3 3 Then P2 and y=Q2 And n3=P4 and y3=Q4 Our equations then become P2+PQ+Q2=a P*+P2Q2+Q=b Equations having no fractional exponents, and are of the same form as in Problem 12. (Art. 91.) Ans. x=81 or 16, y=16 or 81. =12 12 to find the values of x and y. Ans. x=9, y=1. to find the values of x and y. Ans. 2=(a+b) y=(a+b)=) 2 3 8. Given √x+√y: √x-√y :: 4:1, and x-y=16, to find the values of x and y. Ans. x=25, y=9. to find the values of x and y. Ans. x 10 Given x+y: x-y:: 3 : 1) And 23-y3=56 9 or 4, y=4 or 9. to find the values of x and y. Ans. x=4, y=2. CHAPTER V. Problems producing Pure Equations. (Art. 93.) We again caution the pupil, to be very careful not to involve factors, but keep them separate as long as possible, for greater simplicity and brevity. The solution of one or two of the following problems will illustrate. 1. It is required to divide the number 14 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 16: 9. Ans. The parts are 8 and 6. Let x= the greater part. Then 14-x= the less. 14-x :: 16:9. Per question, 14-x : Clearing of fractions, we have 9x2=16(14-x)2 indicating it, the exact value and form of the factors would have been lost to view, and the solution might have run into adfected quadratic equations. The same remark may be applied to many other problems, and many are put under the head of quadratics that may be reduced by pure equations, 2. Find two numbers, whose difference, multiplied by the difference of their squares, is 32, and whose sum, multiplied by the sum of their squares, gives 272. If we put x= the greater, and y= the less, we shall have Multiply these factors together, as indicated, and add the equations together, and divide by 2, and we shall have x3+y3=152 (3) If we take (1) from (2), after the factors are multiplied, we shall have 2xy2+2x2y=240, or xy(x+y)=120 (4) Three times equation (4) added to equation (3) will give a cube, &c. A better solution is as follows: Let x+y= the greater number, and x-y= the less. Then 2x= their sum, and 2y= their difference. Also, 4xy equal the difference of their squares, and 2x2+2y2= the sum of their squares. By the conditions, And By reduction, xy2= 4 2yX4xy= 32 2x(2x2+2y2)=272 x=4 By subtraction, x3 =64 or We give these two methods of solution to show how much depends on skill in taking first assumptions. 3. From two towns, 396 miles asunder, two persons, A and B, set out at the same time, and met each other, after travelling as many days as are equal to the difference of miles they travelled per day, when it appeared that A had travelled 216 miles. How many miles did each travel per day? Let x=A's rate, and y B's rate. Then x-y the days they travelled before meeting. By question, (-y)x=216, and (x--y)y=180. x3 6 Therefore, y=x, which substitute in the first equation, and we have (x-x)x=216, or =216=6×6×6. By evolution, x=36; therefore y=30. 4. Two travellers, A and B, set out to meet each other, A leaving the town C, at the same time that B left D. They travelled the direct road between C and D; and on meeting, it appeared that A had travelled 18 miles more than B, and that A could have gone B's distance in 15 days, but B would have been 28 days in going A's distance. Required the distance between C and D. Let the number of miles A travelled. Divide the denominators by 7, and extract square root, and we have Therefore, x=72; and the distance between the two towns is 126 miles. 5. The difference of two numbers is 4, and their sum, multiplied by the difference of their second powers, gives 1600. What are the numbers? Ans. 12 and 8. 6. What two numbers are those whose difference is to the less as 4 to 3, and their product, multiplied by the less, is equal to 504? Ans. 14 and 6. 7. A man purchased a field, whose length was to its breadth as 8 to 5. The number of dollars paid per acre was equal to the number of rods in the length of the field; and the number of dollars given for the whole was equal to 13 times the number of rods round the field. Required the length and breadth of the field. Ans. Length 104 rods, breadth 65 rods. Put 8x the length of the field. 8. There is a stack of hay, whose length is to its breadth as 5 to 4, and whose heighth is to its breadth as 7 to 8. It is worth as many cents per cubic foot as it is feet in breadth; and the whole is worth at that rate 224 times as many cents as there are square feet on the bottom. Required the dimensions of the stack. Put 5x= the length. Ans. Length 20 feet, breadth 16 feet; heighth 14 feet. 9. There is a number, to which if you add 7, and extract the square root of the sum, and to which if you add 16 and extract the square root of the sum, the sum of the two roots will be 9. What is the number? Ans. 9. Put x2-7= the number. 10. 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 as many as A, he would have received 8 pence. How many had each? Ans. A 40, and B 60. 11. The sum of two numbers is 6, and the sum of their cubes is 72. What are the numbers? Ans. 4 and 2. 12. One number is a2 times as much as another, and the product of the two is ba. What are the numbers? Ans. b and ab. a 13. The sum of the two numbers is 100, the difference of their square roots is 2. What are the numbers? Ans. 36 and 64. Put x= the square root of the greater number, |