EXAMPLES. 2. Transform }; into a continued fraction. Ans. 1 3 + + 3. Transform 22 into a continued fraction. 4. Find the approximate values of 44. Ans. z, }, }, }, 74. 5. Find the first five approximate values of 1831 6. Find the first three approximate values of $29. Ans. Í, }, }, or 2, 23, 24. 1 7. Find the first six approximate values of 132980 Ans. 1', I, 1, 35, 37, 387. 8. What are the first four approximate values of 1.27 ? Ans. +, $, {, tt, 38, or 1, 1$, 11, 1111, 176 RATIO. 310. Ratio is the relation, in respect to magnitude or value, which one quantity or number bears to another of the same kind. 311. The comparison by ratio is made by considering how often one number contains, or is contained in, another. Thus, the ratio of 10 to 5 is expressed by 2, the quotient arising from the division of the first number by the second, or it may be expressed by t = 1, the quotient arising from the division of the second number by the first, as the second or the first number shall be regarded as the unit or standard of comparison. In general, of the two methods, the first is regarded as the more simple and philosophical, and therefore has the preference in this work. NOTE. – Which of the two methods is to be preferred, is not a question of so much importance as has been by some supposed, since the connection in which ratio is used is usually such as to readily determine its interpretation. 312. The two numbers necessary to form a ratio are called 315. the terms of the ratio. The first term is called the antecedent, and the last, the consequent. The two terms taken together are called a couplet ; and the quotient of the two terms, the index or exponent of the ratio. 313. The ratio of one number to another may be expressed either by two dots (:) between the terms; or in the form of a fraction, by making the antecedent the numerator and the consequent the denominator. Thus, the ratio 6 miles to 2 miles may be expressed as 6: 2, or as S. 314. The terms of a ratio must be of the same kind, or such as may be reduced to the same denomination. Thus, cents have a ratio to cents, and cents to dollars, &c.; but cents have not a ratio to yards, nor yards to gallons. A simple ratio is that of two whole numbers; as, 3:4, 8:16, 9:36, &c. 316. A complex ratio is that of two numbers, of which one 1} : 5 317. A compound ratio is the product of two or more ratios. Thus, the ratio compounded of 4 : 2 and 6 : 3 is f X = 2 4, or 4 X 6:2 X 3 = = 4. A compound ratio is generally expressed by writing the ratios composing it, in a column, with the antecedents in one 4:2 vertical line, and the consequents in another ; 'thus, 3 presses a compound ratio. Хоте. If a ratio be compounded of two equal ratios, it is called a duplicate ratio; of three ratios, a triplicate ratio, &c. 318. A ratio is either direct or inverse. A direct ratio is the quotient of the antecedent by the consequent ; an inverse ratio, or reciprocal ratio, as it is sometimes called, is the quotient of the consequent by the antecedent, or the reciprocal of the direct ratio. Thus the direct ratio of 6 to 2 is or 3; and the inverse or reciprocal ratio of 6 to 2 is for }, which is the same as the reciprocal of 3, the direct ratio of 6 to 2. NOTE 1. — - One quantity is said to vary directly as another, when both in. crease or decrease together in the same ratio; one quantity is said to vary is ex versely as another, when the one increases in the same ratio as the other do creases. NOTE 2. — The word ratio, when used alone, means the direct ratio. 319. When the antecedent and consequent of a ratio are equal, the ratio equals 1, and is called that of equality. Thus, the ratio of 6 : 6 = 1, and the ratio of 6 X 4:8 X 3 22 h 1, are ratios of equality. But if the antecedent is larger than the consequent, the ratio is that of greater inequality, and if the antecedent is smaller than the consequent, the ratio is that of less inequality. Thus, the ratio of 15:5=1= 3, is a ratio of greater inequality; and the ratio of 7:14 14 1, is a ratio of less inequality. 320. The ratio of two fractions having a common numerator is the same as the inverse ratio of their denominators. Thus, the ratio of 1: g is j = = 2, which is the inverse ratio of the denominator 4 to the denominator 8. 321. The ratio of two fractions having a common denominator is the same as the ratio of their numerators. Thus, the ratio of 6:is of •*= 2, which is the ratio of the numerator 6 to the numerator 3. 322. The inverse or reciprocal ratio of two numbers denotes what part or multiple the consequent is of the antecedent. Thus, inquiring what part of 4 is 3, or what part 3 is of 4, is the same as inquiring the inverse or reciprocal ratio of 4:3. The inverse ratio of 4:3 is j, and 3 is i of 4. 323. In order to compare one number with another, by ratio, it is necessary that they should not only be of the same kind, but of the same denomination. Thus, to compare 2 days with 12 hours, it is necessary that the days be reduced to hours, before we can indicate the ratio, which is 48 hours : 12 hours. 324. If the antecedent of a ratio be multiplied, or the consequent divided, the ratio is multiplied. Thus, the ratio of 6:3 is 2, but 6 X 2 : 3 is 4; or 6:3 • 2 is 4. 325. If the antecedent of a ratio be divided, or the consequent multiplied, the ratio is divided. Thus, the ratio of 18 : 6 is 3, but 18 3 : 6 is 1; or 18 : 6 X 3 = 1. 326. If both the antecedent and consequent of a ratio be multiplied or divided by the same number, the ratio is not altered. Thus, the ratio of 8 : 2 is 4 ; of 8 X 2:2 X 2 is 4; and of 8 • 2:2 • 2 is 4. REDUCTION AND COMPARISON OF RATIOS. 327. Ratios, being of the nature of fractions, may be reduced, compared, and otherwise operated upon like them. 328. To reduce a ratio to its lowest terms. We cancel in the two terms the 18:9 16 = 2 : 1. common factor 9, and obtain i = 2:1, the answer. Hence Cancel in the given ratio all factors common to its terms. OPERATION. EXAMPLES Ans. š. 2. Reduce to its lowest terms 63 : 72. 3. Reduce to its lowest terms 66 : 24. 4. Reduce to its lowest terms 4 x 6 x 3:8 X 9 X 2. Ans. 1. 5. What are the lowest terms of 19 X 5 X 2 X 3:15 X 12 X 38 ? 329. To reduce a complex or a compound ratio to a simple one. Ex. 1. Reduce 54 : to a simple ratio. Ans. 22 : 3. the 5.4 51 : 1 4 4 form of plex fraction, which, changed to a simple fraction (Art. 242), and reduced to its lowest terms, gives a 22 : 3, the answer required. 8: 5 2. Reduce 7 : 24 Ans. 7:15. OPERATION. We express a com } OPERATION. 8: 5 7 24 3 We express the given ratio in the form of a compound fraction, which, reduced to a simple one (Art. 329), gives 7: = 7:15, the answer required. Hence, to reduce a complex or a compound ratio to a simple one, Proceed as in like operations with fractions. OPERATION. Ans. š. OPERATION. 330. To find the ratio of one number to another. Ex. 1. Required the direct ratio of 108 to 9. Ans. 12. Since 9 is the unit or standard of 108 : 9 = - 138 12 Ans. comparison, we make it the conse quent (Art. 111) and the 108 the antecedent of the ratio, and obtain 108 12 Ans. 2. Required the inverse ratio of 72 to 8. We divide the consequent 72 : 8 inverted j Ans. 8 by the antecedent 72, or, which is the same thing, find the reciprocal of the direct ratio of 72: 8 (Art. 318), by inverting its terms, and thus obtain i Ans. Hence, The direct ratio is found by dividing the antecedent by the consequent, and the inverse ratio by dividing the consequent by the antecedent. NOTE 1. - Ratios expressed by fractions having different denominators must be reduced to a common denominator, in order to be compared; and then they are to each other as their numerators (Art. 323). NOTE 2. – When a ratio is expressed in terms inconveniently large and prime to each other, we may find the approximate values of the ratio expressed in smaller numbers, as in other fractional expressions (Art. 309). EXAMPLES. 3. What is the ratio of 39 to 13 ? Ans. 3. 4. What is the ratio of 2 yards 2 quarters to 9 yards ? |