotient in a line with the time, it will show the rate atop.

3. What ready money will discharge a debt of 1000D. due 4 years hence, at 5D. per cent. per annum, compound interest?

Answer, 822D. 70c. 2m.

ANNUITIES OR PENSIONS, IN ARREARS, AT COM. POUND INTEREST.

CASE I.

When the annuity, or pension, the time it continues, and the rate per cent are given, to find the amount.

RULE.*1. Make 1 the first term of a Geometrical Progression, and the amount of 11. or D. for 1 year at the given rate per cent. the ratio.

2. Carry the series to so many terms as the number of years, and .find its sum.

3. Multiply the sum thus found by the given annuity, and the product will be the amount sought.

Or, multiply the amount of £.1 or D.1 for 1 year into itself so many times as there are years less by 1; then multiply this product by the annuity; and subtract the annuity therefrom. Lastly, divide the remainder by the ratio less 1, and the quotient will be the amount. EXAMPLES.

It is plain that upon the first year's annuity there will be due fo many years' compound interest, as the given number of years lefs 1, and gradually one year lefs, upon every fucceeding year, to that preceding the last, which has but one year's intereft, and the last bears none.

Let r rate, or amount of 11. for 1 year, then the series of amounts of 11. annuity for several years, from the first to the laft, is 1, r, r ,3,&c. to -1; and the

2 3

fum of this, according to the rule in Geometrical Progreffion, will be

= 2

mount of fl.annuity fort years. And all annuities are proportional to their amounts; therefore, 1:

:: 7:

X = amount of any given annuity #.

Let r=rate, or amount of 11. for 1 year, and the other letters as before, then

And from these equations, all the cafes relating to annuities or pensions in atrears, may be conveniently exhibited in logarithmick terms, thus :

L. Ln +Lr2—1 + L.r—1=La.

II. La—L.r2 −1 + Lr—1=La.

which continually divided by till nothing remain, the number of thofe divifions will be equal to : Or, being extracted, (the given time fhewing the power) will be equal to r.

EXAMPLES.

1. What will an annuity of 60l. per annum, payable yearly, amount to in 4 years, at 61. per cent.?

Or, 1+1·06+1·06* + 1·06) *× 60 = £.262 9s. 6+d.

Divide by 1.06-1='06) 15·7482(262·47=£.262 9s. 4 d. Ans.

12

OR, BY TABLE HI.*

Multiply the tabular number under the rate, and opposite to the time, by the annuity, and the product will be the amount.

2. What will an anuity of 60l. per annum amount to in 20 years, allowing 61. per cent. compound interest?

Under 61. per cent. and opposite 20, in table 3d. you will find, Tabular number 36.78559,

=

2207.13540.2207 2s. 81d. Ans.

3. What will a pension of D.75 per annum, payable yearly, amount to in 9 years at 5 per cent. compound interest?

Ans. D.826 99c. 23 m.

4. If a salary of 1001. per annum, to be paid yearly, be forborne 5 years, at 61. per cent. What is the amount? Ans. £.563 14s. 2d. 5. What will wages of D.25 per month, amount to in a year, at per cent. per month? Ans. D.308 38c. 9m.

CASE II.

When the amount, rate per cent. and time are given, to find the annuity, pension, &c.

RULE.-Multiply the whole amount by the amount of 11. or D.1 for a year, from which subtract the whole amount, divide the remainder by that power of the amount of 11. or D.1 for a year, signified by the number of years, made less by unity, and the quotient will be the answer.

Or, find the amount of an annuity of 11. or D.1 for the given time and rate (by Case 1 ;) divide the given sum by this amount; and the quotient will be the annuity required.

EXAMPLES.

1. What annuity, being forborne 4 years, will amount to L.262-47696, at 61. per cent. compound interest?

262.47696=
= amount

1.06 amount of 11. for 1 year.

=

Multiply by

* Table 3 is calculated thus: Take the first year's amount, which is 11. multiply it by 106+1=206=fecond year's amount, which alfo multiply by 106+1=3·1836 third year's amount, &c. and in this manner proceed in calculating tables at any

other rates.

Or, by Table III. the amount of 11. is found to be 4.374616; and the answer is found, as before.

2. What annuity, being forborne 20 years, will amount to D.2207.1354, at 6. per cent. compound interest?

Amount of an annuity of D.1 for 20 years at 6. per cent. per an num= 36.78559. And,

36-78559)2207-13540(D.60, Ans.

2207.1354

0

CASE III.

When the annuity, amount and ratio are given, to find the time. RULE.-Multiply the amount by the ratio, to this product add the annuity, and from the sum subtract the amount; this remainder being divided by the annuity, the quotient will be that power of the ratio signified by the time, which being divided by the amount of 11. for 1 year, and this quotient by the same, till nothing remain, the number of those divisions will be equal to the time Or, look for this number under the given rate in table 1, and in a line with it, you will see the time. Or,

Divide the amount by the annuity; from the quotient subtract 1; from the remainder subtract the ratio; from successive remainders subtract the square, cube, &c. of the ratio, till nothing remain; and the whole number of the subtractions will be the answer. Or, find the quotient in Table III. under the rate, and in a line with it stands the answer.

EXAMPLES.

1. In what time will 601. per annum, payable yearly, amount to £.262-47696, allowing 61. per cent. compound interest, for the forbearance of payment?