8. Prove that 175 +8 49 14 +190 - 54 + 14 > 144 + 13 9. Prove that 216 44 10. Place the proper sign two expressions, (247 +104) and (546 =, >, or <) between these 11. Place the proper sign (=, >, or <) between these two expressions, (11947 +16) and (317104). 13. All operations in Algebra are based upon certain self-evident truths called AXIOMS, of which the following are the most common: 1. If equals are added to equals the sums are equal. 2. If equals are subtracted from equals the remainders are equal. 3. If equals are multiplied by equals the products are equal. 4. If equals are divided by equals the quotients are equal. 5. Like powers and like roots of equals are equal. 6. The whole of a quantity is greater than any of its parts. 7. The whole of a quantity is equal to the sum of all its parts. 8. Quantities respectively equal to the same quantity are equal to each other. SECTION II. ALGEBRAIC OPERATIONS. 14. A THEOREM is something to be proved. 15. A PROBLEM is something to be done. 16. The Solution of a Problem in Algebra consists, 1st. In reducing the statement to the form of an equation; 2d. In reducing the equation so as to find the value of the unknown quantities. EXAMPLES FOR PRACTICE. 1. The sum of the ages of a father and his son is 60 years, and the age of the father is double that of the son; what is the age of each? It is evident that if we knew the age of the son, by doubling it we should know the age of the father. Suppose we let x equal the age of the son; then 2x equals the age of the father; and then, by the conditions of the problem, x, the son's age, plus 2x, the father's age, equals 60 years; or 3x equals 60, and (Axiom 4) x, the son's age, is of 60, or 20, and 2x, the father's age, is 40 Expressed algebraically, the. process is as follows: xson's age, : Let then 2x father's age. 2. A horse and carriage are together worth $450; but the horse is worth twice as much as the carriage; what is each worth? Ans. Carriage, $150; horse, $300. All problems should be verified to see if the answers obtained fulfil the given conditions. In each of the preceding problems there are two conditions, or statements. For example, in Prob. 2 it is stated (1st) that the horse and carriage are together worth $450, and (2d) that the horse is worth twice as much as the carriage; both these statements are fulfilled by the numbers 150 and 300. 3. The sum of two numbers is 72, and the greater is seven times the less; what are the numbers? 4. A drover being asked how many sheep he had, said that if he had ten times as many more, he should have 440; how many had he? 5. A father and son have property of the value of $8015, and the father's share is four times the son's; what is the share of each? Ans. Father's, $6412; son's, $1603. 6. A farmer has a horse, a cow, and a sheep; the horse is worth twice as much as the cow, and the cow twice as much as the sheep, and all together are worth $490; how much is each worth? 7. A man has three horses which are together worth $540, and their values are as the numbers 1, 2, and 3; what are the respective values? Let x, 2x, and 3x represent the respective values. Ans. $90, $180, and $270. 8. A man has three pastures, containing 360 sheep, and the numbers in each are as the numbers 1, 3, and 5; how many are there in each? 9. Divide 63 into three parts, in the proportion of 2, 3, and 4. Let 2x, 3x, and 4x represent the parts. 10. A man sold an equal number of oxen, cows, and sheep for $1500; for an ox he received twice as much as for a cow, and for a cow eight times as much as for a sheep, and for each sheep $6; how many of each did he sell, and what did he receive for all the oxen? Ans. 10 of each, and for the oxen, $960. 11. Three orchards bore 872 bushels of apples; the first bore three times as many as the second, and the third bore as many as the other two; how many bushels did each bear? 12. A boy spent $4 in oranges, pears, and apples; he bought twice as many pears and five times as many apples as oranges; he paid 4 cents for each pear, 3 for each orange, and 1 for each apple; how many of each did he buy, and how much did he spend for oranges? how much for pears, and how much for apples? Ans.{ 25 oranges, 50 pears, and 125 apples. Spent for oranges, $0.75; pears, $2; apples, $1.25. 13. A farmer hired a man and two boys to do a piece of work; to the man he paid $12, to to the other $4 per week; they all time, and received $264; how many work? one boy $6, and worked the same weeks did they Ans. 12 weeks. 14. Three men, A, B, and C, agreed to build a piece of wall for $99; A could build 7 rods, and B 6, while C could build 5; how much should each receive? 15. Four boys, A, B, C, and D, in counting their money, found they had together $1.98, and that B had twice as much as A, C as much as A and B, and D as much as B and C; how much had each? Ans. A 18 cents, B 36, C 54, and D 90. 16. It is required to divide a quantity, represented by a, into two parts, one of which is double the other. 17. If in the preceding example a 24, what are the required parts? 2 a 48 = 16. 18. It is required to divide c into three parts so that the first shall be one half of the second and one fifth of the third. с Ans. 2 c 8 8 5 c and 8 19. Divide n into three parts, so that the first part shall be one third the second and one seventh of the third. 20. A is one half as old as B, and B is one third as old as C, and the sum of their ages is p; what is the age of each? Ans. A's, B's, and C's P 9' 6 р |