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22. If $477.72 be equally divided among 9 men, what will be each man's share? Ans. $53.08.

23. A man bought a barrel of flour for $5.375, 7gal. of molasses for $1.78, 9gal. of vinegar for $1.1875, 1gal. of wine for $1.125, 14lb. of sugar for $ 1.275, and 5lb. of tea for $2.625; what did the whole amount to? Ans. $13.367. the first contained 2g

24. A man purchased 3 loads of hay; tons, the second 37 tons, and the third 1 value of the whole, at $17.625 a ton? 25. How many hogsheads of water cistern which is 15.25 feet long, 8.4 deep?

tons; what was the Ans. $128.88218. will it take to fill a feet wide, and 10 feet Ans. 152hhd. 66 gal.

26. At $13.625 per cwt., what cost 3cwt. 2qr. 7lb. of sugar? Ans. $48.6411. 27. At $125.75 per acre, what cost 37A. 3R. 35rd.? Ans. $4774.570. 28. At $11.25 per cwt., what cost 17cwt. 2qr. 21lb. of rice? Ans. $199.2371⁄2. 29. What cost 73 bales of cotton, each weighing 3.37cwt., at $9.37 per cwt.?

30. What cost 7hhd. 49gal. of wine, at $97.625 per hogshead? Ans. $759.30585. 31. What cost 7yd. 3qr. 3na. of cloth, at $4.75 per yard? Ans. $37.703. 32. What cost 27T. 15cwt. 1qr. 34lb. of hemp, at $183.62 per ton? Ans. $5098.071. 33. What is the cost of constructing a railroad 17m. 3fur. 15rd., at $1725.875 per mile? Ans. $30067.97882.

34. When $624.53125 are paid for 17A. 3R. 15p. of land, what is the cost of one acre?

35. Paid $494.53125 for 19T.

what was the cost per ton?

15cwt. 2qr. 14lb. of hay; Ans. $24.999 172

36. How much land, at $40 per acre, can be obtained for $1004.75?

Ans. 25A. OR. 19p.

37. How many cords of wood can be put into a space 20.5 feet long, 12.75 feet wide, and 7.6 feet high?

Ans. 15 cords 66 cubic feet.

38. How many bushels of corn at $0.623 per bushel must a farmer exchange for 31 yards of sheeting at $0.08 per yard, and 7 yards of broadcloth at $2.75 per yard? Ans. 37,23

of

39. I have expended $42.875 for a quantity of grain, it being corn, at $0.75 a bushel; 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 gentlenian gave of his property to his son James; 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.

281. 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 .i; and 2222+, denoted by .2.

284. A compound repetend is one in which the same set of figures is repeated; as in .135135+, 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 always arise from common fractions, which, when in their lowest 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 decimal resulting from dividing the numerator with ciphers annexed, will not terminate, but will contain one or more figures that constantly repeat.

293. 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, 2 will be equivalent to §,.3 to 3, and so on, till .9 is equal to or 1. Again, 5, and 35, being reduced, give .01, and 001; that is, 'g .001; therefore, .02, and 002, and so on; the same principle holding true in all like cases.

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.01, and

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294. A mixed repetend is equivalent to a complex decimal (Art. 278), or to a complex fraction. tend .2412 is equivalent to the mixed

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Thus, the mixed repedecimal .24%, which is

295. Repetends 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.

.123

OPERATION.
123

4 1

=
999 333 Ans.

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We write the figures of the given repetend with the decimal point omitted for the numerator, and as many nines as places in the repetend for the denominator, of a common fraction (Art. 293), and obtain 123, which, reduced to its lowest terms, = , the answer required.

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138

which, reduced to its

tion by erasing the decimal point and writing the denominator, 100, which is understood; and thus obtain simplest form, gives, the answer required.

100'

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.

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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 125 = 12.52, and 17.56 — 17.567 — 17.5675, &c.

EXAMPLES.

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3. Required the common fraction equal to .6. Ans. § 4. Reduce 1.62 to its equivalent mixed number.

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Ans. 13.

5. Change .53 to an equivalent common fraction.
6. What common fraction is equivalent to .769230?

7. What common fraction is equivalent to .5925? 8. Change 31.62 to an equivalent mixed number.

Ans. 13.

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297. 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 20% be finite or circulating; and if finite, of how many places the decimal will consist. Ans. Finite, of 4 places.

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