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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 ? 1 . Ans. 110179 years.
ANNUITIES OR PENSIONS,
:: CASE I.
a t Compound Interest..
RULE. 1. Make 1 the first term of a geometrical progression, and the amount of $1 or £1 for one year, at the given rate 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 đols. yearly rent, or annuity, be forborne, (or unpaid) 4 years; what will it amount to, at 6 per cent. per annum, compound interest ? 5.1+-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 one posite 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 1l. or 1 dollar annuity, for the given time, which may be found in Table II. ready calculated.
Bence, either the amount or present worth of annuities may be readily found by Tables for that purpose.
If a salary of 60 dollars per annum to be paid year. • forborne 20 years, at 6 per cent. compound in.
; what is the amount? der 6 per cent. and opposite 20, in Table II, you nd, pulur numbers-36,78359
60 Annuity. Ans. $2207,13540=82207, 13cts. 5m.**! Suppose an Annuity of 1001. be 12 years in arrearsi equired to find what is now due, compound interest allowed at 51. per cent. per annum?
Ins. £1591 14s. 3,024d. (by Table III.) What will a pension of 120l. per annum, payable , amount to in 3 years, at 5l. per cent. compound st?
* Ains. £378 6s... find the present worth of Annuities at Compound
3. 117 in ready
III. To "
1. Di noted by
remaind the pres
3. Dit Roted by before the the prese
Gde the annuity, &c. by that power of the ratio siga by the nunber of years, and subtract the quotient he annuity: 'I'his remainder being divided by the ess 1, the quotient will be tire present value of the ty sought.
What ready money will purchase an Annuity of 501.
added to worth of mainder
BY TABLE III. - Under 5 per cent. and even with 4 vears. * We have 3,54595=present worth of 16. for 4 years. i Multiply by 50=Annuity.
Ans. £177,29750=present worth of the annuity.
2. What is the present worth of an annuity.of 60 dols. per annum, to continue 20 years, at 6 per cent. compound interest :
Ans. $688, 194cts. + 3. What is Sol. per annum, to continue 7 years, worth in ready money, at 6 per cent. compound interest ?
Ans. £ 167 9s. 5d. taxa III. To find the present worth of Annuities, Leases, &c.
taken in KEVERSION, at Compound interest. 1. Divide the Annuity by that power of the ratio de noted by the tine of its continuance.
2. Subtract the quotient froin the Annuity : Divide tlie remainder by the ratio less 1, and the quotient will be the present worth to commence imrnediately.
3. Divide this quotient by that power of the ratio de noted 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 Annaity of 501. payable yearly, for 4 years : but not to commence till two years, at 5 per cent. ? 4th power of 1,05=1,215506)50,00000(41,15515
Subtract the quotient=41,13513
Divide by 1,05—1=,05)8,86487
2d. power of 1,05–1,1025)177,297(160,8136*£160 : 16s. 3d. 1gr. present worth of the Annuity in Reversion.
OR BY TABLE III. Find the present value of il. 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.
S6 years, gives 5,075692
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 afterithat time at 6 per cent. ?
. Ans. £233 15s. 9i. 3. What is the present worth of the reversion of a lease 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 78cts. 23om. IV. To find the present worth of a Freehold Estate, or 1 an Annuity to continue forever, at Coinpound Interest.
RULE. As the rate per cent. i8 to 1001. : so is the yearly rent to the value required. EXAMPLES
1. What is the worth of a Freehold Estate of 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 6 per cent. for his money?
Ans. £, 2500 1. To find the present worth of a Freehold Estate, in Reyersion, at Compound Interest.
RULE. . 1. Find the present value of the estate (by the foregoing 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 freebold estate of 40l. per annum to commence two years hence, be put on sale; what is its value, allowing the purchaser 5l. per cent. ?
As $ : 100 : : 40 : 800-present worth if entered on inmediately.
Then, 1,05 =1,1025) 800,001725,62358=7251. 125, 5}dn=present worth of £800 in two years reversion. Ans.
OR BY TABLE III.
Thas in the foregoing example,
40-annuity or rent.
74,376400=present worth of the annuity or rent, for
- (the time of reversion. Fron S0030000=valne of immediate possession. Take 4,3764 present worth of rent.
6725,6236=6725 i 2s. 5 d. Ans. 2. Suppose an estate of 90 dollars per annům, to cormence 10 years hence, were to be sold, allowing the purs chaser 6 per cent., what is it worth?
Ans. $837, 39cts. 2. ! 3. 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 after the said 15 years, computing at the rate of 5 per cent. per annum, compound interest?
Ans. The first term of 15 years is better than the reyersion forever afterwards, by £75 188. 73d.
A COLLECTION OF QUESTIONS TO EXERCISE
THE FOREGOING RULES.