33. What two numbers between 32 and 480, have the former for their greatest common divisor and the latter for their least common multiple? (See Art. 115.) Ans. 96 and 160. 34. At $8 per yard, how many yards of flannel can be bought for $83? 35. How many bushels of wheat at $15 per bushel, will pay for of a barrel of flour at $83 per barrel? 36. If a person can do a job of work in 24 days, what part of it can he do in 16 days? 37. If a person going 4 miles per hour, performs a journey in 10 hours, how many hours would it take him, if he traveled 3 miles per hour? 38. Bought 12 barrels of flour at $10 per barrel, how many barrels more could I have bought for the same money, if I had paid $2 less per barrel? Ans. 412 barrels. 39. Bought 31 bushels of wheat at many bushels less would I have bought if I had paid $76 more per bushel? $13 per bushel, how for the same money, Ans. 97 bushels. 40. What is the greatest breadth of carpet that would cover the floors of three rooms with whole breadths, the rooms being, respectively, 14 feet, 16 feet, and 19 feet broad? (See Arts. 162, and 163). Ans. 23 feet. 41. A miller has 531 bushels of oats, 663 bushels of corn, and 851 bushels of wheat, which he wishes to put in sacks of equal size without mixing. What is the capacity of each sack, and the number of sacks? Ans. 2 bushels, and 77 sacks. 42. What is the least number of cents that will buy either a whole number of oranges at 63 cents each, or of lemons at 5 cents each? (See Arts. 164, and 165.) Ans. 56. 43. What is the least sum of money for which I can buy a number of ducks, at $3 each, a number of geese, at $1 each, or a number of turkeys at $1 each, and how many of each could be bought for the same sum? 5 44. If A can dig a ditch in 6 days, and B can dig it in 9 days, what part of it can they both dig in 1 day? Ans. g. 45. If A can mow a field in 10 days, B in 12 days, and C in 15 days, in how many days can they all mow it? 46. If A can build a wall in 20 days, and A and B in 8 days, how long will it take B, alone, to build it? Ans. 131 days. 47. A, B, and C can reap a field in 10 days, A and C in 15 days, and B and C in 20 days; in how many days could each alone reap it? Ans. A in 20 days, B in 30 days, and C in 60 days. 48. C has 341 sheep in two fields. How many are in each field, provided the first contains as many as the second? 49. A man spent $372, and saved the remainder of his salary which was of it; what was his salary? Ans. $1426. 50. A farmer has of his farm in grain, in grass, and the remainder, 91 acres in timber. How many acres in his farm, and how many in grain and grass respectively? 51. A tree whose length was 133 feet, was broken into two pieces by falling; g of the length of the longer piece equaled of the length of the shorter. What was the length of each piece? Ans. 70 feet, and 63 feet. 52. Divide $2125 between A and B, so that of A's part shall equal of B's. 53. A owns of an oil well, valued at $27547; B owns of the remainder; C owns as much as A and B; and D owns the remainder. How much does each own? Ans. A owns $6357, B $8476, C $6846, and D $5868. 54. A man dying left an estate to be divided among his four sons, A, B, C, D. A was to have of it; B 4% of it; C of it; and D the remainder, which is $12480. What was the estate, and what did each receive? 40 55. If of a farm is cleared, and of it is timber land, how many acres are in the farm, provided there are 32 acres more cleared than timber land, and how many acres are there of each kind? ៖ 56. C's farm cost $3500, and 3 of its cost is of the cost of his house; what was the cost of the house? of 57. A, B, and C bought a drove of sheep. A received them; B of them; and C the remainder. If A had 104 more than B, how many were in the drove, and how many did each receive? Ans. 1560: A 624, B 520, C 416. CHAPTER X. DECIMAL FRACTIONS. Art. 166. A decimal fraction is a fraction whose denominator is a power of 10. (See Art. 44). Thus, o, 100, 1000, 1000, are decimal fractions. 35 629 1234 NOTE. Decimal fractions are only a variety of common fractions. NOTATION OF DECIMAL FRACTIONS. Art. 167. Decimal fractions may be written like common fractions, but the fact that their denominators are decimal permits the fractions to be written without their denominators, and this is usually done. As, in integers, every place, reckoning from left to right toward units, is one-tenth of the value of the preceding place, so places to the right of units may express decimal fractions, each one-tenth of the preceding. ILLUSTRATION. The unit is 1. of a unit may be written To of 10, or Too, may be written. .1 .01 .001 .0001 .00001 .000001 Hence, the numerator of any decimal fraction will, without its denominator, express the value of the fraction, if it is written as many places to the right of the place of units as the fraction has ciphers in its denominator. Art. 168. When decimal fractions are written without their denominators, they are usually called decimals. The decimal point, or separatrix, is a period placed between the units' place and a decimal. Thus, in the mixed number 2, if the fraction is written as a decimal, a period is placed between 2 and 3, making the number 2.3. NUMERATION OF DECIMALS. Art. 169. Proceeding from the decimal point towards the right, the names of the places are tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths, millionths, tenmillionths, hundred-millionths, billionths, ten-billionths, &c. |