versely as another, when the one increases in the same ratio as the other decreases. Note 2. — The word ratio, when used alone, means the direct ratio. 319i 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 = f = 1, and the ratio of 6 X 4 : 8 X 3 = = 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 — = 3, is a ratio of greater inequality; and the ratio of 7 : 14 — = 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 J : § is f -i- § = 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^:^isf-r-f = 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 ichat 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 f, and 3 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 -i- 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 he 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 -r- 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. Ex. 1. Reduce 18 : 9 to its lowest terms. Ans. 2 : 1. Operation. 'We cancel in the two terms the 18:9 = -"j- =$=2:1. common factor 9, and obtain | 2:1, the answer. Hence Cancel in the given ratio all factors common to its terms. Examples. 2. Reduce to its lowest terms 63 : 72. Ans. J. 3. Reduce to its lowest terms 66 : 24. 4. Reduce to its lowest terms 4X6X3:8X9X2. Ans. 5. What are the lowest terms of 19 X ^ X 2 X 3: la X 12 X 38? 329. To reduce a complex or a compound ratio to a simple one. Ex. 1. Reduce 5£ : £ to a simple ratio. Ans. 22 : 3. Operation. 'We express the 5A : * = # = *A = =22:3 Ans. fiven rfo in the • 4 3 6 j form ot a com plex fraction, which, changed to a simple fraction (Art. 242), and reduced to its lowest terms, gives ^ = 22 : 3, the answer required. 8 , 5 ) 2. Reduce 7 - 24 r to a simple ratio. Ans. 7 : 15. OPERATION. 7:2i}=fofA=f X^ = A = 7:15Ans. 3 We express the given ratio in the form of a compound fraction. •which, reduced to a simple one (Art. 829), gives Jg mm 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. Examples. 3. Reduce £ : f to a simple ratio. Ans. 35 : 24. 4. Reduce 13^ : 27 to a simple ratio. Ans. 1 : 2. 5. Reduce 6.25 : 3.125 to a simple ratio. Ans. 2 : 1. 6. Reduce 25 • 10 } t0 a s'mP^e ratio- Ans. 5 : 8. 3: 6 7. Reduce 9 : 27 ^ to a simple ratio. Ans. 3 : 2. 8. Reduce .. 25 5 } t0 a simPle ratio> Ans. 6 : 1. 330. To find the ratio of one number to another. Ex. 1. Required the direct ratio of 108 to 9. Ans. 12. Operation. Since 9 is the unit or standard of 108 : 9 = J§a = 12 Ans. comparison, we make it the consequent (Art. Ill) and the 108 the antecedent of the ratio, and obtain -LIp =12 Ans. 2. Required the inverse ratio of 72 to 8. Ans. Operation. We divide the consequent 72 : 8 inverted = 72 _ h 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 ^ = £ 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? 5. What is the ratio of 21 gallons to J of a hogshead? Ans. 1. 6. What is the ratio of £ of £ of $ 2 : £ of $ 0.50? Ans. $. 7. What is the inverse ratio of 24 : 6? Ans. 8. What part of 36 is 4? Ans. f 9. What part of a farm of 94A. 2R. 16rd. is 11A. 3R.? 10. Which is the greater, the ratio of 17 to 9, or of 39 to 19? Ans. 39:19. 11. By how much does the ratio of 36 X 4 X 3 : 12 X 1C X 2 exceed that of 60 H- (3 X 5) : 20 X 2 ,+, 8? Ans. 4g. 12. What is the inverse ratio of .02 : 2.503? 13. Which is the greater, the ratio of £ of \ : £ of or that of 5 : 4? 14.° The height of Bunker Hill Monument is 220 feet, and that of the great pyramid. Egypt, 500 feet; what is the ratio of the height of the former to that of the latter? Ans. 15.° A certain farm contains 180 acres, and the township of which it forms a part is 36 square miles in extent. What is the ratio of the latter to the former? 16.° Find approximate values for the ratio of 4900 to 11283. Ans. i, ?,ff, Jft, &c. 17.° The ratio of the circumference of a circle to its diameter is 3.141592. Required approximate values for this ratio. Ans. 3, m, m, &c, or 3, 3*, Srffc, 3^% &c. ANALYSIS BY RATIO. 331 i Operations by analysis may often be much abridged by ratio. Thus, frequently, it is more convenient to multiply or divide by the ratio a number bears to a unit of the same kind, than to multiply or divide by the number itself. This form of analysis is much used by business men; and, like that by aliquot parts (Art. 114), is sometimes called Practice. Examples. |