EXAMPLE.

The amount of a sum at 7 per cent. compound interest, for 14 years, is $12892.65: Required the principal lent.

12892.652.57853, tabular number for 14 years,=$5000. Ans. CASE III.

The principal, rate and amount, given to find the time.

RULE.-Divide the amount by the principal, the quotient will be the tabular number, or amount of 1, opposite which, find the number of years, &c.; otherwise, divide the amount of 1, continually by the ratio, till nothing remains; the number of divisions will be equal to the number of years.

Lent out $5000, at 7 per cent. compound interest; on settlement, received $12892.65: Required the time the money was at interest.

12892.65 5000 2.57853, tabular number for 7 per cent. which, on inspecting the table, I find opposite 14 years. Ans. CASE IV.

The principal, time and amount, given to find the rate per

cent.

RULE. Divide the amount by the principal, for the tabular number, then extract that root, denoted by the number of years, for the amount of 1 in one year; this last, less unity, will be the ratio.

Lent $5000, which in 8 years amounts to $7969.24: Required the rate per cent.

7969.24 5000=1.593848, tabular number for 8 years, and /1.593848 1.06, amount of 1 for one year, and 1.06-1.06 ratio. Ans.

Annuities, or Pensions in Arrear, at Compound

Interest.

RULE 1.-Find the amount of the given yearly sum, for the number of years less 1, which will be the last term of a geometrical progression, of which the given sum is the first; then find the sum of that progression and it is the amount of the annuity required,

2. Otherwise, find the amount of 1, for the given years less 1; then find the sum of that progression, whose first term is 1, and last term the said amount; and multiply the sum by the given annuity for the amount required.

EXAMPLE.

An annuity of $320, being 11 years in arrear; required the amount due, compound interest computed at 5 per cent. on every payment.

Showing the amount of £1 or $1, annuity forborne 21 years,

This Table is constructed from Table 1, of compound interest.

Thus to 1.

add 1.07,
2.07,

add 1.1449,

the amount of 1 for 1 year's annuity,
the amount of 1 at compound int.-tab. 1.

amount of 1 annuity at 2 years end,

2d term of table 1.

3.2149, amount of 1 annuity at 3 years end, add 1.225043, 3d term of table 1.

4.439943, amount of 1 annuity at 4 years end, at 7

=

per cent.

USE.

Multiply the annuity by the amount of 1, for the given time; the product will be the amount and answer required.

An annuity of $20 per annum, is forborne 7 years: How much is due at the end of that term, allowing the annuitant 6 per cent. per annum, compound interest?

Tabular number for 6 years, in table 1,1.418519

The foregoing may be explained by the following calculation.

20.

X 1.06

21.20 +20.

due first year.

167.876772 seventh year, or $167.87, as before.

3. If £30 yearly rent be forborne 9 years, what will it amount to in that time, at 6 per cent. per annum, compound interest? Ans. £344 14 94.

4. Suppose a person who has an annuity of £20 per annum, suffers it to be in arrear 15 years, what has he to receive at the end of that term, compound interest being computed at 6 per cent.? Ans. £465 10 43.

5. An annuity of $80 per annum is forborne 17 years: How many dollars has the annuitant to receive on settlement, compound interest computed at 7 per cent. per annum ?

CASE II.

Ans. $2467.20.

The amount, rate and time given, to find the annuity :

RULE.-Divide the amount by the amount of one for the rate and time (see tab. 2nd) the quotient is the annuity.

EXAMPLE.

Amount of an annuity for 17 years, at 7 per cent. is $2467.20 : Required the annuity.

2467.20

30.84 tab. no. for 17 yrs. tab. II.=80 dolls. Ans.

CASE III.

Annuity, rate, and amount given to find the time:

RULE.-Multiply the amount by one,+the rate, and add the annuity; from this sum subtract the amount; divide the remainder by the annuity, the quotient will be the amount of 1 at the given rate, which in table first stands opposite the number of years required—or divide this amount continually by the ratio till nothing remains, the number of divisions will give the number of years, &c.

EXAMPLE.

Amount of an annuity is $2467.20, rate 7 per cent. annuity $80: Required the time.

2467.20 x 1.07-2639.904+80=2719.904

and 2719.904-2467.20—252.704-80 3.1588 tabular number for 17 years, at 7 per cent. in table first.

Construction of the above Table.

Let £1 or $1 be divided by 1 more the ratio, the quotient will be the value of 1 at the year's end, and the first term; then this quotient being divided by the above divisor, gives its value at the end of the second year and second term.

Thus 11.07.934579, value of £1 or $1, due 1 y. hence. And .934579-1.07=.873438, value of 1, due 2 years hence. USE.-Multiply the sum to be discounted by the number answering to the number of years and rate per cent.-the product is the present worth. Otherwise,

Find the amount of 1 for the given time and rate per cent.; divide the sum to be rebated thereby-the quotient will be the present value.

EXAMPLES.

1. Suppose £521 4 114 were to fall due 10 years hence, what is the present worth of the same, discount being allowed at 5 per cent. per annum, compound interest ? Ans. £320.

2. What is the present worth of £629 17 1, due 3 years hence, at 8 per cent. per ann. compound interest? Ans. £500.

3. A legacy of £520 18 7 is left payable in 4 years; but the executor is willing to pay it at the end of one year, on being allowed discount at 5 per cent. per annum, compound interest: This being agreed to, what had he to pay? Ans. £450.