proportion must he mix 40 bushels so that he may sell the mixture at 10 shillings per bushel? Ans. 30 bushels at 9 shillings, and 10 at 13 shillings. 21. What number is that, the treble of which, increased by 12, shall as much exceed 54 as that treble is less than 1443 Ans. 31. 3x412-54=144-5 22. A asked B how much money he had. He replied, if I had 5 times the sum I now possess, I could lend you $60, and then of the remainder would be equal to the dollars I now have. Required the sum which B had. 5x-60x 5 Ans. $24. 23. A, B, and C found a purse of money, and it was mutually agreed that A should receive $ 15 less than one half, that B should have $13 more than one quarter, and that C should have the remainder, which was $27. How many dollars did the purse contain? Ans. $100. 24. Two persons, A and B, 120 miles apart, set out at the same time to meet each other. A goes 3 miles an hour, and B 5 miles. What distance will each have traveled when they meet? Ans. A, 45 miles; B, 75 miles. 25. The first digit of a certain number exceeds the second by 4, and when the number is divided by the sum of the digits, the quotient is 7. What is the number? Ans. 84. 26. A person bought a certain number of acres of land for $180. After reserving two of them, he sold the remainder for $180. Now he found that he had gained on the cost price of each acre one third more per cent. than that cost price. How many acres did he buy? Ans. 12. 27. A prize of $1000 is to be divided between A and B, so that their shares may be in the ratio of 7 to 8. Required the share of each. Ans. A's share, $4663, and B's, $533. 28. A gentleman let a certain sum of money for 3 years, at 5 per cent. compound interest; that is, at the end of each year there was added to the sum due. At the close of the third year there was due him $2315.25. Required the sum let. Ans. $2000. 29. Bought a picture at a certain price, and paid the same price for a frame; if the frame had cost $1 less, and the picture $0.75 more, the price of the frame would have been only half that of the picture. Required the cost of the picture. Ans. $2.75. 30. A and B can do a piece of work together in 7 days, which A alone could do in 10 days. In what time could B alone do it? Ans. 23 days. 31. A gentleman gave in charity $46; a part in equal portions to 5 poor men, and the rest in equal portions to 7 poor women. Now, a man and a woman had between them $ 8. What was given to the men, and what to the women? Ans. The men received $25, and the women $21. 32. A man has two farms, and his stock is worth $183. Now, the stock and his first farm are worth once and two sevenths the value of the second farm; and the stock and the second farm are worth once and five eighths the value of the first farm. What is the value of each farm? Ans. First farm, $384; second farm, $ 441. 33. At 6 o'clock the hands of a watch are opposite the one to the other. When will they next be in like position? Ans. At 5 minutes after 7 o'clock. 31. A vessel can be emptied by three taps; by the first alone it could be emptied in 80 minutes, by the second in 200 minutes, and by the third in 5 hours. In what time will it be emptied if all the taps be opened? Ans. 48 minutes. 35. A fox is pursued by a greyhound, and is 60 of her own leaps before him. The fox makes 9 leaps while the 760 greyhound makes but 6; but the latter in 3 leaps goes as far as the former in 7. How many leaps does each make before the greyhound catches the fox? Ans. The greyhound, 72 leaps; the fox, 108. 36. It is found that 7 men and 3 boys pulling together produce just the same effect as 9 men pulling against 9 boys. What is the relative strength of one of the men to one of the boys? Ans. 6 to 1. 37. It is required to divide the number 43 into two such parts that one of them shall be 3 times as much above 20 as the other wants of 17. bers. 43-x-30-3 (1-x) Required the num- 38. Two horses ran over a mile course; the winner completed the distance in 2 minutes 54 seconds, winning by 2 seconds. How many yards start might have been allowed to the other without risk of losing, supposing the same rates be kept? Ans. 20. 39. Gold is 19 times as heavy as water, and silver 10 times. A mixed mass weighs 4160 ounces, and displaces 250 ounces of water. What proportions of gold and silver does it contain ? Ans. Gold, 3377 ounces; silver, 783 ounces. 40. A alone could perform a piece of work in 12 hours; A and C together could do it in 5 hours; and C's work is of B's. Now, the work has to be completed by noon. A begins work at 5 o'clock in the morning; at what hour can he be relieved by B and C, and the work be just finished in time? Ans. At 10 o'clock. 41. A merchant possesses $5120, but at the beginning of each year he sets aside a fixed sum for family expenses. His business increases his capital employed therein annually at the rate of 25 per cent. At the end of four years he finds that his capital is reduced to $ 3275. What are his annual expenses? Ans. $1280. 161. SIMPLE EQUATIONS CONTAINING TWO UNKNOWN QUANTITIES. INDEPENDENT EQUATIONS are such as cannot be made to assume the same form. If they relate to the same problem, they must, therefore, express essentially different conditions of that prob lem. 162. When the conditions of a problem require the unknown quantities to be denoted by different letters, in order to determine their values, as many independent equations are necessary as there are unknown quantities. For, if we have an equation containing two unknown quantities, and y, as x y = 1, transposing y, we have x = y +1. (1) But the value of y is not known; consequently, from this equation alone the value of x cannot be determined. If, however, we have a second equation, as or, x + y = 7, = x = 7 y, (2) in which the value of x and y are the same as in the first, the second members of (1) and (2) being equal to the same quantity, x, and consequently equal to each other (Art. 46, Ax. 7), give Substituting 3, the value of y, for y in either equation (1) or equation (2), we obtain 4 as the value of x; and the values obtained for the two unknown quantities satisfy the two equations. 163. SIMULTANEOUS EQUATIONS are those in which the unknown quantities are satisfied by the same values. Two unknown quantities, denoted by different letters, require for their determination, as has been shown (Art. 162), at least two independent, simultaneous equations. ELIMINATION. 164. ELIMINATION is the process of combining simultaneous equations, in such a manner as to cause one or more of their unknown quantities to disappear. There are four principal methods of elimination ; — by Comparison, by Substitution, by Addition or Subtraction, and by Undetermined Multipliers. CASE I. 165. Elimination by comparison. 1. Given 5x -3y=9, and 2x+5y=16, to find 9+3y From (1), by transposition and division, x = (3) 2 9+3y 16 5y Combining (3) and (4), by Ax. 7, (4) (5) |