The construction of this table is from an algebraic theorem, given by the learned A. De Moivre, in his treatise of annuities on lives; which may be in words, thus: For half-yearly payments, take a unit from the ratio, and from the square root of the ratio; half the quotient of the first remainder, divided by the latter, will be the tabular number. For quarterly payments, use the 4th root, as above, and take one quarter of the quotient. ANNUITIES AT COMPOUND INTEREST. Case 1. The annuity, time, and rate of interest, given, to find the amount. Rule. Multiply the number under the rate, and opposite the time, in Table III., by the annuity; and the product will be the amount for yearly payments. Note. When the payments are to be made half-yearly or quarterly, the amount for the given time, found as above, multiplied by the proper number in Table V., will be the true amount. Case 2. The annuity, time, and rate, given, to find the present worth. Rule. Multiply the number under the rate, and opposite the time, in Table IV., by the annuity; the product will be the present worth for yearly payments. Note.-When the payments are to be made half-yearly or quarterly, the present worth so found must be multiplied by the proper number in Table V.. Questions. What are annuities at compound interest? When the annuity, time, and rate of interest, are given. by what rule do you find the amount ? What is to be noticed, when the payments are halfyearly or quarterly? When the annuity, time, and rate, are given, to find the present worth, how do you proceed? What is to be noticed, when the payments are halfyearly or quarterly? ANNUITIES IN REVERSION. Sums of money which are payable yearly for a limited period, but which do not commence till after the expiration of a given period, are called annuities in re version. The annuity, time of reversion, time of continuance, and rate, given, to find the present worth of the rever sion Rule. Take two numbers under the given rate in Table IV., that opposite the sum of the two given times; and the number opposite the time when the annuity is to commence, or time of reversion, and multiply their difference by the annuity for the present worth. Note.-When the payments are to be half-yearly or quarterly, use Table V., as before. Questions. What are annuities in reversion? When the annuity, time of reversion, time of continuance, and rate, are given, to find the present worth, by what rule do you work? What is to be noticed, when the payments are halfyearly or quarterly? PERPETUITIES AT COMPOUND INTEREST. Annuities which continue for ever, are called Perpetuities. The annuity and rate given, to find the present worth. Rule. Divide the annuity by the ratio, less 1, for the present worth. Note.-Table V. must be used as in temporary annuities, when the payments are half-yearly or quarterly. Questions. What name is given to annuities which continue for ever? By what rule do you proceed, when the annuity and rate are given to find the present worth? What is to be noted, when the payments are halfyearly or quarterly? COMPOUND INTEREST BY DECIMALS. Examples. 1. What is the interest and amount of £400, for 3 years, at 4 per cent? 1.04 X 1.04X1.04-1.124864 400 449.945600 amount. 400 Ans. 49.9456 interest. 2. What is the amount and interest of £750, at 5 per cent. per annum, for 4 years and 6 months? Ans. Amount, £934 2s.. 10d.; interest, £184 2s. 10d. Case 2. 1. What principal, put to interest, will amount to £695 138. 9d., in 5 years, at 5 per cent.? Ans. £545 18. 9d.+ · 2. What principal must be put to interest, to amount to £260 5s. 3d., at 6 per cent. per annum, for 3 years? Ans. £218 10s. 5d. ANNUITIES AT COMPOUND INTEREST. Case 1. 1. What is the amount of an annuity of 180 dollars, for 9 years, at 5 per cent.? 11.026564 180 882125120 11026564 1984.781520 Ans. 2. What will an annuity of $200 amount to in 5 years, to be paid by half-yearly payments, at 6 per cent. per Ans. $1144 08cts. 2m.+ annum? Case 2. 1. What is the present worth of £50 per annum, for 6 years, at 4 per cent.? 5.24214 50 £262.10700 Ans. 2. What is the present worth of 70 dollars a year, for 5 years, payable yearly, per cent. per annum? Ans. half-yearly, and quarterly, at 6 ANNUITIES IN REVERSION. 1. The reversion of a freehold estate of £60 per annum, for 4 years, to commence 2 years hence: what is the present worth, allowing 4 per cent. for present payment?" 5.24214 1.88609 3.35605 60 Ans. £201.36300 2. What is the present worth of a reversion of a lease for $120 per annum, to continue 9 years, but not to commence till the end of 4 years, at 4 per cent., to the purchaser? Ans. $762 69cts. Im.+ PERPETUITIES AT COMPOUND INTEREST. 1. What is the present worth of an annuity of £150 per annum, to continue for ever, allowing 5 per cent. to the purchaser? 1.05-1.05)150.00 $3000 Ans. 2. What is an estate of 260 dollars per annum, to continue for ever, worth in present money, allowing 6 per cent. to the purchaser? Ans. $4333 33cts. 3m.+ COMBINATION. COMBINATION is used to show how many different ways a less number of things can be combined out of a greater; as, out of the figures 1, 2, 3, the three combinations, 12, 13, and 23, may be formed. Rule. 1. Take a series, proceeding from and increasing by a unit, up to the number to be combined. 2. Take another series, of as many places, decreasing by unity from the number out of which the combinations are to be made. 3. Multiply the first continually for a divisor, and the latter for a dividend; the quotient will be the answer. Questions. What is Combination? |