per

2. What will an annuity of 30L. per annum, payable yearly, amount to in 4 at 5 years, per cent. annum, and what would be the respective amounts, if the payments were to be half yearly or quarterly? Amount for yearly payments is L. 129.30375 for half yearly for quarterly

Ans.

L. 130.9004

L. 131.7035

3. If a salary of 351. per annum to be paid yearly, be omitted for 6 years at 5 per cent. what is the amount? Ans. 241L. 1s. 7d. 2.5+qrs.

CASE 2.

The annuity, time, and rate given, to find the present worth:

RULE.

Divide the annuity by the ratio involved to the time, and subtract the quotient from the annuity; divide the remainder by the ratio less one, and the quotient will be the present worth: Or, by Table IV.

Multiply the number under the rate, and opposite the time by the annuity, and the product will be the present worth.

When the payments are half yearly or quarterly, multiply the present worth so found, by the proper number in Table V.

EXAMPLES.

1. What is the present worth of a pension of 30L. per annum for 5 years, at 4 per cent.?

Number from Table IV. 4.45182

Ans. 133L. 11s. 1d.

×30 annuity.

L. 133.55460

Or, 133L. 11s. 1.104d.

2. What is the present worth of 20L. a year for 6 years, payable either yearly, half yearly, or quarterly, computing at 5 per cent. per annum?

L.

3. What is the yearly rent of 50L. to continue 5 years, worth in ready money, at 5 per cent.? Ans. 216L. 9s. 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.

CASE 1.

The annuity, time of reversion, time of continuance, and rate given, to find the present worth of the annuity in reversion:

RULE.

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 40L. 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 in

volved to

the time.

1.4185191)40.000000000000(28.19842

11.80158

1.06×1.06.06.067416)11.80158(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

X40 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. 622.48dols. 3. What is the present worth of a reversion of a lease of 60L. per annum, to continue 7 years, but not to commence till the end of 3 years, allowing 5 per cent. to the purchaser? Ans. 299L. 18s. 2.112d.

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 140L. per annum, to continue for ever, worth in present money, allowing 4 per cent. to the purchaser?

L.

1.04-1.04)140.00

L. 3500

2. What is the present worth of a freehold estate of 290 dollars per annum, to continue for ever, allowing per cent. to the purchaser?

4

Ans. 7250dols.

PERPETUITIES IN REVERSION.
CASE 1.

The rent of a freehold estate, time of reversion, and rate per cent. given, to find the present worth:

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 50L. per annum, to commence 4 years hence, be put up at sale, what is the present worth, allowing the purchaser 5 per cent.?

Ans. 822L. 14s. 1d. 2qrs. +

Ratio involved to the time ? 1.2155062 of reversion, viz. 4 years S

.05 ratio less one.

.060775310)50(8227. 14s. 1d. 2q.+ 2. What is an estate of 696ḍols. 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. 14873.594dols.

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, a b c, a c b, ba c, bca, cab, cb a, are six different positions of three letters.

RULE.

Multiply all the terms of the natural series continually from one to the given number inclusive; the last product will be the answer required.

R

EXAMPLES.

1. In how many different positions can 6 persons place themselves at a table? 1x2×3×4x5x6-720. 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 12 bells, and in what time may they be rung, allowing 3 seconds to every change.

Ans.

S479001600 changes.

45 years, 195 days, 18 hours.

COMBINATION.

Combination is a rule for discovering how many different ways a less number of things may be combined out of a greater; thus out of the letters a, b, c, are three different combinations of two; viz. ab, ac, bc.

RULE.

Take a series proceeding from and increasing by a unit, up to the number 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 ten?

10×9x8x7x6

=252. Ans.

1x2x3x4x5

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. 11267L. 6s. 4d.

DUODECIMALS.

Duodecimals are fractions of a foot, or of an inch, or parts of an inch, &c. having 12 for their denominator.