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3. Seven gentlemen met at an inn, and were so well pleased with their host, and with each other, that they agreed to tarry so long as they, together with their host, could sit every day in a different position at dinner; how long must they have staid at said inn to have fulfilled their agreement ?
Ans. 110179 years.
ANNUITIES OR RENSIONS,
at Compound Interest.
RULE. 1. Makc 1 the first term of a geometrical progression, and the amount of $1 or £1 for one year, at the given fate per cent. the ratio.
2. Carry on the series up to as many terms as the given 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.
EXAMPLES. 1. If 125 dols. yearly rent, or annuity, be forborne, (or , enpaid) 4 years; what will it amount to, at 6 per cent. per annum, compound interest?
1+1,06+1,1236+1,191016=4,374616 sum of the series.* Then, 4,374616x125=8546,827 the amount sought.
OR BY TABLE II. Multiply the Tabular number under the rate and opposite to the time, by the annuity, and the product will be the amount sought.
*The sum of the series thus found, is the amount of 11. or 1 dollar annuity, for the given time, which may be found in Table II. ready calculated.
Hence, either the amount or present worth of annuities may be readily found by Tables for that purpose.
2. If a salary of 60 dollars per annum to be paid yoar. sy, be forborne 20 years, at 6 per cent. compound inerest; what is the amount ?
L'ndler 6 per cent. and opposite 20, in Table II, you will find, Tabular number=36,78559
Ans. 82207,13540=82207, 13cts. 5m. + 3. Suppose an Annuity of 1001. be 12 years in arrears, it is required to find what is now due, compound interest being allowed at 5l. per cent. per annum?
Aus. £1591 14s. 3,024d. (by Table III.) 4. What will a pension of 1201. per annum, payable yearly, annount to in 3 years, at 5l. per cent. compound interesti
ins. [978 6s. II. To find the present worth of Annuities at Compound
Divide the annuity, &c. by that power of the ratio sig. nified by the number of years, and subtract the quotient from the annuity: 'This remaiuter being divided by the ratio less 1, the quotient will be the present value of the Annuity sought.
1. What ready money will purchase an Annuity of 501 to continue 4 years, at šl. per cent. compound interest ? 4th pervers of the ratio, sa
BY TABLE III.
We have 3,54595 =present worth of 1l. for 4 years. Multiply by 50=Annuity.
Ans. £177,29750=present worth of the annuity.
2. What is the present worth of an annuity of 60 cols. per annum, tu continue 20 years, at 6 per cent. compound interest ?
Ins. 8688, 19 jets. + 3. What is 30l. per annum, to continue 7 years, worth in ready money, at 6 per cent. compound interest ?
Ans. £167 9s. 5d. + III. To find the present worth of Annuitics, Leascs, &c.
taken in REVERSION, at Compound Interest. 1. Divide the Annuity by that puwer of the ratio de. Roted by the time of its continuance.
2. Subtract the quotient from the Annuity : Divide the remainder by the ratio less l, and the quotient will be the present worth to commence immediately.
3. Divide this quotient by that power of the ratio denoted by the time of Reversion, (or the time to come before the Annuity commences) and the quotient will be the present worth of the Annuity in Reversion.
EXAMPLES 1. What ready money will purchase an Annuity cf 501. pavable yearly, for 4 years : but not to commence till two years, at 5 per cent. 4th power of 1,05=1,215546)50,00000(41,13513
Subtract the quotient=41,13513
Divide by 1,05–13,05)8,86487 2d. power of 1,05=1,1025)177,297(160,8156=6160 16s. 30. lyr. present worth of the Annuity in Reversiun.
OR BY TABLE UI. Find the present value of 11. at the given rate for the sum of the time of continuance, and time in reversion added together; from which value subtract the present worth of il. for the time in reversion, and multiply the remainder by the Annuity; the product will be the answer.
Thus in Example 1.
=6 years, gives 5,075698 Time in reversion, =2 years, -- 1,859410
Remainder, 3,216282 X50
Ans. £160,8141 2. What is the present worth of 751. yearly rent, which is not to commence until 10 years hence, and then to continue 7 years after that time at 6 per cent. ?
Ans. £233 15s. 9d. 3. What is the present worth of the reversion of a leage of 60 dollars per annum, to continue 20 years, but not to commence till the end of 8 years, allowing 6 per cent to the purchaser ? Ans. 8431 78ets. Some IV. To find the present worth of a Freehold Estate, or an Annuity to continue forever, at Compound Interest.
RULE. As the rate per cent. is to 100l. : so is the yearly rent to the value required. EXAMPLES.
1. What is the worth of a Freehold Estate af 401. per annum, allowing 5 per cent. to the purchaser ?
As £5 : £100 :: 640 : £800 Ans. 2. An estate brings in yearly 1501. what would it sell for, allowing the purchaser ô per cent. for his money ?
Ans. £2500 V. To find the present worth of a Freehold Estate, in Reversion, at Compound Interest.
RULE. 1. Find the present value of the estate (by the foregomg rule) as though it were to be entered on immediately, and divide the said value by that power of the ratio de noted by the time of reversion, and the quotient will be the present worth of the estate in Reversion. .
EXAMPLES 1. Suppose a freehold estate of 401. per annum to com. mence two years hence, be put on sale; what is its value, ollowing the purchaser 5l. per cent. ?
As 5 : 100 :: 40 : 800 me present worth if entered on immediately.
Then, 1,05=1,1025)800,00(725,62358=725l. 12s. 53d. =present worth of £ 800 in two years reversion. Ans.
OR BY TABLE III. Find the present worth of the annuiiy, or rent, for the time of reversion, which subtract from the value of the immediate possession, anu you will have the value of the estate in reversion.
Thus in the foregoing example,
40=annuity or rent.
74,376400=present worth of the annuity or rent, for
(the time of reversion. from .800,0000=value of immediate possession. I'ake 74,3764=present worth of rent.
6725,6236=£725 12s. 5 d. Ans. 2. Suppose an estate of 90 doliars per annum, to commence 10 years hence, were to be sold, allowing the purchaser 6 per cent. ; what is it worth?
Ans. 8837, 39cts, 2m. S. Which is the most advantageous, a term of 15 years, in an estate of 100l. per annum; or the reversion of such an estate forever alter the said 15 years, computing at the rate of 5 per cent. per annum, compound juterest ?
Ans. The first terin of 15 years is better than the reversion forever afterwards, by 675 18s. 7 d.
A COLLECTION OF QUESTIONS TO EXERCISE
THE PREGOING RULES. 1. I demand the sum of 1748} added to itself?
Ans. 3497. 2. What is the difference between 41 eagles, and 1099 diines ?
Ans. 10cis. 3. What number is that which being multip ied by 21, the product will be 1365 ?