L. Fresent worth for yearly payments, 101.5138. Ans. for half yearly 102.7673. for quarterly 103.3978. 3. What is the yearly rent of 50 L. to continue 5 years, worth in ready, money, at-5 per cent. ? Ans. 216L. 95. 10d. 2.24qrs. ANNUITIES 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. a CASE 1. The annuity, time of reversion, time of continuance, and rate given, to find the present worth of the annuity in reversion: RULE. a Divide the annuity by the ratio involved to the time of continuance, and subtract the quotient from the annuity, for a dividend; multiply the ratio involved to the time of reversion by the ratio, less one, for a divisor; the quotient of this division will be the present worth. Or, . Take two numbers under the given rate in Table IV. viz. that opposite the sum of the two given times, and that against the time of reversion, and multiply their difference by the annuity of the present worth. When the payments are half yearly or quarterly, -use Table V. EXAMPLES 1. What is the present worth of a reversion of a lease of 40 L. per annum, to continue for six years, but not to commence till the end of 2 years, allowing 6 per cent. to the purchaser ? 40 annuity. Ratio involved to the = 1.4185191)40.000000000000(28.19842 } time, 11.80158 1.06 X 1.06 X 06.06741611.80158(175.056+L. Ans. Or by Table IV. First, the sum of the two given times is 8 years, and the time of reversion 2 years ; therefore, Take for 8 years 6.20979 for 2 do. 1.83339 Difference 4.37640 X 40 annuity L. 175.05600 Ans. as before. 2. A person owns a farm which he proposes to let for 8 years, at 100 dollars 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? Ans. 622.48 dols. 3. What is the present worth of a reversion of a lease of 60 L. per annum, to contine 7 years, but not to commence till the end of 3 years, allowing 5 per cent. to the 5 purchaser? Ans. 299 L. 18s, 2.112d. a 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 less one, for the present worth. Note-For perpetual half yearly, or quarterly payments, Table V. is to be applied as in similar cases of temporary annuities. EXAMPLES 1. What is an estate of 140 L. per annum, to continue for ever, worth in present money, allowing 4 per cent. to the purchaser ? L. 1.04—1=.04)140.00 Ans. L. 3500 2. What is the present worth of a freehold estate of 290 dollars per annum, to continue for ever, allowing 4 per cent. to the purchaser? Aus. 7250 dols. PERPETUITIES IN REVERSION. CASE 1. The rent of a freehold estate, time of reversion, and rate per cent. given, to find the present wortli : RULE. Multiply the ratio involved to the time of reversion, by the ratio, less one, for a divisor ; by which divide the yearly payment, the quotient will be the answer. EXAMPLES 1. If a freehold estate of 50 L. per annum, to commence 4 years hence, be put up at sale, what is the present worth, allowing the purchaser 5 per cent. ? Ans. 822 L. 145. Id, 2qrs. + Ratio involved to the time of reversion, viz. 4 years, S 1.2155062 .05 ratio less one. .060775310) 50(8221. 148. 1d. 29:+ 2. What is an estate of 696 dol. per annum, to continue for ever, but not to commence till the expiration of 4. years, worth in present money, allowance being made at 4 per cent. Ans. 14873.594 dol. PERMUTATION. Permutation is a rule for finding how many different ways any given number of things may be varied in position, place, or succession ; thus abc, acb, bac, bca, cab, cba, are six different positions of three letters. RULE. Multiply all the terms of the natural series continually from 1 to the given number inclusive; the last product will be the answer required. EXAMPLES. 1. In how many different positions can 6 persons place themselves at a table ? 1 X2 X3 X-4 X 5 X 6 = 780. Ans. 2. How many days can 7 persons be placed in a different position at dinner? Ans. 5040 days. 3. What number of changes may be rung upon 2 bells, and in what time may they be rung, allowing 3 seconds to every change? 5 47900 1600 changes. Ans. 45 years, 195 days, 18 hours: COMBINATION. Combination is a rule for discovering how many different ways a less number of things may be cornbined out or a greater; thus out of the letters a, b, c, are three different combinations of two: viz. ab, ac, bc. R RULE. Take a series proceeding from and increasing by a anit, up to the nụmber to be combined; and another series of as many places decreasing by a unit, from the number out of which the combinations are to be made, multiply the former continually for a divisor, and the latter for a dividend, the quotient will be the answer. EXAMPLES. 1. How many combinations can be made of 5 letters out of 10? 10 X 9 X 8 X 7 X 6 = 252. Ans. i X2 X 3 X 4 X 5 2. How many combinations can be made of 6 letters out of 10? Ans. 210. 3. What is the value of as many different dozens as may be chosen out of 24, at 1d. per dozen ? Ans. 11267 L. 6s. 4d. DUODECIMALS. · Duodecimals are fractions of a foot, or of an inch, or parts of an inch, &c. having 12 for their denominator. The denominations are, foot, inch, second, third, and fourth. 12 Fourths "'ll make 1 Third ". I Second !!. 1 Inch I. 12 Inches | Foot Ft. ADDITION OF DUODECIMALS. RULE, Proceed as in compound addition. |