u EXAMPLES. 15. What is the present worth of an annuity of 830 per an. num, to continue 7 years at 6 per cent ? Ans. $167.4716. 16. What is the present worth of a pension of $50 per annum, for 8 years. at 5 per cent? Ans. $323.1608. II. Given, p, t, and r, to find -Rule. pXrtXr-pXrt ot-1 EXAMPLES. 17. If an annuity be purchased for $167.4716 to be continued 7 years at 6 per cent : quere the annuity ? Ans. 830. 18. If the present worth of $323.1608 were required for a pension for 8 years to come, at 5 per cent, what was the pen.. sion ? Ans. $50. III. Given, u, p, and r, to find t.-Rule. =gt ptumpar which being continually divided by r, till nothing remains, the number of those divisions will be =t. EXAMPLES. 19. How long may a lease of $30 yearly rent be had for $167.4716 allowing 6 per cent to the purchaser ? Ans. 7 years. 20. If a house is let upon lease for $50 per annum, and the lessee makes present payment of $323.1608, he being allowed 6 per cent : I demand how long the lease is purchased for? Ans. 8 years. ANNUITIES, LEASES, &c. TAKEN IN REVERSION. I. To find the present worth of annuities, &c. in reversion. Rule 1Find the present worth of the annuity, &c. at the given rate, Thus, 1-- :--1 and for the time of its continuance, 2. Change p into a, and find what principal being put to interest will amount to p, at the same rate, and for the time to come, > Thus, =p. before the annuity commences, which will be the present worth of the annuity, &c. EXAMPLES. 21. What is the present worth of a reversion of a lease of $30 per annum, to continue for 7 years, but not to com. mence till the end of 4 years, allowing 5 per cent. to the purchaser ? Ans. $142.9153. 22. There is a lease of a house at $40 per annum, which is yet in being for 2 years, and the lessee is desirous to take a lease in reversion for 6 years, to begin when the old lease u .م=--r: gt shall be expired, what will be the present worth of the said lease in reversion, allowing 6 per cent. to the purchaser ? Ans. $175.0563. II. To find the yearly income of an annuity, &c. taken in reversion. Rule 1.--Find the amount of the present worth, at the given rate, and for the time Thus, pXr+= before the annuity commences. 2. Change a into p, and find what yearly rent, &c. being sold Thus, pXr*xr-pXr* will produce p, at the same rate, and for the time of its continuance,j q-1 EXAMPLES. 23. What annuity to be entered upon 4 years hence, and then to continue 7 years, may be purchased for $142.9153 at 5 per cent. Ans. $30. 24. There is a lease of a house in being for 2 years, and the lessee being minded to take a lease in reversion for 6 years, to begin when the old lease shall be expired, paid down $175.0563: What was the yearly rent of the house, when the lessee was allowed 6 per cent. for present payment? Ans. $40. REBATE OR DISCOUNT: Here s represents the sum to be purchased, t, r, p, as before. I. Given, s, t, and r, to find p. Rule. =p. rt EXAMPLE. 25. What is the present worth of $315.6175, payable 4 years hence, at 6 per cent ? Ans. $250. II. Given, p, t, and r, to find s.--Rule. pXrt=s. EXAMPLE. 26. If a sum of money due 4 years hence produce $250 for the present payment, rebate being made at 6 per cent. What was the sum first due ? Ans. $315.6175. III. Given, s, p, and r, to find t.- :--Rule. =r which being continually divided by r, till nothing remains, the number of those divisions will be =t. EXAMPLE. 27. The present payment of $250 is made for a debt of $315.6175, rebate at 6 per cent." In what time was the debt payable ? Ans. S 4 years. IV. Given, typ, and t, to find r.--Rule. ; =/=r. ***r EXAMPLE 28. A debt of $315.6175 is due years hence, but it is agreed to take $250 now. What is the rate per cent. that the rebate is made at ? Ans. 6 per cent. PURCHASING FREEHOLD OR REAL ESTATES, Is to find the present worth of an annuity to continue forever. 1. Given, U, and r, to find p.-Rule. u EXAMPLES 29. What is the worth of a freehold estate of $50 per ann. allowing 5 per cent. to the buyer ? Ans. $1000. 30. If a freehold estate of $75 yearly rent was to be sold, what is it worth allowing the buyer 6 per cent? Ans. $1250. ptu II. Given, p, and u, to find r.- -Rule. P EXAMPLES. 31. If an estate of $50 per annum be bought for $1000 : what is the rate per cent ? Ans. 5 per cent. 32. If a freehold estate of $75 per annum is sold for $1250 what is the rate per cent allowed ? Ans. . 6 per cent. III. Given, p, and r, to find u. -Rule. pXr-lzu. EXAMPLES 33. If a freehold estate is bought for $1000, and the allowance of 5 per cent. is made to the buyer, what is the yearly rent? Ans. $50. 34. If a freehold estate is sold for $1250 present money, and an allowance of 6 per cent. made to the buyer for the same : Quere the yearly rent? Ans. $75. PURCHASING FREEHOLD ESTATES IN REVERSION I. To find the worth of a freehold estate in reversion. Rule 1.--Find the worth of the yearly} Thus, rent, 2. Change p into a, and find what principal being put to interest will amount -Thus, to a, at the same rate, and for the time to come, before the estate commences, EXAMPLES 35. If a freehold estate of $50 per ann. to commence 4 years hence, is to be sold, what is worth, allowing the purchaser 5 per cent. for present payment ? Ans. $822.70625 u a t r 36. What is an estate of $240 per ann, worth in ready money, to continue for ever, but not to commence till the end of 3 years, allowance being made at 6 per cent ? Ans. $3358.4775 II. To find the yearly rent of an estate taken in reversion, Rule 1-Find the amount of the worth of the estate at the given rate, Thus, pxroza. and the time before it commences. 2. Change a into p, and find pXrXr-pXr what yearly rent being sold Thus, will produce p at the same rate. S EXAMPLES. 37. If a freehold estate, to commence 4 years hence, is sold for $822.70625, allowing the purchaser 5 per cent. What is the yearly income ? Ans. $50. 38. There is a freehold estate bought for $3358.477} but not to commence till the expiration of 3 years, allowing 6 per cent. for present payment : What is the yearly income? Ans. $240. r INSURANCE. Insurance is a security given by the underwriters, to indemnify the insured froin such losses, as are mentioned in the policy of insurance, in consideration of a sum of money called premium ; which varies according to the risk, and is generally at so much per cent. In cases of total loss, the underwriter, is allowed a discount or rebate of 2 per cent. Most of the computations relative to insurance, fall under one or other of the following cases, viz. Let x, represent 100 dollars or pounds. the rate per cont. of the premium. the sum to be covered. - the discount or rebate. P, the whole insurance or premium, I. Given, a, and r, to find p.-RULE. axr EXAMPLES. 1. What will be the whole insurance on 84500, from NewYork to Bombay, at 6 per cent. Ans. $270. 2. What is the insurance on 50001. from London to Hamburgh, at 1} per cent ? Ans. 751. Merchants sometimes cover their property by taking out a policy for such amount as will be equal to the value of the invoice and premium together. N II. Given, s, and r, to find a.-Rule. xX8, EXAMPLES. 3. A merchant ships goods as per invoice, 41731. 12. what amount of a policy will cover the same premium at 4 per cent discount, 2 per cent in case of loss? Ans. 44401. 4. Shipped goods for Constantinople valued at $7590. Premium 6 per cent. discount 2 per cent in case of loss : for what sum must the policy be taken ? Ans. $8250. III. Given, a, r, and d, to find s.-Rule. X-r+dxa EXAMPLES. 5. If a polícy be taken out for 44401. at 4 per cent premium, and 2 per cent discount. Quere, the amount of the in. voice to be covered ? Ans. 41731. 128. 6. A trader insuring an adventure at 6 per cent premium, and 2 per cent discount in case of loss, took out a policy for $8250. Required the invoice amount ? Ans. 87590. IV. Given, a and 3, to find r+d.-Rule. =r+d. EXAMPLES. 7. Suppose a merchant to take out a policy for 44401. to insure 41731. 12s. and cover the premium and discount ; which is now required ? Ans. 4 and 2 per cent. 8. A trader insuring an adventure took out a policy for $8250, to cover $7590. Quere, the rate per cent of the premium and discount? Ans. 6 and 2 per cent. V. When a vessel, or an adventure in goods is continued several voyages, or from one port to another, at the same or different risks, to find the amount that will cover the whole all round. Here n, represents the number of risks, and is the exponent of any given power. x, r, a, s, d, p, as before. &xxn RULE. sa=sum to be insured: x--r xx--r Xx-r, &c. Or in words, multiply the sum to be covered by 100, raised to that power denoted by the number of risks, for a dividend : then subtract each premium from 100, and multiply the remainders for a divisor, the quotient will be the amount of the policy to cover the vessel or adventure the voyage round. EXAMPLES. 9. Suppose a merchant in New York insures a vessel t London, value $5000, at 4 per cent. thence to a port in the West-Indies, at 5 per cent, and thence to Baltimore, where |