27. A and B carried 100 eggs between them to market, and each received the same sum. If A had carried as many as B, he would have received 18 pence for them, and if B had taken only as many as A, he would have received only 8 pence. How many had each ? •: 3x= ± 2y, the negative value of which will not answer the conditions of the problem. 28. What two numbers are those, which being both multiplied by 27 the first product is a square, and the second the root of that square but being both multiplied by 3, the first product is a cube, and the second the root of that cube? whence 9y3 = 27 y3, and y = 3; :: x = 27y2 = 243; .. the numbers are 243, and 3. 29. It is required to find the three sides of a right-angled triangle from the following data. The number of square feet in the area is equal to the number of feet in the hypothenuse the sum in the other two sides; and the square described upon the hypothenuse is less than the square described upon a line equal in length to the two sides, by half the product of the numbers representing the base and area. Let the number of feet in the altitude, and y = the number in the base; :. √(x2 + y2) = the number in the hypothenuse, (Eucl. B. 1. p. 47.) also x2 + y2 = {(x + y)2 − {xy2 = } x2 + 2xy + y2 − {xy3 ; .. by transposition, xy2 = 2xy, and y = 8; hence from the first equation, 4x = √(x2 + 64) + x + 8, and by transposition, 3x-8=√ (x2 + 64) ; .. the sides are 6, 8, and 10 feet, respectively. 30. A Farmer has 2 cubical stacks of hay. The side of one is 3 yards longer than the side of the other; and the dif ference of their contents is 117 solid yards. Required the side of each. cubing the latter equation, x3- 3x2y + 3xу2 — y3 = 27 ; and the sides of the stacks are 5, and 2 yards, respectively. 31. When a parish was enclosed, the allotment of one of the proprietors consisted of two pieces of ground; one of which was in the form of a right-angled triangle; the other was a rectangle, one of the sides of which was equal to the hypothenuse of the triangle, the other, to half the greater side; but wishing to have his land in one Q piece, he exchanged his allotments for a square piece of ground of equal area, one side of which equalled the greater of the sides of the triangle which contained the right angle. By this exchange he found that he had saved ten poles of railing. What are the respective areas of the triangle and rectangle; and what is the length of each of their sides? Let 2x the greater side of the triangle, and y = the less; :: √ (4x2 + y2) = the hypothenuse; and also the greater side of the rectangle, and .. xy and (4+ y2) the less side of the rectangle ; the area of the triangle, = the area of the rectangle; .. 4x2 = xy + x √ (4x2 + y2), or 4x y = √ (4x2 + y2) ; also 8x + 10 = 2 x + y + √ (4 x2 + y2) + 2 x + 2 √ (4x2 + y2), or 4x+10= y + 3 √ (4x2 + y); in which equation substituting the value of ✓ (4x2 + y2) found above; .. 4x + 10 = y + 3 (4x − y) =12x-2y; .. by transposition, 2y = 8x-10, and y=4x-5; .. from the first equation, 5= {4x2 + (4x — 5)"}, ..the sides of the triangle are 3, 4, and 5; the sides of the rectangle are 2, and 5; and the areas of the triangle and rectangle are 6, and 10, respectively. SECTION IX. Examples of the Solution of Problems producing Adfected Quadratic Equations. 1. A MERCHANT sold a quantity of brandy for £39, and gained as much per cent. as the brandy cost him. What was the price of the brandy? by transposition, x2 + 100x = 3900, completing the square, x2 + 100x + 50 2 3900 + 2500 = 6400; ± 80; 2. There are two numbers whose difference is 9, and their sum multiplied by the greater produces 266. What are those |