x 2 Then and + x the cost of the first purchase, the cost of the second purchase. +10= the whole cost and profits. As he sold 5 bushels for $3, he sold each bushel for $3; and, of course, he received for 2 x bushels, that is, for a bushels of each sort, $*. 6 x 5 Therefore, * =+ + 10. 57. Two clerks, A and B, have the same income. A saves of his; but B, by spending $80 a year more than A, at the end of 4 years finds himself $220 in debt. What was their income? 58. After spending of my money, and of the remainder, I had $96 left. How much had I at first? 59. A traveller spent of his money in Boston; } of the remainder in Providence; of what was left in New York; of the balance in Philadelphia, and had $80 left. How much had he at first? 60. Divide 26 into three such parts, that, if the first be multiplied by 2, the second by 3, and the third by 4, the products shall all be equal. 61. Divide 56 into two such parts, that, the larger being divided by 7, and the smaller by 3, the sum of their quotients may be 10. 62. A cistern has three cocks; the first will fill it in 5 hours, the second in 10 hours, and the third will empty it in 8 hours. In what time will the cistern be filled, if all the cocks are running together? 63. A school-boy, being asked his age, replied, that of his age multiplied by1⁄2 of his age, would give a product equal to his age. How old was he? 64. A person has a lease for 99 years; and, being asked how much of it had expired, he replied, that of the time past was equal to of the time to come. How many years had the lease run ? 65. What number is there which may be divided into either two or three equal parts, and the continued product of those parts shall be equal? 66. A shepherd, driving a flock of sheep in time of war, meets with a company of soldiers, who plunder him of half his flock and half a sheep over; and a second, third and fourth company treat him in the same manner, each taking half the flock left by the last and half a sheep over, when but 8 sheep remained. How many sheep had he at first? 67. A gentleman has two horses, and a chaise worth $150. Now, if the first horse be harnessed, the horse and chaise together will be worth twice as much as the second horse; but if the second horse be harnessed, they will be worth three times as much as the first horse. What is the value of each horse? *68. Divide 54 into two such parts, that, if the greater be divided by 9, and the less by 6, the sum of the quotients shall be 7. 69. A farmer sells a quantity of corn, which is tc the quantity left as 4 to 5. After using 15 bushels, he finds he has as much left as he sold. How many bushels had he at first? 70. Divide 84 into two such numbers, that the quo tient of the greater, divided by their difference, may be four. *71. A laborer agreed to work for a gentleman a year, for $72 and a suit of clothes; but at the end of 7 months, he was dismissed, having received his clothes and $32. What was the value of the clothes? 72. A laborer reaps 35 acres of wheat and rye For every acre of rye he receives 5 shillings; and what he receives for an acre of wheat, if it were 1 shilling more, would be to what he receives for an acre of rye as 7 to 3. For the whole he receives £13. How many acres are there of each sort? *73. A man and his wife consumed a sack of meal in 15 days. After living together 6 days, the woman alone consumed the remainder in 30 days. would the sack last either of them alone? How long 74. A company of men, women and children consists of 75 persons. The number of the men exceeds that of the women by 5, and there are 13 more children than adults. Required the number of men women and children. 75. Three adventurers, A, B and C, bought 10170 acres of wild land. By the terms of te contract, B had 549 acres less than A, and C had 987 acres more than B. How many acres had each? 76. A man, being asked how much money he had, replied, "If you multiply my money by 4, add 60 to the product, divide the sum thus obtained by 3, and then subtract 45 from the quotient, the remainder will be the number of dollars I have.” How much money had he? SECTION VI. Two Unknown Quantities. ELIMINATION BY COMPARISON. 1. A fruiterer sold to one person 6 lemons and 3 oranges for 42 cents; and to another, 3 lemons and 8 oranges for 60 cents. What was the price of each? This question can be solved, without difficulty, by means of only one unknown quantity: the solution will be more simple, however, if two unknown quantities are used. When the conditions of a question are such as require two or more unknown quantities, our first object is to obtain an equation involving but one unknown quantity. This process is called Elimina- ◄ tion. There are several methods of elimination, with which the learner should become familiar, as he may often find it convenient to use them all in the same operation. Let the price of a lemon, and y the price of an orange. A. Then 6x+3y= 42, by the first sale. D Again, 3 x + 8 y = 60, by the second sale E. 3 x 60-8 y, by transposition.. Now, as things, which are equal to the same thing, arc equal to each other, we can form a new equation from the value of r, as determined in equations & which will contain but one unknown quantity and F, 9 y = 360 48 y, by multiplication. 1. 48 y 9 y 360 126, by transposition. J. 39 y 234, by uniting terms. K. y 6, the price of an orange. By substituting this value of y in equation c, we or x = 24 = 4, the price of a lemon. When a question involves two unknown quantities, its conditions must admit of two equations, or it cannot be solved. By this method of elimination, we find the value of one of the unknown quantities in each of the equations, and make the expressions of its value, thus found, equal to each other. An equation is thus obtained, involving but one unknown quantity. *2. A gentleman has two silver cups, and a cover adapted to each, which is worth £10. If the cover be put upon the first cup, its value will be twice that of the second; but if it be put upon the second, its value will be three times that of the first. What is the value of each cup? Let and y the value of the first cup, the value of the second cup. Then 2 y = x + 10, by the question. |