7 ку ។ 28. It is required to find two numbers, such that if their product be added to their sum, the result shall be 47; and if their sum be taken from the sum of their squares, the remainder shall be 62. +2x=47 442+24=156 Ans. 7 and 5. 62 NOTE.-In many examples of two unknown quantities, giving rise to symmetrical equations, it will be found convenient to denote one of the unknown quantities by x+y, and the other by x-y. 29. The sum of two numbers is 27, and the sum of their cubes is 5103. What are the numbers? Ans. 12 and 15. 30. The sum of two numbers is 9, and the sum of their fourth powers is 2417. What are the numbers? Ans. 7 and 2. 1. The product of two numbers multiplied by the sum of their squares, is 1248; and the difference of their squares is 20. What are the numbers? Ans. 6 and 4. 32. Two men are employed to do a piece of work, which they can finish in 12 days. In how many days could each do the work 14y=12 alone, provided it would take one 10 days longer than the other? Ans. One in 20 days; the other in 30 days. v 2 33. The joint stock of two partners was $1000; A's money was in trade 2 months, and B's 6 months; when they shared stock and gain, A received $1,140 and B $640. What was each man's stock? Ans. A's, $600; B's, $400. 34. A speculator, going out to buy cattle, met with four droves. In the second were 4 more than 4 times the square root of one half the number in the first; the third contained three times as many as the first and second; the fourth was one half the number in the third, and 10 more; and the whole number in the four droves was 1121. How many were in cach drove? 2 X 2-4 Ans. 1st, 162; 2d, 40; 3d, 606; 4th, 313. 35. Find two numbers, such that if the sum of their squares be subtracted from three times their product, 11 will remain; and if the difference of their squares be subtracted from twice their product, the remainder will be 14. Ans. 3 and 5. 36. Divide the number 20 into two such parts, that the product of their squares shall be 9216. Ans. 12 and 8. W R 37. Divide the number a into two such parts, that the product хоу Yay 38. The greater of two numbers is a кузь _3x2 = 75 times the less, and the product of the two is . Find the numbers. b Ans. and ab. a' 39. A certain number is formed by the product of three consecutive numbers; and if it be divided by each of them in turn, the sum of the quotients will be 74. What is the number? Ans. 120; that is, 4.5.6; or (-5) - (-6). X, 24 40. An engraving, whose length was twice its breadth was mounted 2x cont. on Bristol board, so as to have a margin 3 inches wide, and equal w +6 (2x+6)in'area to the engraving, lacking 36 inches. 184+36 +x+36 the engraving. + 2 -36 Find the width of Ans. 12 inches. 41. A man has two square lots of unequal dimensions, containing 1+y=41oo together 25 A. 100 P. If the lots were contiguous to each other, it would require 280 rods of fence to embrace them in a single in4x + 2y = 28/03 4x+24 =2 closure of six sides. Required the dimensions of the two lots. Ans. 62 rods and 16 rods, 50 rods and 40 rods. 42. A person has £1300, which he divides into two portions, and lends at different rates of interest. He finds that the incomes from the two portions are equal; but if the first portion had been lent at the second rate of interest it would have produced £36, and if the second portion had been lent at the first rate of interest it would have produced £49. Find the rates of interest. ARs. 7-and 6 per cent. 361x11x12 Six:x:243. A sets out from London to York, and B at the same time from York to London, both traveling uniformly. A reaches York 25 hours, and B reaches London 36 hours, after they have met on the road. Find in what time each has performed the journey. Ans. A, 55 hours; B, 66 hours. 44. A owns a village lot, in square form, containing 36 square rods; B owns the adjacent lot on the same street, which is also a square, but greater than A's. Now if A should purchase all the 45. There are three numerical quantities having the following Z relations to cach other;-the sum of the squares of the first and second, added to the first and second, is 32; the sum of the squares of the first and third, added to the first and third, is 42; and the sum of the squares of the second and third, added to the second and third, is 50. Required the quantities. 2 г 1st, 3 or -4; 46. What is the side of that cube which contains as many solid units as there are lincar units in the diagonal through its opposite corners. Ans. √3. 47. It is required to find two quantities such that their sum, their product, and the sum of their squares, shall all be equal to cach other. Ans. 1(3±√3), and 1(3‡V—3). 48. Find those two numbers whose sum, product, and difference of their squares, are all equal to each other. Ans. (3/5), and (1±1/5). 49. Find two numbers, such that their product shall be equal to the difference of their squares. and the sum of their squares shall be equal to the difference of their cubes. Ans. ±†√5, and √(5±√/5). ] SECTION VI. PROPORTION, AND THE THEORY OF PERMUTATIONS AND COMBINATIONS PROPORTION. 315. Two quantitics of the same kind may be compared, and their numerical relation determined, by finding how many times one contains the other. This mode of comparison gives rise to ratio and proportion. 316. Ratio is the quotient of one quantity divided by another of the same kind regarded as a standard of comparison. There are two methods of indicating the ratio of two quantities. 1st. By writing the divisor before the dividend, with two dots between them; thus, a: b indicates the ratio of a to b, where a is the divisor and the dividend. 2d. In the form of a fraction; thus, the ratio of a to may be written ს α 317. A Compound Ratio is the product of two or more ratios. Thus, 318. The Duplicate Ratio of two quantities is the ratio of their squares. 319. The Triplicate Ratio of two quantities is the ratio of their cubes. 320. Proportion is an equality of ratios. Thus, if two quantities, a and b, have the same ratio as two other quantities, c and d, the four quantities, a, b, c, d, taken in their order, are said to be proportional. Proportion may be written in two ways; thus, a: b::c: d, which is read, a is to b as c is to d; or thus, which may be read the ratio of c to d. as the a: b = c:d, other, or, the ratio of a to b is equal to The second method of writing proportion is rec ommended as the more appropriate. 321. A Couplet is the two quantities which form a ratio. 322. The Terms of a proportion are the four quantities which are compared. 323. The Antecedents in a proportion are the first terms of the two couplets; or the first and third terms of the proportion. 324. The Consequents in a proportion are the second terms of the two couplets; or the second and fourth terms of the proportion. 325. The Extremes in a proportion are the first and fourth terms. 326. The Means in a proportion are the second and third terms. 327. When the first of a series of quantities has the same ratio to the second, as the second has to the third, as the third to the fourth, and so on, the several quantities are said to be in continued proportion, and any one of them is a mean proportional between the two adjacent ones. Thus, if abb:cc:d=d: e, then a, b, c, d, and e are in continued proportion, and b is a mean proportional between a and c, c a mean proportional between 6 and d; and so on. 328. One quantity is said to vary directly as another when the two quantities, by reason of their mutual dependence, have always a constant ratio, so that if one be changed the other will be changed in the same proportion. Thus, for illustration, suppose, in the purchase of a commodity, a certain quantity, A, costs a certain sum, B. Now if the price of unity remain the same, it is evident that 24 will cost 2B; 34 will |