mixes 74 ounces of the former with 14 ounces of the latter, he obtains a mixture 10 carats fine. What was the fineness of each mass? Ans. The one 9 carats, the other 15 carats. Prob. 6. A farmer has a certain number of oxen, and proven- . der for a certain number of days. If he sells 75 oxen, his provender will last 20 days longer; but if he buys 100 more oxen, his provender will be exhausted 15 days sooner. How many oxen has he, and how many days will the provender last? Ans. 300 oxen, and the provender will last 60 days. Prob. 7. A certain number of laborers remove a pile of stones in 6 hours from one place to another. If there had been 2 more laborers, and if each laborer had each time carried 4 pounds more, the pile would have been removed in 5 hours; but if there had been 3 less laborers, and if each laborer had each time carried 5 pounds less, it would have required 8 hours to remove the pile. How many laborers were there, and how much did each carry at one time? Ans. There were 18 laborers, and each carried 50 pounds. Prob. 8. A heavy wagon requires a certain time to travel from A to B. A second wagon, which every 4 hours travels 5 miles less than the first, requires 4 hours more than the first to go from A to B. A third wagon, which every 3 hours travels 8f miles more than the second, requires 7 hours less than the second to make the same journey. How far is A from B, and what time does each wagon require to travel this distance? Ans. From A to B is 60 miles; the first wagon requires 12 hours, the second 16, and the third 9 hours. Prob. 9. I have two equal sums to pay, one after 9, and the other after 15 months. If I settle them both at once, at the same rate of discount, I must pay for the first sum $1208, and for the second $1160. How much was each sum, and at what per cent. was the discount reckoned ? Ans. $1280, and the discount was 74 per cent. Prob. 10. A small square lies with one angle in the angle of a larger square. The excess of the side of the larger square above that of the smaller is 118 feet; the excess of the square itself is 26,432 square feet. What are the contents of each of the two squares ? Ans. The one 29,241, the other 2809 square feet. Prob. 11. It is required to find two numbers whose sum, dif- . ference, and product are in the ratio of the numbers 5, 1, and 18. Ans. 9 and 6. Prob. 12. Two numbers are in the ratio of 7 to 3, and their difference is to their product as 1 to 21. What are the numbers? Ans. 28 and 12. Prob. 13. Three towns, A, B, and C, lie at the angles of a triangle. From A by B to C is 164 miles; from B by C to Ą is 194 miles; and from C by A to B is 178 miles. How far are A, B, and from each other? Ans. From A to B 74 miles, from B to C 90, and from C to A 104 miles. Prob. 14. A railway train, after traveling for one hour, meets with an accident which delays it one hour, after which it proceeds at three fifths of its former rate, and arrives at the terminus three hours behind time; had the accident occurred 50 miles further on, the train would have arrived 1 hour and 20 minutes sooner. Required the length of the line. Ans. 100 miles; original rate 25 miles per hour. Prob. 15. A railway train, running from New York to Albany, meets on the way with an accident, which causes it to 1. diminish its speed to th of what it was before, and it is in consequence a hours late. If the accident had happened b miles nearer Albany, the train would have been c hours late. Find the rate of the train before the accident occurred. Ans. miles с per hour. a-C Prob. 16. Three boys are playing with marbles. Said A to B, Give me 5 marbles, and I shall have twice as many as you will have left. Said B to C, Give me 13 marbles, and I shall have three times as many as you will have left. Said C to A, Give me 3 marbles, and I shall have six times as many as you will have left. How many marbles had each boy ? Ans. A had 7, B 11, and C 21 marbles 10 cents per Prob. 17. It is required to divide the number 232 into three parts such that, if to the first we add half the sum of the other two, to the second we add one third the sum of the other two, and to the third we add one fourth the sum of the other two, the three results thus obtained shall be equal, What are the parts? Ans. The first 40, the second 88, and the third 104. Prob. 18. Four towns, A, B, C, and D, are situated at the angles of a quadrilateral figure. When I travel from A by B and C to D, I pay $6.10 passage-money; when I travel from A by D and C to B, I pay $5.50. From A by B to C, I pay Ι the same as from A by D to C; but from B by A to D, I pay 40 cents less than from B by C to D. What are the distances of the four towns from each other, supposing I paid in each case mile? Ans. From A to B 21, from B to C 17, from C to D 23, and from D to A 15 miles. Prob. 19. Four players, A, B, C, and D, play four games at cards. At the first game A, B, and C win, and each of them doubles his money; at the second game A, B, and D win, each of them doubling the money he had at the commencement of that game; at the third game A, C, and D win; and at the fourth game B, C, and D win; and at each game each winner won as much money as he had at the commencement of that game. They now count their money, and find that each has $64. How much had each before commencing play? Ans. A had $20, B had $36, C had $68, and D had $132. Prob. 20. A and B start together from the foot of a mountain to go to the summit. A would reach the summit half an hour before B, but, missing his way, goes a mile and back again need. lessly, during which he walks at twice his form reaches the top six minutes before B. C starts twenty minutes after A and B, and, walking at the rate of two and one seventh miles per hour, arrives at the summit ten minutes after B. Find the rates of walking of A and B, and the distance from the foot to the summit of the mountain. Ans. 21, 2; distance 5 miles. pace, and : Prob. 21. Find three numbers such that if six be subtracted from the first and second, the remainders will be in the ratio of 2:3; if thirty be added to the first and third, the sums will be in the ratio of 3:4; but if ten be subtracted from the second and third, the remainders will be as 4:5. Ans. 30, 42, 50. Prob. 22. A and B engage to reap a field of wheat in twelve days. The times in which they could severally reap an acre are as 2:3. After some days, finding themselves unable to . finish it in the stipulated time, they call in C to help them, whose rate of working was such that, if he had wrought with them from the beginning, it would have been finished in nine days. Also, the times in which he could have reaped the field with A alone, and with B alone, are in the ratio of 7:8. When was C called in? Ans. After six days. x-2-3 + EQUATIONS OF THE SECOND DEGREE WITH ONE UNKNOWN QUANTITY. *c+18, X–18 5 Ex. 1. Given + to find the values of x. 2+2 Ans. x=+14. 5 5 Ex. 2. Given +49-V -49=7, to find the values x2 ot' x. Ans. x= #. 5 2 Ex. 3. Given + to find the values of x. Ans. x=+v10. aéta Ex. 4. Given x+vat x2 = to find the values of x. 2 Va+ana! Ans. x=+1a-1). : 3m 3m2 + m2—3=m+1-V 2, to find the values of x. Ans. x==m. 560 560 Ex. 6. Given Ex. 6. Given V m 2c2 -+29- . .22 34=7, to find the val. x2 Ans. x=+4. ues of x. 1-V 1 1 V3 Ex. 7. Given to find the val. 1-V1-22 1+11-22 ues of x. Ans. x= +1. Ex. 8. Given 27(7 — «2—43=77–3(7—2), to find the val. ues of x. Remark. Put 7-x=y; first find the value of y, and thence the value of x. Ans. x=5 or 9. a-Vaz_ @c? Ex. 9. Given =b, to find the values of x. a+Vaz-22 2avī Ans. x=+ 1+6° Vã+Vx-a al Ex. 10. Given to find the values of x. 3-VZa X-a' a( 16) . 1826 Ans. x= Ex. 11. Given Va+z, va . + to find the values of x, Va Va Ans. x=32 Vab—62. V1+3 71Ex. 12. Given to find the values of x. 1+11+1-11 Ans. x=+1V3. B.-COMPLETE EQUATIONS OF THE SECOND DEGREE. Ex. 13. Given 557x=58011+882, to find the values of x. Ans. x=567 or 124. Ex. 14. Given (7x)2—7x=1, to find the values of x. Ans.x=0.2311477 or -0.0882905. Ex. 15. Given 12x2=21+52, to find the values of x. Ans. x=1} or -116 Ex. 16. Given 57—1832 +145=0, to find the values of X. Ans. x=4 or -13. 5x 4 (2 21 35 the values of x. Ans. =-1 or Ex. 17. Given 3.0 (x+1)=E(22®+x—1)=f(x+1), to find |