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Rule for quarterly payments. Take a fourth part of the ratio, a fourth part of the annuity, and four times the number of
years; that is, reduce the years to quarters, and work as before.
9. If 280 dollars annuity, payable quarterly, was unpaid five years, what will it amount to in that time, at 5 per cent per annum?
Ans. 1566 dollars 25 cents.
To shew the difference of the payments.
by half yearly payments 1557
ANNUITIES OR PENSIONS AT COMPOUND
To find the amount of an annuity, or pension, in arrears, by compound interest.
Rule 1. Make 1 the first term of a geometrical progression, and the amount of one dollar for one year, at the given rate per cent. the ratio. Carry on the series up to as many terms as the given number of years, and find its sum, which multiply by the given annuity, and the product will be the amount required.
6 1,000000 1,000000 || 17 | 25,840366 28,212380 2 2,050000 2,060000 || 18 28,132385 30,905653 33,152500 3,183600 1930,539004 33,759992 4 4,310125 4,374616 || 20 33,065954 36,785592 5 5,525631 5.637193 | 21 35,719252 39,992727 6 6,801913 6,97531922 38,505214 43,392291
8,142009 8,393838 | 23 41,430475 46,995828 8 9,549109 9,897468 | 24 44,501999 50,815578 9 11,026564 11,491316 25 | 47,727099 54,864512 10 12,577892 13,180770 26 51,113454 59,156332 11 14,206787 | 14,971643 | 27 57,669126 63,705765 12 15,917126 16,869942 | 28 58,402583 68.528112 13 17,712982 18.882130 29 62,322712 73,639798 14 19,598632 21,015066 30 66,438847 79,058186 15 21,578564 23,275969 3170,760790 84,801677 16 23,657492 25,672528
RULE 2, by Table 1st. Multiply the tabular number, under the rate per cent. and opposite to the given number of years, by the annuity or rent, and the product will be the amount sought. The table shews the amount of one dollar annuity, forborne or unpaid, at 5 and 6 per cent. per annum, compound interest, for 31 ycars, or under.
8,383844 28 14,898127
6 10.477260 10,827603 11,158116 11.469921 11,764077 12.041582 12,303380 12,550357 12,783356 13,003166 13,210534 13,406164 13,590721 13,764831 13,929086
Table 2nd shews the present worth of one dollar annuity, to continue for 31 years, at 5 and 6 per cent. per annum, compound interest.
1. If 150 dollars yearly rent, or an annuity, be forborne or unpaid four years, what will it amount to in that time, at 6 per cent. per annum, compound interest ?
Operation by Rule 1,
1,06x1,06= 1,123600 Amount of one dollar for a year 1,060000 First term
1,000000 Sum of the series
$ cts, m. 4,574616x 150=656,192400=656 192+ Ans. Note. Look for the amount of one dollar, &c. in the tahle of compound interest, and you will find 1,06, &c.
By Table ist, opposite 4 years, and under 6 per cent. . Jeannum, is
4,374616X150=656 dolls. 19 cts. 2+ m. Ans.
2. If a salary or pension be forborne 20 years, which is 60 dollars, how much will it come to, at 6 per cent. per annum, compound interest? Ans. 2207 dolls: 13+ cts.
3. Suppose an annuity of 400 dollars be 12 years in arrears, it is required to find what is due now, compound in. terest being allowed at 5 per cent. per annum?
Ans. 6366 dolls. 85+ cts. by table 1st. 4. What will a pension of 480 dollars per annum, payable yearly, amount to in three years, at 5 per cent. per annum, compound interest? Ans. 1513 dolls. 20 cts. To find the present worth of annuities at compound interest.
Rule. Divide the annuity, &c. by that power of the ratio, signified by the number of years, and subtract the quotient from the annuity. This remainder being divided by the ratio less one, the quotient will be the present value of the annuity required.
5. What ready money will purchase an annuity of 200 dollars, to continue four years, at 5 per cent. compound interest?
Foarth power of the ratio, 1,2r5506; then divide $200 by 1,215506, subtract that quotient from $200, and divide the remainder by ,05, and you will have 709 dollars 19-+ cents, which is the answer.
By Table 2nd. Under 5, and even with 4 years, you have the number, 3,545950, equal to the present worth of one dollar for four years, which multiplied by the annuity of 200 dollars, will give 709 dollars 19-4 cents as the answer.
6. What is the present worth of an annuity of 60 dollarg per annum, to continue 20 years, at 6 per cent. compound interest?
Ans. 688 dollars 19+ cents. 7. What is 120 dollars per annum, to continue 7 years, worth in ready money, at 6 per cent. compound interest ?
Ans. 669 dollars 88+ cents.
To find the present worth of annuities, leases, &c. taken in
reversion, at compound interest. Rule. Divide the annuity by that power of the ratio, denoted by the time of its continuance, subtract the quotient from the annuity, divide the remainder by the ratio less one, and the quotient will be the present worth, to commence immediately. 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.
8. What really inoney will purchase an annuity of 200 dollars, payable fearly, for four years, but not to commence till two years, at 5 per cent. 4th power of 1,0531,215506) 200,00000(164,53052
2nd power of 1,05=1,1025)709,188 Present worth of the annuity in reversion $643,25 to answer. By Table 2nd. Find the present worth of one dollar, at the given rate, for the sum of the time of continuance, and time in reversion. Add these together, and from which value subtract the present worth of one dollar, for the time in reversion. Multiply the remainder by the annuity, and the product will be the answer.
Time of continuánce 4 years
$ 643,25,6,400 9. What is the present worth of 300 dollars yearly rent, which is not to commence until ten years hence, and then to continue seven years after that time, at 6 per cent. ?
Ans. 935 dollars 15 cents. 10. What is the present worth of a reversion of a lease of 60 dollars per annum, to continue 20 years, but not to commence until the end of 8 years, allowing 6 per cent. to the purchaser
Ans. 431 dolls. 78 cts. 2-4 m.
To find the present worth of a freehold estate, or an annuity
to continue for ever. Rule. As the rate per cent. is to 100 dollars, so is the annuity or yearly rent to the value required.
11. What is the worth of a freehold estate of 200 dollars per annam, allowing 5 per cent. to the purchaser?
As 5 : 100 :: 200 84000 Ans. 12. An estate brings in yearly 600 dollars; what would it sell for, allowing the purchaser 6 per cent. for his money?
Ans. 10000 dollarsa
To find the present worth of a freehold estate in reversion.
RULE. Find the present value of the estate by the foregoing rule, as though it was to be entered upon immediately, and divide the said value by that power of the ratio, denoted by the time of reversion, and the quotient will be the present worth of the estate in reversion.
13. A freehold estate, which brings in 160 dollars yearly, is to be sold, but does not commence for two years. What is its value, allowing the purchaser 5 per cent.
As 5 : 100 : : 160 to $3200, the present worth, if entered on immea diately. Then 2nd power of the ratio, denoted by the time of reversion, 1,1025 = 3200 = 2902 dollars 49-7 cents, the present worth of 3200 dolls. in two years reversion.
By Table 2nd. Find the present worth of the annuity or rent, for the time of reversion, which subtract from the value of the immediate possession, and you will have the value of the estate in reversion.
Present worth of one dollar for two years 1,859410
Ans. % 2902 49 5
Note. 3200 dollars is the value, if immediately entered. 14. If an estate, which brings 90 dollars per year, to commence ten years hence, were to be sold, what would it bring, allowing the purchaser 6 per cent. ?
Ans. 837 dollars 39+ cents. 15. Which is most advantageous, an estate of 400 dollars per annum, in a term of 15 years, or the reversion of such an estate for ever, after the said term of 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 re
version for ever afterwards, by 303 dolls. 72+ cts.
PERMUTATION Is the changing and shewing in how many different positions any given number of things may be placed. To find the number of permutations or changes that can be made of any given number of things, all different from each other.