its continuance, at the given rate per cent. To do which, there are given U, T, and R to find P, by the following theorem: 2. Find what sum will amount to the present worth of the annui ty, &c. at the same rate and for the time before it commences, and that will be the present worth of the annuity in reversion; to do which, change Pinto A, and proceed by the following theorem: 1. What is the present worth of the reversion of a lease of $80 a year, to continue 4 years, but not to commence till the end of 2. years, allowing $5 per cent. to the purchaser? 4 $=u. 1.05|=1,21550625)80.00000000000(65.816=u÷r $283.68cts.the present worth for the time of continuance. Now I change the present worth for the time of continuance into A, and proceed by the second theorem. cts.a. $cts.m. 1.05=1.1025)283.6800000(257.30,6-P, or the present worth in. reversion. Ans. 1350 rem. 2. I have rented a plantation for 5 years at $250 a year, but am not to get possession till the end of 3 years; what is the reversion worth in ready money, allowing discount at $5 per cent. per annum? Ans. $834.20cts.+36528 rem. 3. There is a lease on a certain tract of land worth $240 a year, which is to continue 4 years, and the lessee is desirous to take a lease in reversion for 6 years, to begin when the old lease terminates; I demand the present worth of the said lease in reversion, allowing the purchaser $5 per cent. per annum discount? Ans. $1002.18cts. 9m.+100681875 rem. 4. There is a tenement now building which will be worth $480 a year, and I have a mind to lease it for 7 years, but I cannot get possession till the expiration of 5 years; what is the reversion worth in ready money, allowing discount at $5 per cent, per annum? Ans. $2176.21cts.+3300871875 rem. CASE 2. When the present worth, time, and ratio are given, to find the annuity, &c. RULE. 1. Find the amount of the present worth of the yearly sum, at the given rate per cent., and for the time before the annuity commences, to do which there are given P, T, and R to find A, by the subsequent theorem: 2. Find what yearly sum will produce the present worth of the annuity, &c. at the same rate and for the time of continuance, and that will be the annuity, &c. required, to do which, change A into. P, and proceed by the subsequent theorem: 1. What annuity, to be entered on 2 years hence, and then to continue 4 years, may be purchased for $257.30cts. 6m.+1350rem. ready money, allowing discount at $5 per cent. per annum? 2 $ cts. m. 257.306=p. 1.05 1.1025=r, raised to t before the reversion.. = 2836798650 1350 rem. added. $283.6800000=a before the commencement, which I ohange into p, and proceed by the second theorem. -1.21550625 17.240500000000(80$=u. Ans. 2. I have paid $334.20cts.+36528 rem. for the rent of a plantation (in reversion) 5 years, but am not to get possession till the end of 3 years; what is the yearly rent worth, allowing $6 per cent. discount for prompt payment? Ans. $250. 3. A certain lessee in reversion has paid his lessor $1002.18cts. 9m.+100681875 rem. in hand, for a lease of 6 years, but he is not to have possession till the termination of 4 years; what is the yearly rent worth, allowing discount at $5 per cent. per annum? Ans. $240. 4. Several lessees in reversion have paid the lessor $2176.21cts. +3300871875 rem. in hand for the lease of a certain tenement 7 years, but they are not to have possession till the end of 5 years; Í demand the yearly rent, allowing discount at $5 per cent. per an-Ans. $480. num? CASE 3. When the annuity, present worth, and ratio, are given, to find the time of its continuance. RULE. 1. Find the amount of the present worth, at the given rate per cent., and for the time before the annuity &c. commences, by the subsequent theorem: t Viz. pxr=a. 2. Find the time that will be necessary, for the annuity, &c. to produce the amount for the present worth, at the same rate per cent. and that will be the time required; therefore, change A into P, and then proceed by the subsequent theorem: Viz. u* pu-pr then involve r till it is *The remainder in the first operation of case the 1st, must always be substracted from u, before the division is performed. EXAMPLES.. 1. The present worth of a certain lease in reversion, to commence 2 years hence, is $257.30cts.6m.+1350 remainder, and the yearly rent is $80-what is the time of its continuance, allowing the lessee $5 per cent. per annum discount for prompt payment ? 2 1.05 $ c. m. 257.30,6=p 1.1025=r t 283.6798 65 0=pxr 135 0 last rem.. added. $283.6800000= a, or the amount of the pre- sent worth for the time before the annuity commences, which I' change into P, and proceed by the second theorem. Add $283.68cts.=p. 80.00cts. u. $283.68cts. Pmultiply. 1.05= $297.8640 pr. n=$ 65.816) 80.00000000000 24065000 the 1st rem. subtracted. 79.99975935000 (1.21550625=r, which is equal to the fourth power of r, therefore the time of continuance is 4 years, the answer required. 2. I have paid $384.20 cts.+36528 rem. ready money, for the rent of a plantation, but I am not to have possession till the expiration of 3 years-the yearly rent is $250, what is the time of continuance, allowing discount at $6 per annum? Ans. 5 years. 3. The present worth of a certain annuity in reversion, to commence 4 years hence, is $1002.18cts. 9m.+100681875 rem.; the yearly income is $240; I demand the time of its continuance, allowing discount at $5 per cent. per annum? Ans. 6 years. 4. Several lessees in reversion have paid their landlord $2176.21 cts.+3300871875 rem, in ready money, for the lease of a certaintenement, but they are not to have possession till the end of 5 years; the yearly rent is worth $480; what is the time of continuance, allowing discount at $5 per cent. per annum for prompt payment? Ans. 7 years. OF PERPETUITIES. Perpetuities are perpetual annuities, or such estates as are bought to continue forever. NOTE. U represents the annuity, or yearly rent; R, the ratio, or amount of one dollar, for one year, and P the present worth. CASE 1. When U and R are given, to find P. 1. Suppose a freehold estate of $150 a year, is to be sold; what is it worth in ready money, allo wing the buyer $5 per cent. for prompt payment? r=1.05 $=u -1.05 ) 150.00 Ans. $3000-p 2. What is an estate of $285 a year, to continue forever, worth in present money, allowing the purchaser $6 per cent. for prompt payment? Ans. $4750 CASE 2. When P and R are given, to find U. RULE.-PXR-1=U. EXAMPLES. 1. Ifa freehold estate be bought for $3000, and the allowance of $5. per cent. is made to the buyer, what is the yearly rent? Ans $150. $3000=p Ans. $150.00=u. 2. If an estate be sold for $4750, present money, and $6 per cent. is allowed to the purchaser for prompt payment, what is the yearly rent? Ans. $285, 1. If a real estate of $150 a year be sold for $3000 in cash, what rate per cent was allowed to the buyer for prompt payment? $3000=p. p=3000)3150.00(1.05=r=$5 per cent. 2. If a freehold estate of $285 a year be sold for $4750 in cash, what rate per cent. was allowed to the purchaser for his ready mohey? Ans. $6 per cent. OF PERPETUITIES IN REVERSION. NOTE.--T represents the time before the reversion, U and R as before. CASE 1. When U, R, and T are given, to find P. 1 RULE.—U÷rxr-1=P. EXAMPLES. 1. If a freehold estate of $150 a year, to commence 3 years hence, be put up to sale, what is the present worth in cash, allowing the buyer $5 per cent. for prompt payment? |