two more, and then he had 22 left. he at first ? How many had Ans. 100. 9. If from 3 times a certain number we subtract 8, half the remainder will be equal to the number itself diminished by 2. What is the number? Ans. 4. 10. Ten years ago, a boy's age was t of his father's; now, it is 4 of it. What is the father's age now? Ans. 60 years. 11. The sixth part of a number added to its eighth part gives 56. What is the number? Ans. 192. 12. Two boys had, together, 35 marbles. One fourth of the number that the first had was equal to one third of the number that the second had. How many had each ? Ans. 20, and 15. 13. A man spent half of his money, and afterwards lost one third of what he had left, when he found that he had remaining $30. How much had he at first ? Ans. $90. 14. What number is that, from which if 5 be subtracted, one half the remainder is equal to 15? Ans. 35. 15. Divide $116 amongst three persons, so that the second shall have two thirds as much as the first, and the third shall have two fifths as much as the second. Ans. $60, $40, and $16. 16. A wheat field yielded 72 bushels, which was divided between landlord and tenant in such a way, that for every five bushels that the landlord received, the tenant got seven. How many bushels did the tenant receive ? Ans. 42. 17. The half of a number exceeds its third part by 8. What is the number? Ans. 48. 18. A sum of money is divided between A., B., and C., so that A. has $8; B. has as much as A., together with one fifth as much as C.; and C. has as much as A. and B. together. How much has C. ? Ans. $20. 19. There are 180 sheep in two flocks. If 20 are taken from the second and added to the first flock, the first flock will then contain twice as many as the second. How many sheep are there in each flock ? Ans. 100, and 80. 20. A post stands f in the mud, 4 in the water, and 10 feet in the air. What is its entire length ? Ans. 28 feet. 21. There are two numbers whose difference is 8, and the first is 5 times the second. What are the numbers ? Ans. 10, and 2. 22. A merchant gains 14 per cent. on his capital, when he finds that he has $8436. What was his capital ? Ans. $7400. 23. A. has 3 times as much money as B.; but if A. were to give to B. $100, B. would then have 3 times as much as A. How much have they each ? Ans. A. $150, and B. $50. 24. A laborer was engaged for 30 days, on condition that for every day he labored, he was to receive $2, and for every day he was idle, he was to forfeit $1. At the end of the time he received $21. How many days did he labor ? Ans. 17. 25. A. is twice as old as B., but 10 years ago he was three times as old. How old is B. now? Ans. 20 years. 26. Find that number which, being increased by 9, the sum divided by 2, the quotient diminished by 9, the result will be 20. Ans. 45. 27. Divide the number 37 into three parts, such that the first shall be 3 less than the second, and the second 5 greater than the third. Ans. 12, 15, and 10. 28. A man spends f of his income for board, f of the remainder for clothing, and has remaining $70. What is his income ? Ans. $630. 29. Divide 1000 into two parts, so that one of them shall be of the other. Ans. 375, and 625. 30. A person after spending 50 dollars more than half of his income, had remaining 125 dollars more than a third of it. How much was his income? Ans. $1050. 31. In a naval action f of a fleet was taken, f of it sunk, and 2 ships burnt; of the remainder were afterwards lost in a storm, when 24 ships were left. How many ships were there in the fleet ? Ans. 60. 32. A sum of 990 dollars was divided between A., B., and C.; B. received $ as much as A.; and C., á as much as A. and B. together. How many dollars did each receive ? Ans. A., 300; B., 240; and C., 450. 33. A courier A. starts 1165 of his own steps ahead of a courier B., and takes 5 steps whilst B. takes but 4; now if 3 steps of B. are equal to 4 of the courier A., how many steps must B. make to overtake A.? Ans. 13980. 34. The hands of a clock are together at 12 o'clock; when are they next together ? Ans. At 1 h. 541 m. 35. A grazier spent te of his money for horses, for oxen, and is of the remainder for sheep, when he had 980 dollars left. How many dollars had he originally ? Ans. 2400. 36. Divide the nnmber 240 into two parts, so that ny times the first shall equal 5 times the second. Ans. 100, and 140. 37. In a garrison of 2400 men, there are 3 times as many cavalry as artillery, and twice as many infantry as artillery and cavalry together. How many are there of each kind ? Ans. 200 artillery, 600 cavalry, and 1600 infantry. 38. Divide 21000 dollars between A., B., C., and D., so that A.'s part shall be of B.'s; Bi's part of Co's; and C.'s part 4 of D.'s. How many dollars will each receive ? Ans. A., 3200; B., 4800; C., 6000; D., 7000. 39. A capital was put out at 63 per cent for one year, when the capital and interest together amounted to 1917 dollars. How many dollars were there in the capital ? Ans. 1800. 40. A boatman rows with the tide 42 miles in 3 hours. In returning, the tide is but f as strong, and it takes 104 hours to row the same distance. At what rate per hour did the tide run in each case ? Ans. 6, and 4 miles. 41. A cistern can be filled by two cocks; the first would fill it in 70 minutes, and the second in 80 minutes. In how many minutes would they both fill it together? Ans. 373. II. EQUATIONS OF FIRST DEGREE, CONTAINING MORE THAN ONE UNKNOWN QUANTITY. Explanation. 83. If we have a single equation, containing two unknown quantities, as we may find the value of one of them in terms of the other, as follows: 14 – 3y (1) 2 . Now, if the value of y is unknown, that of 2 will also be unknown; hence, from this equation alone, the value of x cannot be determined. If now, we have a second equation, 3x + 2y = 11, we may, in like manner, find the value of a in terms of y, 11 - 2y (2) 3 X = If the values of x and y are the same in equations (1) and (2), we shall have their second members equal to each other, giving the equation, |