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 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=1.568569)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. RULE. Multiply the amount of one dollar for one year at the given rate per cent., involved to the time of reversion, by the ratio, for a divisor; by which divide the yearly payments; the quotient will be the answer. EXAMPLES. 1. Suppose a freehold estate of 140 dollars per annum, to commence three years hence, is to be sold, what is it worth, allowing the purchaser 7 per cent. ? Ans. $1632.59,5. 2. What is an estate of 290 dollars per annum, to continue for ever, but not to commence till the expiration of 4 years, worth in ready money, allowing the purchaser four Ans. $6197.33. per cent. ? PROMISCUOUS EXAMPLES. 1. Let the product of two numbers be 576, and one of them 16-I demand the other Ans. 36. 2. Required the gain per cent. on buying a bushel of potatoes for 50, and selling them for 56 cents ? Ans. 12. 3. A person was 17 years of age 29 years since, and suppose he will be drowned 23 years hence ; pray, in what year of his age will this happen? Ans. in his 69th year. 4. What is the length of a road 33 feet broad, to contain just one acre ? Ans. 1320ft.=80 poles. 5. If 12 boarders drink in 7 days half a barrel of beer, how long would the same quantity last, if 2 boarders more came among them? Ans. 6 days. 6. Required the cubic feet in a grindstone 40 inches diaméter, and 5 inches thick; likewise its value, at $1.50 per cubic foot. Ans. 3.63 cubic feet; value $5.44,5. 7. If A has ; of of the half of a trading schooner and cargo, worth $16131.70, and sells his brother B of of his interest therein, at prime cost; what did it cost the brother, and what did his cousin P pay at the same rate and time, for 8 of the remainder ? 8. What is the interest of 456 dollars for one year, at six Ans. $27.36. 9. Required the square root of 729. Ans. 27. 10. A owes B 136, to be paid in 10 months, $96 in 7 per cent. ? months, and $260, to be paid in 4 months—what is the equated time for the payment of the whole ? Ans. 6 months and 7 days+ 11. Two ships, A and B, sailed from a certain port at the same time; A sailed north 8 miles an hour, and B east 6 miles an hour--what was their distance apart at the end of 12 hours ? Ans. 120 miles. 12. If I buy coffee at 20 cents per lb., how must I sell it to gain 10 per cent. ? Ans. 22cts. 13. If a family of 9 persons spend 450 dollars in 5 months, how much would be sufficient to maintain them 8 months, if 5 persons more were added to the family? Ans. $1120. 14. Reduce lqr. and 7lbs. of wheat flour to the decimal of a cwt., and find its value at $4 per cwt. Ans. .3125 of a cwt. ; $1.25 value. 15. A has scantling that is 3 inches by 6 inches, and B wishes to purchase 75 feet; admitting it is in one stick, what is its length ? Ans. 50 feet. 16. Required the greatest common measure of 1224 and 1080. Ans. 72. 17. What is the cube root of 1728? Ans. 12. 18. A has į share of a ship, of which he sells to B 4, and B sells out of his share to C, half of his interest—the question is, what part had each man in said ship? Ans. A, B, and C. 19. What will 500 dollars amount to in 4 years, at six per cent. per annum, compound interest ? Ans. $631.23,848+ 20. Required the value of a load of Spanish oak bark, that measures 14.5 feet in length, 3.8 feet in width, and 2.5 feet in height, at 10 dollars per cord ? Ans. $10.76+ 21. What is the interest of 75 dollars for 25 days? Ans. $.30,8. 22. What is the measurement of a stick of scantling 30 feet in length, 8 inches in breadth, and 6 inches in depth ? Ans. 120 feet. 23. What will an annuity or annual rent of 75 dollars amount to in 3 years, at 6 per cent. compound interest ? Ans. $238.77. 24. A owns f of £ of a ship, and sells of his share to B, for 100 dollars-what was the whole ship worth at that rate? Ans. $1200. |