thief in one hour? how long before he will overtake the thief? A. 2 miles; 15 hours. 54. A person, looking at his watch, was asked what o'clock it was; he replied it was between 4 and 5; but, a more particular answer being requested, he said the hour and minute hands were then exactly together. What was the time? In 1 hour the minute hand passes over 12 spaces, while the hour hand passes over only 1 space; that is, the minute hand gains upon the hour hand 11 spaces in 1 hour; conse quently, it must gain 12 spaces before both will be together. 60 m. ÷ 11 = 5—5m., gain in 1 hour; then,5,5m. × 4 hours =219 m. past 4 o'clock, Ans. Or, the ratio being of an hour, hence, 4 hours 21 minutes past XTI 211 4, Ans., as before. 11 I 55. At 12 o'clock the hour and minute hands of a clock aro exactly together; when will they be together again? Ans. 1 h. 5 m. 27 sec. 56. If 10 men can perform a piece of work in 25 days, how many men will accomplish another piece of work, four times as large, in a fifth part of the time? A. 200. 57. A can do a piece of work in 8 days, and B in 12; in what time would both finish it by working together? days, day, work, work. 8:1: 1 : 12:1 :: 1 : 1+12=24. Then, 4:1:1: 44, Ans. 58. What number is that, from which, if you take 3, the remainder will be ? +1=1, Ans. 59. What number is that, from which, if you take, the remainder will be ? Ans. §. 60. What number is that, from which, if you take of of 24, the remainder will be ? Ans. = 11% • 2 & 9 10 61. What number is that, which, being divided by, the quotient will be ? A. 19. 62. What number is that, which, being multiplied by, the product will be 31? A. 339. 63. What number is that, from which, if you take of itself, the remainder will be 12? 1, or 4, — §, = 1, remainder. Then, the remainder 12, being 4 × 12= 48 times greater than the remainder, the number itself will be 48 times greater than 1. A. 48. 64. What number is that, to which, if you add of of itself, the sum will be 39? of 3 =, and the number itself 18; then, † +18=18; v 1 of 333 ΤΟΣ to the whole number of it be added, the sum will be 18; consequently, 39 is 13 of the number. Ans. 30. 65. What number is that, to which if you add of itself, the sum will be 18? A. 12. 66. A owns of a vessel, B, C, and D the remainder; D's part is $100: can you tell me how many dollars is each man's part, and what part of the vessel D owns ? Ans. A's part, $100; B's, $200; C's, $400 ; and D's part is 67. There is a beam, of which is in the ground, 5 9 in the water, and the rest, being 2 feet, out of water; how long is the beam? A. 16 feet. 68. The third part of an army was killed, the fourth part taken prisoners, and 1000 fled; how many were in this army? how many killed? how many taken captives? 12 5 }+1=√2, of the whole army; then, as more makes, or the whole army, 12 = 1000; and if be 1000, how much is †, or the whole? Ans. 2400, the whole army; 800 killed, 600 captives. 69. Suppose that there is a mast erected, so that of its length stands in the ground, 12 feet in the water, and of its length in the air, or above water; I demand the whole length. Reducing the fractions to the least common denominator, gives 125 +13 =17; therefore 12 fået 8 1. A. 216 feet. 70. In an orchard of fruit-trees, of them bear apples, pears, plums, 40 of them peaches, and 10 cherries; how many trees does the orchard contain? 50+10. Ans. 600. 71. A man spent one third of his life in England, one fourth in Scotland, and the remainder, which was 20 years, in the United States; to what age did he live? A. 48 years. 72. The number of scholars in a certain school is as follows: of the pupils study geography, grammar, arithmetic, and 10 learn to read: what number is pursuing each branch of study? A. 30 in geography, 80 in grammar, 120 in arithmetic, and 10 learn to read. 73. The double and the half of a certain number, increased by 7 more, make 100; what is that number? A. 37. 74. A man, having purchased a drove of cattle, was driving them to market, when he was met by a gentleman, who inquired of him where he was going with his 100 head of cattle? Šir, said he, I have not near 100, but if I had as many more as I now have, as many more, and 7 cattle and, I should have a hundred. How many had he? A. 37. 75. Five eighths of a certain number exceed of the same number by 36; what is that number? #—4=; hence 36 is of the number sought. A. 160. 76. What number is that, which, being increased by 1, 1, and of itself, the sum will be 131? Ans. 737. The eleven foregoing questions are usually performed by a rule called Position, but this method of solving them by fractions is preferable. 77. A hare starts up 12 rods before a hunter, and scuds away at the rate of 10 miles an hour; now, if the hunter does not change his place, how far will the hare get from the hunter in 45 seconds? A. 52 rods. 78. If a dog, by running 16 miles in one hour, gain on a hare 6 miles every hour, how long will it take him to overtake her, provided she has 52 rods the start? A. 97) seconds. 79. A hare starts 12 rods before a greyhound, but is not perceived by him till she has been up 45 seconds; she scuds away at the rate of 10 miles an hour, and the dog after her at the rate of 16 miles an hour; what space will the dog run before he overtakes the hare? A. 138 rods, 3 yards, 2 feet. 80. A gentleman has an annuity of $2000 per annum; I wish to know how much he may spend daily, that, at the year's end, he may lay up 90 guineas, and give 20 cents per day to the poor of his own neighbourhood? A. $4,128. 81. What is the interest of $600 for 120 days? (12) For 2 days? (20) For 10 years, 10 mo. and 10 days? (391) For 5 years, 5 mo. and 5 mo. and 5 days? (19550) For 6 years, 6 mo., and 6 days? (23460) For 4 years, 4 mo. and 4 days? (15640) A. Total, $989,70. 82. What is the present worth of $3000, due 2 years hence, discounting at 6 per cent. per annum? . $2608,695+. A. 83. Suppose A owes B $1000, payable as follows; $200 in 4 mo., $400 in 8 mo., and the rest in 12 m and the rest in 12 mo.; what is the equated time for paying the whole? A. 8 months. 84. How many bricks, 8 inches long, 4 inches wide, and 24 inches thick, will it take to build a house 84 feet long, 40 feet wide, 20 feet high, and the walls to be 1 foot thick? The pupil will perceive that he must deduct the width of the wall, that is, 1 foot, from the length of each side, because the inner sides are 1 foot less in length than the outer sides. A. 105408 bricks. APPENDIX. ALLIGATION. ↑ LXXXII. Alligation is the method of mixing several simples of different qualities, so that the compound, or composition, may be of a mean or middle quality. When the quantities and prices of the several things or simples are given, to find the mean price or mixture compounded of them, the process is called ALLIGATION MEDIAL. 1. A farmer mixed together 2 bushels of rye, worth 50 cents a bushel, 4 bushels of corn, worth 60 cents a bushel, and 4 bushels of oats, worth 30 cents a bushel: what is a bushel of this mixture worth? In this example, it is plain, that, if the cost of the whole be divided by the whole number of bushels, the quotient will be the price of one bushel of the mixture. RULE. Divide the whole cost by the whole number of bushels, &c., the quotient will be the mean price or cost of the mixture. 2. A grocer mixed 10 cwt. of sugar at $10 per cwt., 4 cwt. at $4 per cwt., and 8 cwt. at $7 per cwt.: what is 1 cwt. of this mixture worth? and what is 5 cwt. worth? A. 1 cwt. is worth $8, and 5 cwt. is worth $40. 3. A composition was made of 5 los. of tea, at $1 per lb., 9 lbs. at $1,80 per lb., and 17 lbs. at $1 per lb.: what is a pound of it worth? A. $1,546-7+. 4. If 20 bushels of wheat, at $1,35 per bushel, be mixed with 15 bushels of rye, at 85 cents per bushel, what will a bushel of this mixture be worth? A. $1,135,7+. 5. If 4 lbs. of gold of 23 carats fine be melted with 2 lbs. 17 carats fine, what will be the fineness of this mixture? A. 21 carats. ALLIGATION ALTERNATE. ¶ LXXXIII. The process of finding the proportional quantity of each simple from having the mean price or rate, and the mean prices or rates of the several simples given, is called Alligation Alternate; consequently, it is the reverse of Alligation Medial, and may be proved by it. 1. A farmer has oars, worth 25 cents a bushel, which he wishes to mix with corn, worth 50 cents per bushel, so that the mixture may be worth 30 cents per bushel; what proportions or quantities of each must he take? In this example, it is plain, that, if the price of the corn had been 35 cents, that is, had it exceeded the price of the mixture, (30 cents,) just as much as it falls short, he must have taken equal quantities of each sort; but, since the difference between the price of the corn and the mixture price is 4 times as much as the difference between the price of the oats and the mixture price, consequently, 4 times as much oats as corn must be taken, that is, 4 to I, or 4 bushels of oats to 1 of corn. But since we determine this proportion by the dif ferences, hence these differences will represent the same proportion. These are 20 and 5, that is, 20 bushels of oats to 5 of corn, which are the quantities or proportions required. In determining these differences, it will be found convenient to write them down in the following manner: OPERATION. 30 { $,25 $,50 I 20 Ans. It will be recollected, that the difference be tween 50 and 30 is 20, that is, 20 bushels of oats, which must, of course, stand at the right of the 25, the price of the oats, or, in other words, opposite the price that is connected or linked with the 50; likewise the difference between 25 and 30= 5, that is, 5 bushels of corn, opposite the 50, (the price of the corn.) The answer, then, is 20 bushels of oats to 5 bushels of corn, or in that proportion. By this mode of operation, it will be perceived, that there is precisely as much gained by one quantity as there is lost by another, and, therefore, the gain or loss on the whole is equal. The same will be true of any two ingredients mixed together in the same way. In like manner the proportional quantities of any number of simples may be determined; for, if a less be linked with a greater than the mean price, there will be an equal balance of less and gain between every two, consequently an equal balance on the whole. It is obvious, that this principle of operation will allow a great variety of answers, for, having found one answer, we may find as many more as we please, by only multiplying or dividing each of the quantities found by 2, or 3, or 4, &c.; for, if 2 quantities of 2 simples make a balance of loss and gain, as it respects the mean price, so will also the double or treble, the , or part, or any other ratio of these quantities, and so on to any extent whatever. PROOF. We will now ascertain the correctness of the foregoing operation by the last rule, thus: 20 bushels of oats, at 25 cents per bushel, 5 25 corn, at 50 = $5,00 = $2,50 25)7,50(30 75 0 Ans. 30 cts., the price of the mixture. Hence we derive the following RULE. I. Reduce the several prices to the same denomination. II. Connect, by a line, each price that is less than the mean rate, with one or more that is greater, and each price greater than the mean rate with one or more that is less. III. Place the difference between the mean rate and that of each of the simples opposite the price with which they are connected. IV. Then, if only one difference stands against any price, it expresses the quantity of that price; but if there be several, their gum will exvress the quantity. |