NOTE. - Before clearing of fractions, transpose 74 and unite it 29. Reduce = 6 22 a b c d be+ acab 3 + x Ans. x7. PROBLEMS PRODUCING EQUATIONS OF THE FIRST DEGREE CONTAINING BUT ONE UNKNOWN QUANTITY. 107. The problems given in this Section must either contain but one unknown quantity, or the unknown quantities must be so related to one another that if one becomes known the others also become known. 108. With beginners the chief difficulty in solving a problem is in translating the statements or conditions of the problem from common to algebraic language; i. e. in preparing the data, and forming an equation in accordance with the given conditions. 1. If three times a certain number is added to one half and one third of itself, the sum is 115. What is the number? Clearing of fractions, 18 x + 3x + 2 x = 690 In this problem there is but one unknown quantity, which we represent by x. 2. There are three numbers of which the first is 6 more than the second, and 11 less than the third; and their sum is 101. What are the numbers? Let x (or some one of the latter letters of the alphabet) represent the unknown quantity; or, if there is more than one unknown quantity, let x represent one, and find the others by expressing in algebraic form their given relations to the one represented by x. With the data thus prepared form an equation in accordance with the conditions given in the problem. Solve the equation. The three steps may be briefly expressed thus: 1st. Preparing the Data; 2d. Forming the Equation; 3d. Solving the Equation. 3. The sum of three numbers is 960; the first is one half of the second and one third of the third. the numbers ? What are Ans. 160, 320, and 480. 4. Find two numbers whose difference is 18 and whose sum 112. Ans. 47 and 65. 5. A man being asked how much he gave for his horse said, that if he had given $70 more than three times as much as it cost, he would have given $445. How much did his horse cost him? 6. A man being asked how many sheep he had, replied that if he had as many more, and two thirds as many, and three fifths as many, he should have 8 more than three times as many as he had. How many sheep had he? and B in such a manner 7. Divide $575 between A that B may have two thirds as much as A. 8. A father divided his estate among his three children so that the eldest had $1440 less than one half of the whole, the second $500 more than one third of the whole, and the youngest $250 more than one fourth of the whole. What was the value of the estate? 9. A gentleman meeting five poor persons, distributed $7.50, giving to the second twice, to the third three times, to the fourth four times, and to the fifth five times as much as to the first. How much did he give to each? |