1. first term of the series third term 4.310125 = sum of the series 300= annuity $1293.03,7,5 Ans. Subtract 1. Interest of $1 for 1 year at 5 per cent.=.05).21550625 4.310125 Annuity = 300 Ans. by the second rule $1293.03,7,5100 2. What will 50 dollars annual rent amount to, at 4 per cent., if it be forborne 7 years? Ans. $394.91 + 3. What will an annuity of 50 dollars per annum, payable yearly, amount to in 4 years, at 6 per cent. per annum; and what would be the respective amounts, if the payments were to be half yearly or quarterly? Ans. Amount for yearly payments, $218.73+ do. for half yearly do. 221.96,3+ do. for quarterly do. 223.59,9+ CASE 2. The annuity, time, and rate given, to find the present worth. RULE. 1. Divide the annuity by the amount of one dollar for one year, at the given rate per cent., and the quotient will be the present worth of one year's annuity. 2. Divide the annuity by the square of the last divisor, (the amount of one dollar, &c.) and the quotient will be the present worth of two years. 3. In like manner, find the present worth of each year by itself, and the sum of all these will be the present value of the annuity sought. Or, divide the annuity by the amount of one dollar ANNUITIES AT COMPOUND INTEREST. 168 one year, at the given rate, involved to the time, and subtract the quotient from the annuity; divide the remainder by the interest of one dollar, for one year, at the given rate ; the quotient will be the present worth. EXAMPLES. 1. What is the present worth of an annuity of 40 dollars, to continue three years, discounting at 5 per cent. per annum, compound interest ? Amt. of $1 one y. at 5 per cent=1.05)40.(38.095=present worth for one year. 1.05*1.05=1.1025)40.(36.281 = two ys. 1.05 X 1.05% 1.05=1.157625)40.234.556=three y. Ans. Whole present worth, $108.93,2 Annuity. 40.0000 annuity 1.052 1.051.05=1.157625)40.000000(34.5535 quot. Divisor 1.0541 =,05)5.4465 per annum? Ans. $108.93 2. A, B, and C, each hold a tenement that they lease for 20 dollars each, for six years; A covenanted with his tenant to receive his rent yearly, B was to have his rent half yearly, and C's rent was to be paid quarterly ; required the present value of each lease, computing discount at five per cent. Ans. Present worth of A's lease, $101.5138 do. B's 102.7672 do. C's 103.3977 3. What is the present worth of a yearly rent of $300 for two years ? Ans. $550.01,7. Note. When the payments are half yearly or quarterly, multiply the present worth so found by the proper number in the above table. Or, find the present worth for the whole years, as before, then find the present worth for the given parts of a year, and it will be the whole present worth. 4. Required the present worth of one dollar for one year at 5 per cent. Ans. $.9523+ CASE 3. When the present worth, time, and rate are given, to find the annuity, rent, &c. RULE. 1. Involve the amount of $1 for 1 year at the given rate (called the ratio) to one power more than is denoted by the number of years. 2. From this product subtract the next lowest power, which is that power denoted by the number of years. 3. Divide the remainder by that power of the ratio denoted by the time, deducting therefrom a unit (or one.) 4. Multiply this quotient by the present worth, and the product will be the annuity, pension, rent, &c. EXAMPLE What annuity, to continue 4 years, will $207.904 pur. chase, compound interest, at 6 per cent.? Ans. $60 nearly. CASE 4. When the annuity, present worth, and ratio are given, to find the time. RULE. Subtract the product of the present worth and the amount of $1 for 1 year at the given rate per cent., from the sum of the present worth and annuity; then divide the annuity by the remainder, and the quotient will be that power of the ratio, denoted by the number of years, which, being divided by the ratio, (or the amount of $1 for 1 year at the given rate) and tnis quotient by the same, till nothing remains, the number of divisions will show the time. EXAMPLE. How long may a lease of $300 yearly rent, be had for $2132.34, allowing 5 per cent., compound interest, to the purchaser ? Ans. 9 years. ANNUITIES, LEASES, &c. TAKEN IN REVERSION AT COMPOUND INTEREST. Annuities taken in reversion, are certain sums of money, payable yearly for a limited period, but not to commence till after the expiration of a certain time. The annuity, time of reversion, time of continuance, and rate per cent. given, to find the present worth. RULE. Divide the annuity by the amount of $1 for 1 year at the given rate involved to the time of continuance, and subtract PERPETUITIES AT COMPOUND INTEREST. 170 the quotient from the annuity, for a dividend; multiply the amount of $1 for 1 year involved to the time of reversion, and by the interest of $1 for 1 year (which is the equitable ratio of the per cent. given) for a divisor, which dividing the dividend by, will give the present worth. EXAMPLES. 1. A person owns a farm, which he proposes to let for 8 years at $100 per annum, but cannot give possession till after the expiration of two years—what is the present worth of such a lease, allowing 4 per cent. for present payment? Amount of $1 for 1 year at 4 per cent., involved to the 100. 8th power, the time=1.368569)100.000000(73.069 26.931 1.04x 1.04x.04=.043264)26.931000($622.48 Ans. 2. There is a legacy of 50 dollars per annum for 4 years, left to a young woman of 13 years of age; the time of payment is to commence when she arrives at 18 years ; but wanting a sum of money, is disposed to sell the same at 4 per cent-I demand the present worth. Ans. $147.12+ 3. I have the promise of a pension of 300 dollars per annum for 5 years; but it does not commence till the end of 4 years, and I am willing to dispose of the same for present payment, at the rate of 5 per cent.--I demand the present worth. Ans. $1068.563+ PERPETUITIES AT COMPOUND INTEREST. PERPETUITIES are such annuities as continue for ever. CASE 1. The annuity and rate* given, to find the present worth. RULE. Divide the annuity by the ratio, (i. e. the interest of one dollar for one year at the given rate) for the present worth. For half yearly or quarterly payments, use the proper numbers in the above table of annuities, as in former cases of annuities. * See note 1 in Simple Interest to reduce the rate per cent. to the ratio. EXAMPLES. 1. Suppose an annuity in a feeehold estate of 140 dollars per annum to continue for ever, is to be sold, what is its value in ready money, allowing 7 per cent. to the purchaser ? Ans. $2000. 2. What is an estate of 290 dollars per annum to continue for ever, worth in ready money, allowing 4 per cent. to the buyer? Ans. $7250. CASE 2. The present worth and rate per cent. given, to find the annuity (or yearly rent.) RULE. Multiply the present worth by the ratio, and the quotient will be the annuity. EXAMPLES. 1. If a freehold estate (or annuity) is bought for $2000, allowing 7 per cent to the buyer, what is the yearly rent or annuity ? Ans. $140. 2. If an estate be sold for $7250 ready money, and 4 per cent. is allowed to the buyer for the same, I demand the yearly rent or annuity. Ans. $290. CASE 3. The present worth and annuity given, to find the ratio, or rate per cent. RULE. Divide the sum of the annuity and present worth, by the present worth, the quotient will be the amount of one dol. lar for one year, subtract unity or one from the quotient and you have the ratio required. NOTE.-To reduce the ratio to its proper rate per cent., see note 1 in Simple Interest. EXAMPLES. 1. If a real estate or annuity of $140 per annum, be sold for $2000 ready money, I demand the rate per cent. Ans. 7 per cent. 2. If an annuity in a freehold estate of $290 per annum, be bought for $7250, I demand the ratio or rate per cent. allowed. Ans. .04 ratio or 4 per cent. |