Let x represent a side of the greater, and y a side of the less gar den; then 22 and y2 represent the respective areas of the two gardens. The Equations then are x2+y2=1025; From these Equations, r will be found 25 rods; and y=20 rods. 22. Find two numbers such, that their sum, their product, and the difference of their squares shall all be equal to one another. Let x+y represent the greater number, and x-y the less; then 2x represents their sum; x2-y2 represents their product; and 4xy represents the difference of their squares. By the conditions of the, problem, the Equations are 23. A merchant received $12 for a quantity of linen, and an equal sum, at 50 cents less per yard, for a quantity of calico, which exceeded the quantity of linen by 32 yards. What was the quantity of each? Let x represent the number of yards of linen, and y the number of yards of calico; then 12 y represents the price of the linen per yard; represents the price of the calico per yard. Since 50 cents is equal to problem, we shall have of a dollar, by the conditions of the From these Equations, x=16, and y=48 yards. 24. Find two numbers whose sum multiplied by the greater shall be equal to 192, and whose difference multiplied by the less shall be equal to 32. Let x represent the greater, and xy the less number; then Substituting for 22 in this Equation, its value as found above, Clearing this Equation of fractions by multiplying it by the least common multiple of the denominators, Since x2= 192 1+y' we have x2=192÷÷÷13=144; hence x= =12. Since the less number is represented by xy, we have for its value, 25. Three merchants gained $1444; of which their respective shares were such that B's added to the square root of A's, made $920; but if added to the square root of C's, it made $912. What was the share of each? for Let x represent A's share, and y B's share; then 1444-x-y represents C's share. We shall then have the following Equations: y+√x=920; y+1444-x-y=912. Transposing y in the first Equation, and squaring, x=846400-1840y+y2. Proceeding in a similar manner with the second Equation, 1444-x-y=831744-1824y+y2. Adding together the last two Equations, 1444-y-1678144-3664y+2y2. This Equation will give y-$900; and substituting this value of y y in the third Equation, we find x=$400. Hence, C's share was 1444-900-400-$144. 26. The sum of three numbers in harmonical progression is 13, and the product of the two extremes is 18. What are the numbers? Let x and y, respectively, represent the two extremes; then х 2xy x+y represents the mean term, (184). The Equations will then be Substituting for xy in this Equation its value from the second Eq'n, (x+y)2+36=13(x+y). This Equation is quadratic with reference to x+y, and will give Substituting this value of 2 in the second Equation, we find y=3; =4, is the value of the mean teria. then x 183=6; and 2×6×3 27. There is a rectangular field whose length is to its breadth as 4 to 3. A part of this field, which is equal to of the whole, being in meadow, there remain for ploughing 1296 square rods. What are the dimensions of the field? Let x represent the length, and y the breadth; then xy represents the area of the field. We shall then have for the first Equation And since is in meadow, there remains for ploughing; then from which, and the first Equation, we shall find a=48 rods, and y= 36 rods. 28. The sum of three numbers in geometrical progression is 21, and the sum of their squares 189. What are the numbers? Substituting these values of x and √ay in the first Equation, 36 12 From this Equation, y=12; hence =3, the value of x. Then. 3x12=6, the mean term. 29. A and B set out from two places which are distant 110 miles, and traveled towards each other. A went 5 miles an hour; and the number of hours in which they met was greater, by four, than the number of miles C went per hour. What was B's rate of traveling? Let x represent the number of miles B went per hour; then +4 represents the number of hours each was traveling; 5(x+4) represents the number of miles A traveled; and x(x+4) represents the number of miles B traveled. The Equation then is 5(x+4)+x(x+4)=110; whence 6 miles. 30. The arithmetical mean between two numbers exceeds the geometrical mean by 13, and the geometrical mean exceeds the harmonical mean by 12. What are the numbers? Let x and y represent the numbers; then x+y 2 represents the arithmetical mean, (179); Vxy represents the geometrical mean, (189); and 2xy x+y represents the harmonical mean, (184). By the conditions of the problem, Clearing this Equation of fractions, and squaring both sides, 4xy=(x+y)2-52(x+y)+676. Equating the two values of ry from the second and third Equations, Clearing this Equation of fractions, transposing, &c., 4xy=(x+y)2—50(x+y). |