61. What is the freight of 60480 pounds of cotton from Charleston to Liverpool, at $4 per ton ? MISCELLANEOUS EXAMPLES. 62. How many bottles, holding 1 pint and 2 gills each, are required for bottling 4 barrels of cider ? 63. How much will 46 bushels of oats cost, at 4 pence 2 farthings for every two quarts ? 64. A brewer sold 96 hogsheads of beer for £388 16 s. What was the price of 1 pint at the same rate ? 65. A certain tippler spent 12 cents a day for ardent spirit, during 39 successive weeks, and then died, the victim of his folly. What did the spirit all cost ? 66. Bought five loads of wood; the first containing 1 cord 32 cubic feet, the second 1 cord 64 cubic feet, the third 112 cubic feet, the fourth 1 cord 28 cubic feet, and the fifth 1 cord 20 cubic feet. How many cords were there in the whole ? 67. Bought goods to the amount of £25 13s. 10d. 2qr.; and afterwards sold goods to the same man, amounting to £30 10s. 4 d. 2qr. What is the balance of money in my favor? 68. A farmer sold five lots of land, at $9 an acre; the first lot containing 30 A. 2 R. 2r., the second 41 A. 3R. 8r., the third 14 A. 1 R. 10r., the fourth 25 A. 36 r., and the fifth 54 A. 6r. What did the whole amount to ? 69. How many cubic inches in a brick 8 inches long, 4 inches wide, and 2 inches thick ? 70. How many cubic inches in the cube of 2 inches ? ..... in the cube of 3 inches? ..... in the cube of 4 inches ? ..... in the cube of 5 inches ? 71. If the cube of 4 inches be taken from the cube of 1 foot, how many cubic inches will remain ? 72. If the cube of 4 inches be taken from the cube of 2 feet, how many cubic inches will remain ? 73. A young man, on commencing business, was worth £643 10s.; the first year he cleared £ 54 11 s. 7 d. 2 qr.; the second year, £87 Os. 10 d. lqr.; but the third year he lost £1967 s. 11 d. 3qr. How much was he then worth ? 74. A gentleman had a hogshead of wine in his cellar, from which there leaked out 17 gal. 3qt. 1 pt. How much then remained ? 75. A man started on a journey of 20 miles 6 fur. 29 r., and stopped to rest at a house, 4 m. 4 fur. 20 r. from the place of starting. How far had he still to go? C.76. In a pile of wood, 96 feet long, 5 feet high, and 4 feet wide, how many cords ? 77. How much would 13 hogsheads of sugar cost, at 8 cents per pound; allowing each hogshead to contain, 8cwt. 3qr. 24 lb.? 78. A cent weighs 8 pennyweights 16 grains. What is the weight of 100 cents ? 79. How many yards of cloth are there in 19 pieces; each piece containing 27 yd. 3qr. 2 na. ? 80. If a man sell 2 bl. 1 kil. 1 fir. 6 gal. 2qt. 1 pt. of beer in one week, how many barrels would he sell in 26 weeks ? 81. If 1 pint and 3 gills of wine will fill a bottle, how much will fill a gross, or 12 dozen bottles ? 82. A father left an estate worth £ 5719 17 s., to be divided equally among 11 children. How much was each one's share ? 83. Sixteen men own 24 tierces of molasses, in equal shares. What is one man's share ? 84. A company of 23 men bought 1850 acres 10 rods of wild land, and divided it equally among them. How much land had each man ? 35. What must be the length of a lot of land, that is 5 rods wide, in order that the lot shall contain 1 acre ? Observe in the above question, that 1 acre contains 160 square rods; and, that this number of square rods is the product of the two factors that denote the width and length of the lot. See PROBLEM V, page 21. 86. What must be the depth of a house lot, that measures 72 feet on the front, to contain 9432 square feet ? 87. What must be the length of a stick of hewn timber, that is 10 inches wide and i ft. 3in. deep, in order that the stick shall contain 1 ton ? Observe in this question, that the number of cubic inches in a ton, is the product of the three factors which denote, in inches, the width and depth and length of the stick. See PROBLEM VIII, page 22. 88. What must be the length of a pile of wood that is 4 feet wide and 3 feet high, in order that the pile shall contain 1 cord, that is, 128 cubic feet? 89. Suppose a pile of wood to be 11 feet long and 3 feet wide; how high must it be, to contain 2 cords 4 feet of wood and 10 cubic feet ? X. FRACTIONS. A FRACTION signifies one or more of the equal parts into which a unit, or some quantity considered as an integer, or whole, is divided. A fraction is expressed by two numbers or terms, written one above the other, thus, . The lower term -called the denominator - denotes the number of equal parts into which the integer is divided; and the upper term — called the numerator— indicates what number of those equal parts the fraction expresses. We may not only consider a fraction as a certain number of parts of a unit, but, may also view it as a part of a certain number of units. Thus, may either be considered as 2-thirds of 1, or, 1-third of 2; for 1-third of 2 is the same quantity as 2-thirds of 1. Hence, if the numerator of a fraction be viewed as an integer, and divided into as many equal parts as the denominator indicates, the fraction may be regarded as expressing one of these parts. Thus, if 4 be divided into 5 equal parts, the fraction 4 expresses one of these parts. Fractions generally have their origin from the division of a number by another which does not measure it; the excess of the dividend, above what can be measured by the divisor, being the numerator, and the divisor being the denominator, as shown in Art. VI. If the numerator of a fraction be made equal to the denominator, the fraction becomes equal to unity; thus 4=1. If the numerator be greater than the denominator, the fraction is equal to as many units as the denominator is contained times in the numerator; for example 2=3. Hence, a fraction may be viewed as an unexecuted division; the divisor being written under the dividend. It follows, also, that since any number divided by 1 gives the same number in the quotient, any number may be expressed as a fraction by making 1 its denominator. For example, 17 may be expressed thus, 4. The following propositions concerning fractions, should be distinctly noticed. PROPOSITION I. As many times as the numerator is made greater, so many times the fraction is made greater; and, as many times as the numerator is made smaller, so many times the fraction is made smaller. Hence, a fraction is multiplied by multiplying the numerator, and divided by dividing the numerator. PROPOSITION II. As many times as the denominator is made greater, so many times the fraction is made smaller; and as many times as the denominator is made smaller, so many times the fraction is made greater. Hence, a fraction is divided by multiplying the denominator, and multiplied by dividing the denominator. PROPOSITION III. When the numerator and denominator are both multiplied, or both divided by the same number, the quantity expressed by the fraction is not thereby changed. A PROPER FRACTION is a fraction whose numerator is less than its denominator; as 1 : An IMPROPER FRACTION is a fraction whose numerator equals, or exceeds its denominator; ass, A number consisting of an integer with a fraction annexed, as 14ā, is called a MIXED NUMBER. A COMPOUND FRACTION is a fraction of a fraction; as of of 1 of. A COMPLEX FRACTION is that which has a fraction either in its numerator, or in its denominator, or in both of them; thus, 5. & 41. REDUCTION OF FRACTIONS. REDUCTION OF FRACTIONS consists in changing them from one form to another, without altering their value. CASE I. To reduce a fraction to its lowest terms; that is, to change the denominator and numerator to the smallest numbers that will express the same quantity. RULE. Divide both terms of the fraction by their greatest common measure, and the two quotients will be the lowest terms of the fraction. See PROB. IX, page 22. When the greatest common measure is readily perceived, the fraction may be reduced mentally. For instance, the greatest common measure of the terms of the fraction 12 , is 4, and the only notation necessary in the reduction, is, 1 =. Dividing the terms of a fraction by a common measure which is not the least, will reduce it in some degree, and when thus reduced, it may be reduced still lower by another division, and so on, till no number will measure both the terms. For example, to reduce 12 , divide by ?, and the result is ; again, divide by 3, and the result is 3. Here the fraction is known to be in its lowest terms, because the terms are prime to each other. 1. Reduce 112 to its lowest terms, by repeatedly dividing the terms by any common measure. 2. Reduce 1919 to its lowest terms, by dividing the terms by their greatest common measure. 3. Reduce each of the following fractions to its lowest terms. . 376. 38 2. 123: 3392. - CASE II. To reduce a whole number to an improper fraction. RULE. Multiply the whole number by the proposed denominator, and the product will be the numerator. When the quantity to be reduced is a mixed number, the numerator of the fraction in the mixed number must be added to the product of the whole number, and their sum will be the numerator of the improper fraction. |