cents a pound respectively, how much of each kind must he take to form a mixture worth 10 cents a pound? Note 2.-In examples like the preceding, we compare two kinds together, one of a higher and the other of a lower price than the required mixture; then compare the other two kinds in the same manner. In selecting the pairs to be compared together, it is necessary that the price of one article shall be above, and the other below the price of the mixture. Hence, when there are several articles to be mixed, some cheaper and others dearer than the mixture, a variety of answers may be obtained. Thus, if we compare the highest and lowest, then the other two, the mixture will contain 1 part at 8 cts.; 1 part at 9 cts.; 1 part at 11 cts.; and 1 part at 12 cts. Again, by comparing those at 8 and 11 cts., and those at 9 and 12 cts. together, we obtain for the mixture 1 part at 8 cts.; 2 parts at 11 cts. ; 2 parts at 9 cts.; and 1 part at 12 cts. Other answers may be found by comparing the first with the third and fourth; and the second with the fourth, &c. 61. A goldsmith having gold 16, 18, 23, and 24 carats fine, wished to make a mixture 21 carats fine: what part of each must the mixture contain? 62. A farmer had 30 bu. of corn worth 6s. a bu., which he wished to mix with oats worth 3s. a bu., so that the mixture may be worth 4s. per bu.: how many bushels of oats must he use? Note 3.—In this example, it will be perceived, that the price of the mixture, with the prices of the several articles and the quantity of one of them are given, to find how much of the other article the mixture must contain. Analysis.-Reasoning as above, we find that the mixture (without regard to the specified quantity of corn) in order to be worth 4s. per bu., must contain 2 bu. of oats to 1 bu. of corn. Hence, if 1 bu. of corn requires 2 bu. of oats to make a mixture of the required value, 30 bu. of corn will require 30 times as much; and 2 bu.×30—60 bu., the quantity of oats required. 63. A merchant wished to mix 100 gallons of oil worth 80 cts. per gallon, with two other kinds worth 30 cts. and 40 cts. per gallon, so that the mixture may be worth 60 cts. per gallon: how Inany gallons of each must it contain ? 64. A merchant has Havana coffee at 12 cts. and Java at 18 cts. per pound, of which he wishes to make a mixture of 150 lbs., which he can sell at 16 cts. a pound: how much of each must he use? Note 4.-In this example, the whole quantity to be mixed, the price of the mixture, and the prices of the several articles are given, to find how much of each must be taken. Analysis.-On 1 lb. of the Havana it is obvious he will gain 4 cts., and on 1 lb. of the Java he will lose 2 cts.; therefore to balance the 4 cts. gain he must put in 2 lbs. of Java; that is, the mixture must contain 1 part of Havana to 2 parts of Java. Now if 3 lbs. mixture require 1 lb. Havana, 150 lbs. mixture, (the quantity required,) will require as many pounds of Havana as 3 is contained times in 150, viz: 50 lbs. But the mixture contains twice as much Java as Havana, and 50 lbs X2=100 lbs. Ans. 50 lbs. Havana, and 100 lbs. Java. 65. It is required to mix 240 lbs. of different kinds of raisins, worth 8d., 12d., 18d., and 22d. a pound, so that the mixture may be worth 10d. a pound: how much of each must be taken? 66. If 10 horses consume 720 quarts of oats in 6 days, how long will it take 30 horses to consume 1728 quarts ? 0 1. 6 Analysis. Since 10 horses will consume 720 qts. in 6 days, 1 horse will consume of 720 qts. in the same time; and 5 of 720 qts. is 72 qts. And if 1 horse will consume 72 qts. in 6 days, in 1 day he will consume of 72 qts., which is 12 qts. Again, if 12 qts. last 1 horse 1 day, 1728 qts. will last him as many days as 12 qts. are contained times in 1728 qts., viz: 144 days. Now if 1 horse will consume 1728 qts. in 144 days, 30 horses will consume them in of the time; and 144 d.÷30-4‡. 1 0 Ans. 30 horses will consume 1728 qts. in 44 days. 468. This and similar examples are usually placed under the rule of Compound Proportion, or Double Rule of Three. 67. If 15 horses consume 40 tons of hay in 30 weeks, how many horses will it require to consume 56 tons in 70 weeks ? 68. If 8 men can make 9 rods of wall in 12 days, how long wil it take 10 men to make 36 rods? 69. If 35 bbls. of water will last 950 men 7 months, how many men will 1464 bbls. of water last 1 month? 70. If 13908 men consume 732 bbls. of flour in 2 months, in how long time will 425 men consume 175 bbls. ? 71. If the interest of $30 for 12 months is $2.10, how much is the interest of $1560 for 6 months? 72. If the interest of $750 for 8 months is $28, how much is the interest of $16425 for 6 months? 73. A man being asked how much money he had, replied that †, 2, and § of it made $980: what amount did he have? 24 Analysis. It is plain that ++3=42. (Art. 202.) The question then resolves itself into this: $980 are of what sum ? Now if $980 are 42 of a certain sum, is of $980; and $980 -49$20, and 24 is $20×24=$480. Ans. 3 4 9 PROOF. of $480 $320; of $480-$360; and of $480 =$300. Now $320+$360+$300=$980, according to the conditions of the question. 469. This and similar examples are placed under the rule of Position. The shortest and easiest method of solving them is by Analysis. 74. A sailor having spent of his money for his outfit, deposited † of it in a savings bank, and had $50 left: how much had he at first? 1. 4 75. A man laid out of his money for a house, for furniture, and had $1500 left; how much had he at first? 76. A man lost of his money in gambling, in betting, and spent in drinking; he had $259 left: how much had he at first ? 77. What number is that and 3 of which is 102 ? 78. What number is that, 79. What number is that itself, the sum will be 164? 4 1 80. What number is that of which exceeds 4 of it by 18? 81. A post stands 40 feet above water, in the water, and † in the ground what is the length of the post? 82. What will 376 yds. of muslin cost, at 2s. and 6d. per yd.? Analysis.-2s. 6d. £. Now if 1 yd. costs £1, 376 yds. will cost 376 times as much; and £1×376=£47. Ans. 83. If 1 yard of silk costs 50 cents, what will 256 yards cost? Analysis.—50 cts.$. Now if 1 yd. costs $4, 256 yds. will cost 256 times as much; and $×256=$128. Ans. 470. Examples like the preceding, in which the price of a single article is an aliquot part of a dollar, &c., are usually classed under the rule of Practice. Practice is defined by a late English author to be "an abridged method of performing operations in the rule of proportion by means of aliquot parts; and it is chiefly employed in computing the prices of commodities.” OBS. After giving several tables of aliquot parts in money, weight, and measure, the same author proceeds to divide his subject into twelve subdivisions or cases, and gives a specific rule for each case, to be committed to memory by the pupil. It is believed, however, that so many specific rules are worse than useless. They have a tendency to prevent the exercise of thought and reason, while they tax the time and memory of the student with a multiplicity of particular directions for the solution of a class of examples, which his common sense, if permitted to be exercised, will solve more expeditiously by Analysis. TABLE OF ALIQUOT PARTS OF $1, £1, AND 1s. Note. If the price itself is not an aliquot part of $1, or £1, &c., it may sometimes be divided into such parts as will be aliquot parts of $1, £1, &c., or which will be aliquot parts of each other. Thus, 874 cts. is not an a'iquot part of $1, but 873 cts.50+25+12 cts. Now 50 cts.=$1; 25 cts.=$4; and 121 cts. $1. Or thus: 50 cts. $1, 25 cts. of 50 cts., and 124 cts.=1 of 25 cts. 84. What will 680 bu. of wheat cost, at 87 cts. per bushel? Analysis.—It is plain, if the price were $1 per bu., the cost of 680 bu. would be $680. Hence, Were the price 50 cts. the cost would be of $680, which is $340 ៩ 66 25 cts. 66 of $680, which is $170 of $680, which is $ 85 But since the price is 50+25+12 cents, the cost must be $595 Or, thus: $1X680=$680, the cost at $1 per bushel. 85. What cost 478 yards of cashmere, at 50 cts. per yard? 86. What cost 1560 lbs. of tea, at 75 cts. per pound? 87. What cost 2400 gals. of molasses, at 37 cts. per gal.? 88. What cost 1800 yds. of satinet, at 621 cts. per yard ? 89. At 25 cts. per bushel, what cost 1470 bu. of oats? 90. At 334 cts. a pound, what cost 1326 lbs. of ginger? 91. At 64 cts. per roll, what cost 3216 rolls of tape? 92. At 8 cts. per pound, what cost 4200 lbs. of lard? 93. At 121⁄2 cts. per dozen, what cost 1920 doz. of eggs? 94. At 16 cts. a pound, what cost 4524 lbs. of figs? 95. At 663 cts. per yard, what cost 1620 yds. of sarcenet? 96. What cost 840 bu. of rye, at $2 per bushel? 97. What cost 690 yds. of cloth, at 6s. 8d. per yard? Analysis.-At £1 per yard the cost would be £690. But 6s. 8d. is £; therefore the cost must be of £690, which is £230. Ans. 98. What cost 360 gals. of wine, at 16s. per gallon? Analysis.-16s.=10s.+5s.+1s. Now 10s. £; 5s.=£; of 5s. If the price were £1 per gal., the cost of 360 gals. would be £360. At 10s., £, it will be of £360, or £180 of £180, or £ 90 |