CHAPTER XXI. RATIO, PROPORTION, AND VARIATION. 316. The relative magnitude of two numbers is called their ratio, when expressed by the fraction which the first is of the second. Thus, the ratio of 6 to 3 is indicated by the fraction, which is sometimes written 6:3. 317. The first term of a ratio is called the antecedent, and the second term the consequent. When the antecedent is equal to the consequent, the ratio is called a ratio of equality; when the antecedent is greater than the consequent, the ratio is called a ratio of greater inequality; when less, a ratio of less inequality. 318. When the antecedent and consequent are interchanged, the resulting ratio is called the inverse of the given ratio. Thus, the ratio 3:6 is the inverse of the ratio 6: 3. 319. The ratio of two quantities that can be expressed in integers in terms of a common unit is equal to the ratio of the two numbers by which they are expressed. Thus, the ratio of $9 to $11 is equal to the ratio of 9:11; and the ratio of a line 23 inches long to a line 3 inches long, when both are expressed in terms of a unit of an inch long, is equal to the ratio of 32:45. 320. Two quantities different in kind can have no ratio, for then one cannot be a fraction of the other. 321. Two quantities that can be expressed in integers in terms of a common unit are said to be commensurable. The common unit is called a common measure, and each quantity is called a multiple of this common measure. Thus, a common measure of 24 feet and 33 feet is of a foot, which 10 is contained 15 times in 24 feet, and 22 times in 33 feet. Hence, 24 feet and 3 feet are multiples of of a foot, 24 feet being obtained by taking of a foot 15 times, and 33 feet by taking of a foot 22 times. 322. When two quantities are incommensurable, that is, have no common unit in terms of which both quantities can be expressed in integers, it is impossible to find a fraction that will indicate the exact value of the ratio of the given quantities. It is possible, however, by taking the unit sufficiently small, to find a fraction that shall differ from the true value of the ratio by as little as we please. Thus, if a and b denote the diagonal and side of a square, If, then, a millionth part of b is taken as the unit, the value of the α ratio lies between 1414213 and 414214, and therefore differs from 000000 10000009 either of these fractions by less than 100000. By carrying the decimal further, a fraction may be found that will differ from the true value of the ratio by less than a billionth, trillionth, or any other assigned value whatever. 323. Expressed generally, when a and b are incommensurable, and b is divided into any integral number (n) of equal parts, if one of these parts is contained in a more than m times, but less than m + 1 times, then The error, therefore, in taking either of these values for made to decrease indefinitely, and to become less than any assigned value, however small, though it cannot be made absolutely equal to zero. 324. The ratio between two incommensurable quantities is called an incommensurable ratio. 325. THEOREM. Two incommensurable ratios are equal if, when the unit of measure is indefinitely diminished, their approximate values constantly remain equal. m Let ab and a': b' be two incommensurable ratios whose true values lie between the approximate values and m+1 n n when the unit of measure is indefinitely dimin1 ished. Then they cannot differ by so much as n Now the difference (if any) between the fixed values a:b and a': b' is a fixed value. Let d denote this fixed value. 1 But if d has any value, however small, -, which by n hypothesis can be made less than any value, however small, can be made less than d. Therefore, d cannot have any value; that is, d = 0, and there is no difference between the ratios ab and a':b'; therefore, ab a': b'. = 326. A ratio will not be altered if both its terms are multiplied by the same positive number. 327. A ratio will be altered if its terms are multiplied by different positive numbers; and will be increased or diminished according as the multiplier of the antecedent is greater than or less than that of the consequent. 328. A ratio of greater inequality will be diminished, and a ratio of less inequality increased, by adding the same positive number to both its terms. 329. A ratio of greater inequality will be increased, and a ratio of less inequality diminished, by subtracting the same positive number from both its terms. 330. Ratios are compounded by taking the product of the fractions that represent them. Thus, the ratio compounded of a: b and c d is found by taking the : The ratio compounded of a: b and a : b is the duplicate ratio a2 : b2, and the ratio compounded of a: b, a: b, and a : b is the triplicate ratio 13 : b8. 331. Ratios are compared by comparing the fractions that represent them. 1. Write the ratio compounded of 3:5 and 8:7. Which of these ratios is increased, and which is diminished by the composition? 2. Compound the duplicate ratio of 4: 15 with the triplicate of 5: 2. 3. Show that a duplicate ratio is greater or less than its simple ratio according as it is a ratio of greater or less inequality. 4. Arrange in order of magnitude the ratios 3:4; 23: 25; 10:11; and 15: 16. 5. Arrange in order of magnitude a2 a2 a+bab and a2+ b2: a2 — b2, if a >b. Find the ratio compounded of: 6. 3:5; 10: 21; 14: 15. 7. 7:9; 102: 105; 15: 17. |