8 = ac X= 700 3.30 69. Given 124 + 3x—6– +53, to find a. 3 Ans. x= =103 3.3 73 3.3 7.3 70. Given + =-15, to find x. 5 10 Ans. x=663 71. Given + + =7x—734+, to find x. Ans. x=120. 9x+7 6x+2 72. Given =36, to find x. Ans. x=9. 2 7 5x-7 3x-2 73. Given to find X. Ans. x=67. 3 7 5x+7 74. Given + =19 to find x. 3 Ans. x=7. de 75. Given ax+c=bx+d, to find x. Ans. x= -6° 76. Given ax + =a'+bx, to find x. Ans. x=a+b. ac 77. Given (a +-x)(b +x)—a(b + c)= 6 : dx ac(1+3ab) 78. If -14 =3ab. c-ad ab ab-1 79. If =bc+d+ X = bc+d' ma—2) 80. If at a(m+3a—3c) 3a+x mtc-a adfn+befn+bden+bdfm 81. If + + + -=p. x= bx dx bdfup x=16 82. If(3x+2y=118 *+5y=191 ) y=35. x=144 83. If Skx+y=42 5x+3y=43 ) y=216. x= 6 84. If x+z=38 y=14 y+z=46) -32. (y+tx=41 85. If <+Iz=202 ly+z=34 5y7-288 x=13 86. If 5.0 — y+37= 227 y=24 7x+by+ = %=62. - C. a с e m y=20 x=18 y=32 z=10. 86. IFS 2x+59–7=_288 297 PROBLEMS PRODUCING SIMPLE EQUATIONS. a ma na na 87. How much money have I in my pocket, when the fourth and fifth of the same together amount to lls. 3d. Ans. L.1, 5s. 88. In a company of 266 persons, consisting of officers, merchants, and students, there were four times as many merchants, and twice as many officers, as students. How many were of each class ? Ans. 38 students, 152 merchants, and 76 officers. 89. Divide the number a into thrée such parts, that the second may be m times, and the third n times as great as the first. Ans. 1+m+n' 1+m+n' i+m+ni 90. A father gives to his five sons L.1000, which is to be divided among them according to their ages, so that each of the four elder may receive L.20 more than his next younger brother. How much will the youngest receive? Ans. L.160. 91. A courier who goes a miles a-day, started n days before another, who goes b miles a-day. In what time will the second overtake the first? Ans. days. 92. A person paid the sum of L.2, 13s. in crowns and shillings, using 17 pieces in all. How many of each sort was used ? Ans. 9 crowns and 8 shillings. 93. One of my acquaintances is now 40 years old, and his son 9 years. In how many years will this man, who is now more than four times as old as his son, be only twice as old ? Ans. In 22 years. 94. A transcriber was asked how many sheets he wrote weekly: he answered, “I only work four hours a day, and cannot finish 70 sheets, which I could wish to do; but if I could work ten hours a-day, then I should write weekly exactly as many above 70 sheets as I now write less than that number.” How many sheets did he write weekly? Ans. 40. 95. Find two numbers, such that if the first be multiplied by 2, and the second by 5, the sum of the products will be 31; and if the first be multiplied by 7, and the other by 4, the sum of the products will be 68. Ans. First 8, and second 3. 96. It is required to find a fraction, such that if 3 be subtracted from the numerator and denominator, the value of the remaining fraction will be 1, and if 5 be added to M a ab 6+1° each of its terms, its value will be t. What is the fraction? Ans. 1 97. The sum of two numbers is =a, the difference of their squares =b. What are the numbers ? a +6 al Ans. and 2a 2a 98. The sum of two numbers is =a, and their quotient is =6. What are the numbers? Ans. and 6+1' 99. A, B, and C, owe L.2190 amongst them, and no one of them can pay this sum alone. But when they unite, it can be done in the following ways: by B putting of his property to all A's; or by C putting & of his property to that of B; or by A putting of his property to that of C. How much did each possess ? Ans. A L.1530, B L.1540, C L.1170 100. Required to find three numbers, which possess the following properties:-If 6 be added to the first and second, the sums are to one another as 2 to 3; if 5 be added to the first and third, then the sums are to one another as 7 to 1l; but if 36 be subtracted from the second and third, the remainders are to one another as 6 to 7. What are the numbers ? Ans. 30, 48, and 50. 24 X = amb or ab 106. If x+1=30—4. x=5, or -2. a+b 107. If a*(1+B?x)=3(2a*x+b). ab 12a 4a 108. If Na+x+ wamx= 57a+ 26 109. If (a−b)x2-(a+b)x+26=0. x=1, or 12 110. If -4 x=+5, or +4,2 11l. If _72=8. x=2, or -]. 3a or 5 a 4 x=+2, or + 114. If XY-= y=+6, or + 115. If _2xy-y= 31 x=+10. ava—ya x=+2. 116. If ) x+y x+y = y=+1. PROBLEMS PRODUCING QUADRATIC EQUATIONS. 117. What number is it, whose half multiplied by its third part, gives 864? Ans. 72. 118. It is required to find a number, such that if we first add it to 94, then subtract it from 94, and afterwards multiply this remainder by the former sum, the product may be 8512. What number is it? Ans. 18. 119. What two numbers are those whose product is 144, and whose quotient is 9 ? Ans. 36 and 4. 120. What two numbers are those whose product is =a, and whose quotient is =b? Ans. Vab and V 121. A gentleman left L.210 to three servants, to be divided in geometrical progression, so that the first shall have L.90 more than the last. Find their legacies. Ans. L.120, L.60, L.30. 122. A certain capital yields 4 per cent.; if we multiply the number of pounds in the capital by the number of pounds in the interest for five months, we obtain 1170413. What is the capital? Ans. 2650. 123. There are two numbers, one of which is greater than the other by 8, and whose product is 240. What numbers are they? Ans. 12 and 20. 124. It is required to find a number, whose square exceeds its simple power by 306 ? Ans. 18. 125. It is required to find a number, such that if we multiply its third part by its fourth part, and to the product add five times the number required, this sum exceeds the number 200 by as many as the number sought is less than 280 ? Ans. 48. 126. A person buys some pieces of cloth, at equal prices, for L.60. Had he got 3 more pieces for the same sum, each piece would have cost him a pound less. How many pieces did he buy? Ans. 12. 127. A person dies, leaving children, and a fortune of L.46,800, which, by the will, is to be divided equally among them. It happens, however, that immediately after the death of the father, two of his children also die. If, consequently, each child receives L.1950 more than he or she was formerly entitled to by the will, how many chil. dren were there? Ans. Eight 128. Two retailers jointly invest L.500 in business, to which each contributes a certain sum; the one let his money remain five months, the other only two, and each received L.450 capital and profit. How much did each advance ? Ans. One L.200, the other L.300. 129. What two numbers are those whose sum is 41, and the sum of whose squares 901 ? Ans. 26 and 15. 130. A capital of L.5000 stands at 4 per cent. compound interest. What will it amount to in forty years ? Ans. L.24005, 2s. 14d. 13]. How long must L.3600 remain at 5 per cent. compound interest, so that it may become as much as L.5000, at 4 per cent. for twelve years? Ans. 16 years, 136 days. 132. What capital, at 4 per cent., will fifteen years hence be equal in value to L.4500, at 6 per cent., for nine years ? Ans. L.4221.483. 133. A town contains 20,000 inbabitants, and we know that the population has regularly increased in yearly. What was its population ten years ago ? Ans. 14,882. 134. In how many years will the population of a place become ten times as great as it is at present, if the yearly increase amount to three persons in a hundred ? Ans. 78 years nearly. 135. What is the present value of an annuity of L.20, to continue for forty years, reckoning interest at the rate of 6 per cent. Ans. L.300, 18s. 6d. 136. What annuity, improved at the rate of 4 per cent. per annum, compound interest, will at the end of twelve years amount to L.500, 178. 2d.? Ans. L.33, 6s. 8d. |