The Elements of Algebra

Prob. 55. A detachment of soldiers from a regiment being ordered to march on a particular service, each company furnished twice as many men as there were companies in the regiment; but these being found insufficient, each company furnished five more men, when their number was found to be increased in the ratio of 5 to 4. How many companies were there in the regiment? Ans. 10.

Prob. 56. A certain sum of money, being put out at interest for 4 months, amounts to $1275. The same sum, put out at the same rate for 20 months, amounts to $1375. Required the sum and the rate per cent. Ans. $1250, at 6 per cent.

Prob. 57. Divide the number 56 into two such parts that the quotient of the greater divided by the less may be to the quotient of the less divided by the greater as to 3. Ans. 35 and 21. Prob. 58. There is a number consisting of two digits, the second of which is greater than the first; and if the number be divided by the sum of its digits, the quotient will be 4; but if the digits be inverted, and that number be divided by a number greater by 4 than the difference of the digits, the quotient will be 9. Required the number.

Ans. 36.

Prob. 59. There is a fraction whose numerator being doubled, and the denominator diminished by 2, the value becomes; but if the denominator be tripled, and the numerator increased by 3, its value becomes

Prob. 60. A father divided his estate among his five children, giving the first four $26,000; the last four,

$30,000; the last three with the first, $29,000; the first three with the last, $27,000; and the first two with the last two, $28,000. What was the sh of each?

Ans. The first, $5000; the second, $6000; the third, $7000; the fourth, $8000; and the fifth, $9000.

Prob. 61. There is an island 15 miles in circumference, and four men start together to travel the same way about it. A goes 4 miles per hour, B 5 miles, C 6 miles, and D 7 miles. How far will each travel before they all meet again at the spot from which they started?

Ans. A goes 60 miles; B 75 miles; C 90 miles; and D 105 miles.

Prob. 62. A certain number, consisting of twc places of figures, is equal to the difference of the squares of its digits; and if 36 be added to it, the digits will be inverted. What is the number?

Ans. 48.

Prob. 63. A general ranging his army in the form of a solid square, finds he has 119 men to spare; but increasing each side by one man, he wants 100 men to fill up the square. How many soldiers had he?

Ans. 12,000.

Prob. 64. A square court-yard has a rectangular gravel-walk round it. The side of the court is 2 yards more than 6 times the breadth of the gravel-walk, and the number of square yards in the walk exceed the number of yards in the periphery of the court by 196. Required the breadth of the gravel-walk.

Ans. 3 yards.

Prob. 65. A and B set out from two towns, which

were distant 204 miles, and traveled the direct road till they met. A went 8 miles per hour, and the number of hours at the end of which they met was greater by 3 than the number of miles which B went in an hour. How many miles did each go?

Ans. A 96 miles, and B 108 miles.

Prob. 66. A farmer bought a certain number of sheep for $60; and if he had bought 10 more for the same money, they would have cost him one dollar a piece less. What was the number of sheep?

Ans. 20.

Prob. 67. A merchant sold a quantity of lace for $56, and gained as much per cent. as the lace cost him. What was the cost of the lace?

Ans. $40.

Prob. 68. A merchant sold a quantity of goods for a dollars, and gained as much per cent. as the goods cost him. What was the cost of the goods?

Ans. √2500+100a-50.

Prob. 69. Two merchants, A and B, enter into partnership with a joint stock of $12,000. A's capital was employed 5 months, and B's 8 months. When the stock and gain were divided, A received $7150. and B $6120. What was each man's stock?

Ans. A's was $6600, and B's $5400.

Prob. 70. A merchant bought a number of pieces of cloth for $225, which he sold again at $16 by the piece, and gained by the bargain as much as one piece cost him. What was the number of pieces?

Ans. 15.

Prob. 71. A merchant bought a number of pieces of

cloth for a dollars, which he sold again at m dollars by the piece, and gained by the bargain as much as one piece cost him. What was the number of pieces?

Prob. 72. A grocer sold 16 pounds of mace and 20 pounds of cassia for $24; but he sold 10 pounds more of cassia for $6 than he did of mace for $5. What was the price of a pound of each?

Ans. Mace one dollar, and cassia forty cents. Prob. 73. Two gentlemen, A and B, hired a pasture, into which A put two horses, and B as many as cost him $4 a week. Afterward B put in two additional horses, and found that he must pay $4.50 a week. At what rate was the pasture hired?

Ans. $6 a week. Prob. 74. What number is that to which, if 1, 4, and 10 be severally added, the first sum shall be to the second as the second to the third?

Ans. 2.

Prob. 75. It is required to find three numbers such that the product of the first and second may be 24, the product of the first and third 32, and the sum of the squares of the second and third 100.

Ans. 4, 6, and 8. Prob. 76. What are the values of x in the equation 3x2-3x+6=51? Ans. x= or 1.

Prob. 77. What are the values of x in the equation x2-x+3=45? Ans. x=7 or -6.

Prob. 78. What are the values of x in the equation

5x2-4x+3=159?

Ans. x=6 or

Prob. 79. What are the values of x in the equation