29. Solve (i.) x3 + y3 = 18 √√2, x + y = 3√2. 30. A rectangular field is 100 yards wide. If it were reduced to a square field by cutting an oblong piece off one end, the ratio of the than the ratio of the piece cut off to the remainder would be less by 31. Extract the square root of 34. Two places, A and B, are 168 miles apart, and trains leave A for B and B for A simultaneously; they pass each other at the end of one hour and fifty-two minutes, and the first reaches B half an hour before the second reaches A. Find the speed of each train. 35. Solve (i.) 3x2+4x+2√x2 + x + 3 (ii.) x + y = 706, x + y = 8. = 30 - x2. 36. Form the equation whose roots are the squares of the sum and of the difference of the roots of 37. Employ the method of Arts. 319, 320 in showing that 38. Solve (a + b)5 — a5 — b5 = 5 ab(a + b) (a2 + ab + b2). xy(3x+y)=10, 27 x8+ y8 35. = CHAPTER XXXIII. RATIO, PROPORTION, AND VARIATION. 336. DEFINITION. Ratio is the relation which one quantity bears to another of the same kind, the comparison being made by considering what multiple, part, or parts, one quantity is of the other. The ratio of A to B is usually written A: B. The quantities A and B are called the terms of the ratio. The first term is called the antecedent, the second term the consequent. 337. Ratios are measured by Fractions. To find what multiple or part A is of B, we divide A by B; hence the A ratio A: B may be measured by the fraction B' and we shall usually find it convenient to adopt this notation. In order to compare two quantities, they must be expressed in terms of the same unit. Thus, the ratio of $2 2 × 100 40 to 15 cents is measured by the fraction 15 or 3 NOTE. Since a ratio expresses the number of times that one quantity contains another, every ratio is an abstract quantity. and thus the ratio a: b is equal to the ratio ma: mb; that is, the value of a ratio remains unaltered if the antecedent and the consequent are multiplied or divided by the same quantity. 339. Comparison of Ratios. Two or more ratios may be compared by reducing their equivalent fractions to a common denominator. Thus, suppose ab and x:y are two ; hence the ratio ab is α ratios. Now, ay x bx and = greater than, equal to, or less than the ratio x: y according as ay is greater than, equal to, or less than bx. 340. The ratio of two fractions can be expressed as a a с ratio of two integers. Thus, the ratio is measured by b d ad or ; and is therefore equivalent to the ratio ad : bc. d bc 341.. If either, or both, of the terms of a ratio be a surd quantity, then no two integers can be found which will exactly measure their ratio. Thus, the ratio √2:1 cannot be exactly expressed by any two integers. 342. If the ratio of any two quantities can be expressed exactly by the ratio of two integers, the quantities are said to be commensurable; otherwise, they are said to be incommensurable. Although we cannot find two integers which will exactly measure the ratio of two incommensurable quantities, we can always find two integers whose ratio differs from that required by as small a quantity as we please. and it is evident that by carrying the decimals further, any degree of approximation may be arrived at. 343. Ratios are compounded by multiplying together the fractions which denote them. Ex. Find the ratio compounded of the three ratios 2a3b, 6 ab: 5 c2, c: a. 344. When two identical ratios, a : b and a : b, are compounded, the resulting ratio is a b2, and is called the Similarly, as : b3 is called the tripliAlso, at: b is called the subduplicate ¿a duplicate ratio of a: b. cate ratio of a ratio of a b. b. : EXAMPLES (1) The duplicate ratio of 2 a 3b is 4 a2 : 9 b2. (2) The subduplicate ratio of 49 : 25 is 7 : 5. (3) The triplicate ratio of 2x: 1 is 8 x3: 1. 345. A ratio is said to be a ratio of greater inequality, or of less inequality, according as the antecedent is greater or less than the consequent. 346. If to each term of the ratio 8: 3 we add 4, a new ratio 12: 7 is obtained, and we see that it is less than the former because is clearly less than §. This is a particular case of a more general proposition which we shall now prove. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by adding the same quantity to both its terms. x(a - b); b(b + x) Now α a + x b b+x = b(b+x) and ab is positive or negative according as a is greater or Similarly, it can be proved that a ratio of greater inequality is increased, and a ratio of less inequality is diminished, by taking the same quantity from both its terms. 347. When two or more ratios are equal, many useful propositions may be proved by introducing a single symbol to denote each of the equal ratios. The proof of the following important theorem will illustrate the method of procedure. where p, q, r, n, are any quantities whatever. By giving different values to p, q, r, n many particular cases of this general proposition may be deduced; or they may be proved independently by using the same method. For instance, if a result which may be thus enunciated: When a series of fractions are equal, each of them is equal to the sum of all the numerators divided by the sum of all the denominators. Ex. 2. Two numbers are in the ratio of 5:8. If 9 be added to each they are in the ratio of 8:11. Find the numbers. Let the numbers be denoted by 5x and 8x. |