39. I have expended $ 42.875 for a quantity of grain, T3ff of it being corn, at $0.75 a bushel; fa of it wheat, at $ 2 a bushel; and the balance oats, at $ 0.40 a bushel, to the amount of $ 3.50. Required the number of bushels of each kind purchased. 40. If a mason, in constructing a drain 250.35 feet long, begin with a width of 8 inches, and increase ^ of an inch in every foot of length, how many times the width of the beginning of the drain will its end be? Ans. 16.646875. 41. A gentleman gave £ of his property to his son James; J of it to his son William; ^ of the remainder to his daughter Mary; and the balance to his wife. It appeared that Mary received $ 2243.26 less than James. What was the amount divided, and how much did each receive? Ans. Amount, $ 13459.56; James, $3364.89; William, $4486.52; Mary, 1121.63; wife, $ 4486.52. CIRCULATING DECIMALS. 281i A Circulating Decimal is a decimal in which one or more figures are continually repeated in the same order. Thus, in reducing ^ to an equivalent decimal, on annexing ciphers and dividing by the denominator, the result obtained, .333-)-, is a circulating decimal; for, however far the division might be carried, the same figure would continue to be repeated without the decimal terminating. Such decimals are sometimes called infinite, or repeating; and, for sake of distinguishing, those decimals that terminate are sometimes termed finite. 282^ A repetend is a figure, or a series of figures, continually repeated. To mark a repetend, a point (.) is placed over a single repeating figure, or over the first and last of a series of repeating figures. Thus, in .3, the point denotes that the 3 is a repetend; and in .72, that the 72 is a repetend. 283. A single repetend is one in which only one figure is repeated; as in .1111-(- denoted by .1; and 2222-f-, denoted by .2. 284. A compound repetend is one in which the same set of figures is repeated; as in .135135-U, denoted by .135, and .30363036+, denoted by 3036. 285- A pure repetend is one which contains only the figures of the repetend; as, .3, .02, and .123. 286. A mixed repetend is one in which a repetend is preceded in the same fraction by one or more figures. The figures preceding the repetend are called the finite part. Thus, .416 is a mixed repetend, of which the figure 6 is the repetend, and the figures 41 the finite part; also, 1.728 is a mixed repetend, of which the figures 28 are the repetend, and the figures 1.7 the finite part. 287. A perfect repetend is a pure repetend containing the same number of figures as there are units in its denominator less one. Thus, ^ reduced to a decimal gives .142857, which, as it contains as many figures as there are units in the denominator, 7, less one, is a perfect repetend. 288. Similar repetends are those which begin at the same distance from the decimal point; as .3 and .6; or 5.123 and 3.478. 289. Dissimilar repetends are those which begin at different distances from the decimal point; as .986 and .4625; or .5925 and .0423436. 290. Conterminous repetends are those which terminate at the same distance from the decimal point; as .631 and.465, or .0753 and .4752. 291. Similar and conterminous repetends are those which both begin and end at the same distance from the decimal point; as .354 and .425; or .5757 and 5723. 292. Repetends alioays arise from common fractions, which, when in their loioest terms, contain in their denominator other factors than 2 and 5. For when a common fraction is in its lowest terms, its numerator and denominator are prime to each other (Art. 219), and the annexing of one or more ciphers to the numerator makes the same a multiple of 10, but does not render it divisible by any factor, except 2 and 5, the factors of 10. Therefore, when the denominator of a fraction, in its lowest terms, contains other factors than those of 10, the decimsi resulting from dividing the numerator with ciphers annexed, will not terminate, but will contain one or more figures that constantly repeat. 293i A pure repetend is always equivalent to a common fraction whose numerator is the repeating figure or figures, and whose denominator as many places of nines as there are repeating figures. For, by reducing £ to a decimal, we obtain as its equivalent the repetend .1; and since .1 is equivalent to J, .2 will be equivalent to §, .3 to §, and so on, till .9 is equal to | or 1. Aga in, and T^tt, being reduced, give .01, and 001; that is, = .01, and g^g- = .001 ; therefore, g2g = .02, and g- = 002, and so on; the same principle holding true in all like cases. 294. A mixed repetend is equivalent to a complex decimal (Art. 278), or to a complex fraction. Thus, the mixed repetend .2412 is equivalent to the mixed decimal .24£§, which is 24££ equal to the complex fraction -y^ 295. Mepetends are of the same denomination only when they are similar and conterminous. For then alone, by having a common denominator, do they express fractional parts of the same unit. REDUCTION OF REPETENDS. 296. To reduce a repetend to an equivalent common fraction. Ex. 1. Reduce .123 to an equivalent common fraction. Ans. ,jvj Operation. We write the figures of the given ,i23 = i^S = J^j'j Ans. repetend with the decimal point omitted for the numerator, and as many nines as places in the repetend tor the denominator, of a common fraction (Art. 293), and obtain iff, which, reduced to its lowest terms, = Alj, the answer required. 2. Reduce .138 to an equivalent common fraction. Ans. OPERATION. The mixed repetend .138 1 3 4 is equivalent to the mixed .1 3 8 = y-f = , — tffc Ans. decimal .13$ (Art. 294), * which we readily change to the form of a complex fraction by erasing the decimal point and writing the denominator, 100, 131 which is understood; and thus obtain — * , which, reduced to its simplest form, gives -fa, the answer required. Hence, If the given repetend be Simple, make the repeating figure or figures the numerator, and take as many nines as the repetend has figures for the denominator. If the given repetend be Mixed, change it to an equivalent complex fraction, and that fraction to its simplest form. Note. — Any circulating decimal may be transformed into another decimal, having a repetend of the same number of figures; as, .78 — .787, and .534 — .5345 - .53453. Thus, when such expressions as 12.5 or 17.56 occur, they may be also transformed; as 12 5 = 12.52, and 17.56 = 17.567 -- 17.5675, &c. Examples. 3. Required the common fraction equal to .6. Ans. | = §. 4. Reduce 1.62 to its equivalent mixed number. Ans. 5. Change .53 to an equivalent common fraction. 6. What common fraction is equivalent to .769230? Ans. 7. What common fraction is equivalent to .5925? 8. Change 31.62 to an equivalent mixed number. Ans. 31 iff 9. Reduce .008497133 to an equivalent common fraction. 297i To determine the kind of decimal to which a given common fraction can be reduced. Ex. 1. Required to find whether the decimal equal to /t'a be finite or circulating; and if finite, of how many places the decimal will consist. Ans. Finite, of 4 places. |