A general, disposing of his army into a square, finds he has 284 soldiers over and above; but increasing each side with one soldier, he wants 25 to fill up the square; how many soldiers had he? Ans. 24000. There is a prize of 212l. 14s. 7d. to be divided among a captain, 4 men, and a boy; the captain is to have a share and a half; the men each a share; and the boy of a share; what ought each person to have? Ans. The captain 54l. 14s. d. each man 367. 9s. 4 d. and the boy 121. 3s. 1 d. A cistern, containing 60 gallons of water, has 3 unequal cocks for discharging it; the greatest cock will empty it in one hour, the second in 2 hours, and the third in 3; in what time will it be emptied, if they all run together? Ans. 32 minutes. In an orchard of fruit trees, of them bear apples, pears, plumbs, and 50 of them cherries: how many trees are there in all? Ans. 600. A can do a piece of work alone in 10 days, and B in 13; if both be set about it together, in what time will it be finished? Ans. 519 days. A, B, and C are to share 1000007. in the proportion of,, and, respectively; but C's part being lost by his death, it is required to divide the whole sum properly between the other two. Ans. A's part is 571421. and B's 428571. Summation of a continued Fraction. "When in the course of a calculation we meet with a fraction whose numerator and denominator are pretty large, and have no common factor, we seek approximate values of this fraction, which are expressed by more simple numbers, with a view to forming a more exact idea of it. 216 If we have, for example, the fractional number 1103, we obtain, at first, the whole number, and there results 1 and 19. Now, to form a more simple idea of the fraction 1, we endeavour to compare it with a part of unity, in order that we may have only one term to be inquired into, and for this purpose we divide the two terms by 216; we find 1 for the quotient of the numerator, and 4 for that of the denominator; this last quotient, being between 4 and 5, shows that the fraction is between and . By stopping at this point, we see that the second approximate value of the expression 3 is 1 and, or . But this value is too great, for the true value would be equal to 1 plus 1 divided by 4 added to, which is written thus; 1 1103 To form an exact idea of the expression 1 1 23 4 210 4 216 ry to consider it as indicating the quotient of the whole number 1 divided by the whole number 4 accompanied by the fraction 23 1 If we divide the two terms of by 23, the quotient will be 9223 ; neglecting the which accompany the whole number 9, there will be only instead of, and consequently, 1, will 1103 4층 be a third approximate value of 13, a value which will be too small, since 9 being less than the true quotient of 216 by 23, the fraction will be greater than that which ought to accompany 4, and consequently the divisor 4 will be greater than the exact divisor 4,23, and the quotient quotient. Arith. 17 4층 smaller than the true By reducing the whole number 4 with the fraction which accompanies it, and performing the division according to the method of art. 74, we obtain; and we have 1 and 3 or for the third approximate vulue of 1103. The exact expression of this value being 11 4 9 923 , if we divide the two terms of by 9, we shall have 1 a value too great; for the fraction being greater than 4 1 91 20 2 1 whose place it occupies, will form, by being joined with 9, a denominator too great; the fraction joined to 4 will consequently be too small, and the denominator being too small, the fraction itself will be too great. By reducing, at first, 9 to a fraction, we have 1 ; will 9 then be, and the approximate value will become 11 472 ; now gives, which joined to unity becomes 14, or for the fourth approximate value of 113. we divide the two terms of the last fraction by 5, and obtain 1, and thence 1. 1 1 and it will be seen as before, that this value is less than the true value. The fraction reduces itself to ; and since the preceding 21 911 gives, the next preceding becomes equal to ; so that the fifth approximate value is 1 473 Dividing, finally, by 4 the two terms of the fraction 2; which was neglected above, we have for a quotient and by sup 11; pressing the fraction, we obtain the new value 1. greater than the true value. If we reduce, successively, all the denominators to a fraction, to obtain the simple fraction which it represents, we shall find 1,4% or 4. By restoring the fraction to the side of the last denominator, we form the expression 1 1 4. 1 93 which being reduced like the preceding, reproduces the fractional number 03 887 We may pursue the same process with any other fraction, and obtain a series of approximate values, alternately greater and less than the true value, if it is a fraction properly so called, or alternately less and greater, if, as in the preceding example, the numerator exceeds the denominator. The developments above found for the expression 1103 are called continued fractions, which may be defined in general thus:-Fractions whose denominator is composed of a whole number and a fraction, which fraction has for a denominator also a whole number and a fraction, &c." |