the remainder for £55, by which he gained 10 per cent. How many yards did he buy, and what did it cost him per yard? Ans. 50 yards at 25s. per yard. 38. Bought a horse for a certain price, and found that if I could sell him for 60 guineas, I should gain twice as much per cent. as I should if I sold him for 50 guineas. Required his prime cost. Ans. 40 guineas. 39. A merry young fellow in a short time spent onefifth of his fortune; by the advice of his friends he gave £2200 for a place in the Guards; his profusion continued till he had only 880 guineas left, which he found was three-twentieths of his money after the commission was bought. Pray what was his fortune at first? Ans, £10450. 40. Three boys met a girl carrying apples to the market the first took half what she had, but returned to her ten; the second took one-third, but returned two; and the third took away half those she had left, but returned one: she had then twelve apples left. How many had she at first? Ans. 40. 41. By sliding out the bottom of a ladder, which was placed upright against a wall, 10 feet from the wall, I observed the top to fall one yard. Required the length of the ladder. Ans. 18 feet 2 inches. 42. A mason having to form a stone for a circular window top, to rise 20 inches, what radius must he take to strike the arc, the breadth of the window being 4 feet 2 inches? Ans. 25.625 inches. 43. Required the area of that square, the paving of which, at two shillings per square yard, cost as much as inclosing it at six shillings per yard, lineal measure. Ans. 144. 44. Required the side of that cube, whose solidity contains as many cubic inches, as the sides do square inches. A Ans. 6. 45. Required the side of that cube, whose superficies is to its solidity as 6 to 11. Ans. 11. 46. Required the diameter of that globe, whose solidity and superficial content are equal to each other. » Ans. 6. 47. Required the diameter of that globe, whose superficies is to its solidity as 2 to 3. Ans. 9. 48. Required the side of that equilateral triangle, whose area cost as much paving, at eight pence per foot, as pallisading the three sides did at a guinea per yard. Ans. 72.746, SIMPLE EQUATIONS OF TWO OR MORE UNKNOWN QUANTITIES. In the solution of equations containing two or more unknown quantities, there must be as many independent equations as there are unknown quantities, in order that the unknown quantities may have determinate values, which may be found by the following rules: RULE I. 1. When the given question contains two equations, each containing two unknown quantities, as in the first example, find the value of the same unknown quantity, in terms of the rest of the equation, in each equation, then, by the 5th axiom in Def. 4, things which are equal to the same things are equal to each other, a new equation will be formed containing one unknown quantity only, which may be solved by the rules already given. 2. But when the given question contains three equations, with three unknown quantities, as in example 2d. find the value of the same unknown quantity, in each equation, in terms of the rest, as in example 2d., then, by axiom 5, two new equations may be formed, each containing two unknown quantities, which may be solved by the first part of the rule. 3. Also, when the given question contains four equations, each containing four unknown quantities, find the value of the same unknown quantity, in terms of the rest of the equation, then, as above, three new equations may be formed, each containing three unknown quantities; therefore, by the second part of the rule, the value of each may be found, and so on, for any other number of equations. 1. Given 3x+2y=18,} to find the values of x and y. Here, from the first equation, 3x=18-2y; .. x= 18-2y (v) *; From the second equation, 2x=5+y; taining one unknown quantity only. Multiplying this equation by 6, the least common multiple of 2 and 3, 15+3y=36—4y; by transposition, 3y+4y=36—15, or 7y=21;. y=3. Substituting this value of y, which is 3, in the equa18-2x3 12 tion (A), x= 4; 3 Or, if the value of y be substituted in the equation (B) NOTE. When there are more unknown quantities than independent equations, some of these quantities cannot be determined, except in terms of the rest; but if there be more independent congruous equations than unknown quantities, some of the equations are unnecessary. Also, from the second equation, y=2x−5; (B) multiplying by 2, 4x-10-18-3x; substituting this value of x, in the equation (A), or if this value of x be substituted in the equation (B), y=2x4—5—8—5=3; whence, x=4, and y=3. 2. Given 3x+2y+4z=28, to find the values of x, 6x-3y+2x=11, and 9x+6y-82=4, Here, from the first equation, 4z=28-3x-2y; from the second equation, 2z=11—6x+3y; y, and z. ; (B) 11-6x+3y also, from the third equation, 8z=9x+6y—4; -28-3x-2y 11-6x+3y multiplying this new equation by 4, 28-3x-2y-22-12x+6y; 2 by transposition, 28-22-3x-12x+2y+6y, |