The following are enunciations of several questions which are connected with the preceding.

Required the number of years for which we must put a sum a, at compound interest, at 5 and 10 per cent, in order to double that

sum.

(Ans. At 5 per %, 14 years 2 months; at 10 per %, 7 years 3 months.)

Required the sum which must be put at interest at the present time, in order to produce for 12 years, at the end of each year, α sum of 1500 francs, so that the whole principal and interest may be repaid at the end of the twelve years, the interest being seven and a half per cent per year.

(Ans. 11602,91.)

A person has bought a property of 100,000 francs, which is to be paid for in fifteen equal payments at compound interest; the rate for each interval of payment is 5 per cent. Required what will be the amount or the quota of each payment.

(Ans. 9634,22.)

A certain number of men a increases every year by the hundredth part of what it was the preceding year; required in how many years the number will become ten times greater.

(Ans. 231 years nearly.)

Suppose that from a barrel of 100 pints of wine, we draw each day a pint, which we replace by a pint of water; required, 1. how much wine will remain in the barrel, when we have replaced the fiftieth pint; 2. in how many days the wine will be reduced to one half, one third, or one fourth.

Ans. to the first part of the question, 601 pints.

Ans. to the second part, 69 days for the half, 109 days for the third, and 138 days for the fourth.

NOTE.

IV. Of Continued Fractions.

166. CONTINUED FRACTIONS originate in the approximate value of fractions whose terms are considerable, and prime to each other. In order to be better understood, let there be proposed the fraction 15, of which it is easy to show that the two terms are prime to each other, and which for that reason is (157) irreducible.

By leaving this fraction under this form, it becomes difficult to obtain a just idea of it; but if, by means of a known principle, we divide its two terms by 159, which does not change its value, it becomes

nominator,

(H)' or by performing the division indicated in the de

This being premised, let us neglect for the moment the fraction ; the fraction, which results, is a little greater than the pro

posed fraction, since we have diminished the denominators.

On the other hand, if, instead of neglecting, we replace this 1 1 fraction by 1, which gives or this new fraction is, in its 3+1 4'

turn, smaller than the proposed fraction, since we have increased the denominator.

15 493

Whence we may conclude that the fraction is comprehended between and . This gives already a sufficiently exact idea of the fraction.

If we wish a greater degree of approximation, we have only to perform the operation with, as we have done with 1, and the proposed fraction becomes

so that the proposed fraction is still comprehended between and

The difference of these two last fractions, reduced to the same denominator, is

Then the error we may commit by taking for the value of the proposed fraction is less than

By performing the operation upon 15, as we have done with the preceding, we have

and the proposed fraction may be put under the forin

Let us neglect; the number 1, or 1, is greater than 1, then

The first is too small and the second too large.

Now the difference of these two fractions is — or di

8

so that the error which we commit, by taking either, or 1, for the value of the proposed fraction, is less than

T•

We see how, by this series of operations, we succeed in finding, in more simple terms, fractions which give the approximate values of another fraction of which the terms are very considerable. The expression

In general, we understand by continued fraction, a fraction which has for its numerator unity, and for its denominator a whole number, plus a fraction which has also unity for its numerator, and for its denominator a whole number, plus a fraction, and so on.

Frequently the proposed fractional number is greater than unity. So that in order to make the definition of a continued fraction more general, we must say : A continued fraction is an expression composed of a whole number, plus a fraction which has for its numerator unity, and for its denominator, &c.

(a, b, c, d,... being whole numbers).

167. By reflecting on the method which has just been pursued in order to reduce 15 to a continued fraction, we see that we first divided 493 by 159, which gave 3 for a quotient, and for a remainder 16. We then divided 159 by 16, which gave for a quotient 9, and for a remainder 15; we then divided 16 by 15, which gave 1 for a quotient, and 1 for a remainder. Thence it is easy to determine the following process, in order to reduce a fraction or a fractional number to a continued fraction.

Proceed with the two terms of the proposed fraction, as if to find their greatest common divisor (49). Continue the operation till there is obtained a remainder equal to zero, and the successive quotients at which we shall arrive, will be the determinators of the fractions which constitute the continued fraction.

In the hypothesis in which the proposed number is greater than unity, the first quotient represents the entire portion which enters into the expression of the continued fraction.

We may, according to this method, reduce to continued fractions the two numbers and

The following is the form of operations