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To find the present worth of an Annuity, or Freehold Estate, in Reversion, at Compound Interest.

RULE.*

1. Find the present worth of the annuity, as if it were to be entered on immediately.

2. Find the present worth of the last present worth, discounting for the time between the purchase and commencement of the annuity, and it will be the answer required.

EXAMPLES.

1. The reversion of a freehold estate of 791. 4s. per annum, to commence 7 years hence, is to be sold: what is it worth in ready money, allowing the purchaser 4 per cent. for his money?

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and 1'045] = 1360862) 1760 ̊000(1293‍297 = 12931. 5s. 114d. = present worth of 17601. for 7 years, or the whole present worth required.

* This rule is sufficiently evident without a demonstration. Those, who wish to be acquainted with the manner of computing the values of annuities on lives, may consult the writings of Mr. DEMOIVRE, Mr. SIMPSON, and Dr. PRICE, all of whom have handled this subject in a very skilful and masterly manner. Dr. PRICE'S Treatise on Annuities and Reversionary Payments is an excellent performance, and will be found a very valuable acquisition to those, whose inclinations lead them to studies of this nature.

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2. Which is most advantageous, a term of 15 years in an estate of 1001. per annum, or the reversion of such an estate forever, after the expiration of the said 15 years, computing at the rate of 5 per cent. per annum, compound interest?

Ans. The first term of 15 years is better than the reversion forever afterward, by 751. 18s. 74d.

3. Suppose I would add 5 years to a running lease of 15 years to come, the improved rent being 1861. 7. 6d. per annum; what ought I to pay down for this favour, discounting at 4 per cent. per annum, compound interest? Ans. 4601. 14s. 12d.

POSITION.

POSITION is a method of performing such questions, as cannot be resolved by the common direct rules, and is of two kinds, called single and double.

SINGLE POSITION.

Single Position teaches to resolve those questions, whose results are proportional to their suppositions.

RULE.*

1. Take any number and perform the same operations with it, as are described to be performed in the question.

* Such questions properly belong to this rule, as require the multiplication or division of the number sought by any proposed number; or when it is to be increased or diminished by itself, or any parts of itself, a certain proposed number of times. For in this case the reason of the rule is obvious; it being then evident, that the results are proportional to the suppositions.

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2. Then say, as the result of the operation is to the position, so is the result in the question to the number required.

EXAMPLES.

1. A's age is double that of B, and B's is triple that of C, and the sum of all their ages is 140: what is each person's age?

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2. A certain sum of money is to be divided between 4 persons, in such a manner, that the first shall have of it; the second; the third ; and the fourth the remainder, which is 281.: what is the sum ?

3. A person, after spending 601. left: what had he at first?

Ans. 1121.

and of his money, had

Ans. 1441.

4. What number is that, which being increased by 11, and of itself, the sum shall be 125 ?

Ans. 60.

5. A person bought a chaise, horse, and harness for 601.; the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness: what did he give for each?

Ans. 131. 6s. 8d. for the horse, 61. 13s. 4d. for the har ness, and 401. for the chaise.

6. A vessel has three cocks, A, B, and C; A can fill it in 1 hour, B in 2, and C in 3: in what time will they all fill it together? Ans. hour.

NOTE. 1 may be made a constant supposition in all questions; and in most cases it is better than any other number.

DOUBLE POSITION.

Double Position teaches to resolve questions by making two suppositions of false numbers.

RULE.*

1. Take any two convenient numbers, and proceed with each according to the conditions of the question.

2. Find how much the results are different from the result in the question.

3. Multiply each of the errors by the contrary supposition, and find the sum or difference of the products.

4. If the errors be alike, divide the difference of the pro

* The rule is founded on this supposition, that the first error is to the second, as the difference between the true and first supposed number is to the difference between the true and second supposed number: when that is not the case, the exact answer to the question cannot be found by this rule.

That the rule is true, according to the supposition, may be thus demonstrated.

Let A and B be any two numbers, produced from a and b by similar operations it is required to find the number, from which is produced by a like operation.

Put x = number required, and let N-A―r, and N—B=s. Then according to the supposition, on which the rule is founded, r: 8 :: x—a: x-b, whence, by mult plying means and extremes,rx-rb=sx—sa ; and, by transposition, rx—sx=rb—

sa; and, by division, x=

rb-sa

r8

number sought.

Again, if r and & be both negative, we shall have - -7: :: x—a : x—b, and therefore—rx+rb=➡sx+sa; and ræ

8x=rb-sa; whence x=

rb-sa

-, as before.

In like manner, if r or s be negative, we shall have x=

by working as before, which is the rule.

rb+sa

r+s

NOTE. It will be often advantageous to make 1 and 0 the suppositions.

ducts by the difference of the errors, and the quotient will be the answer.

5. If the errors be unlike, divide the sum of the products. by the sum of the errors, and the quotient will be the an

swer.

NOTE. The errors are said to be alike, when they are both too great or both too little; and unlike, when one is too great and the other too little.

EXAMPLES.

1. A lady bought tabby at 4s. a yard, and Persian at 2s. a yard; the whole number of yards she bought was 8, and the whole price 20s.: how many yards had she of each sort? Suppose 4 yards of tabby, value

16s.

Then she must have 4 yards of Persian, value 8

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So that the first error is + 4

Again, suppose she had 3 yards of tabby at 12s.
Then she must have 5 yards of Persian at 10

Sum of their values 22

So that the second error is + 2

Then 4-2-2= difference of the errors.

Also 4×2=8= product of the first supposition and second error.

And 3x4-12= product of the second supposition by the first error.

And 12-8=4= their difference.

Whence 4+2=2= yards of tabby, the answer.

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And 8-2-6 yards of Persian,

2. Two persons, A and B, have both the same income; A saves of his yearly; but B, by spending 50l. per annum more than A, at the end of 4 years finds himself 1001. in debt what is their income, and what do they spend per annum?

Ans. Their income is 1251. per annum; A spends 1001. and B 150l. per annum.

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