372. A Direct Ratio arises from dividing the antecedent by the consequent. 373. An Inverse or Reciprocal Ratio is obtained by dividing the consequent by the antecedent. The reciprocal of a ratio equals 1 divided by the ratio. When the numerator and denominator of a fraction are interchanged the fraction is said to be inverted; and in the same way, when the antecedent and consequent are interchanged the ratio is called an inverse ratio. Thus, the direct ratio of 5 to 15 is of 5 to 15 is 15 = 3. 1 ; and the inverse ratio 374. A Simple Ratio is the ratio between two terms; as, 3:12. 375. A Compound Ratio is the ratio of the corresponding terms of two or more simple ratios. Thus, 3:9:: 5:15. 376. When the multiplication is performed in a compound ratio, the result is a simple ratio. Thus, the compound ratio formed from the simple ratios 8:4 and 9: 12 may be expressed: 377. In comparing numbers with each other, they must be of the same kind. If the terms of a ratio are denominate numbers, they must be reduced to the same unit value. 378. The ratio of two fractions is obtained by dividing the first by the second; or by reducing them to a common denominator, when they will be to each other as their numerators. Thus, the ratio ofis÷ 3 = 15 = 1, which is the same as the ratio of the numerator 3 to the numerator 6 of the equivalent fractions and 1%. Since the antecedent is a dividend and the consequent a divisor, any change in either or both terms. will be governed by the general principles of division (86). We have only to substitute the terms antecedent, consequent, and ratio, for dividend, divisor, and quotient, and these principles become simple. PRINCIPLES.-I. Multiplying the antecedent multiplies the ratio; dividing the antecedent divides the ratio. II. Multiplying the consequent divides the ratio; dividing the consequent multiplies the ratio. III. Multiplying or dividing both antecedent and consequent by the same number does not alter the ratio. These principles may be embraced in one law. GENERAL LAW. A change in the ANTECEDENT produces a LIKE change in the ratio; but a change in the CONSEQUENT produces an OPPOSITE change in the ratio. 379. Since the ratio of two numbers is equal to the antecedent divided by the consequent, it follows that, 1. The antecedent is equal to the consequent multiplied by the ratio; and that, 2. The consequent is equal to the antecedent divided by the ratio. 12. What is the ratio of 3 gal. to 2 qt. 1 pt. ? Ans. 44. 13. What is the ratio of 6.3s. to 8s. 6d. ? 14. What is the ratio of 5.6 to .56? Ans. 63 85 Ans. 10. 15. What is the ratio of 19 lb. 5 oz. 8 pwt. to 25 lb. 11 oz. 4 pwt.? 16. What is the inverse ratio of 12 to 16? 17. What is the inverse ratio of to? 18. What is the inverse ratio of 53 to 174? 19. If the consequent is 16 and the ratio 22, the antecedent? Ans. 3. Ans. 14. Ans. 15. Ans. 3. what is Ans. 364. Ans. 45. 20. If the antecedent is 14.5 and the ratio 3, what is the consequent ? 21. If the consequent is 7 and the ratio 3, what is the antecedent? Ans. 22. If the antecedent is and the ratio, what is the consequent ? 23. What is the ratio of 84 to 60? 24. What is the ratio of 1 to 26? 3 Ans. 33. Ans. 217. Ans. 5 21 25. What is the ratio of 7 to 21? Ans.. 28. Find the reciprocal of the ratio of 3 qt. to 43 gal. 29. If the antecedent is 15 and the ratio, what is the consequent ? Ans. 183. 30. If the consequent is 31 and the ratio 7, what is the antecedent? Ans. 223. 31. If the antecedent is of and the consequent .75, what is the ratio? 32. If the consequent is $6.12 and the ratio 25, what is the antecedent? Ans. $153.125. PRAC. AR. - - 21 PROPORTION. 380. Proportion is an equality of ratios. Thus, the ratios 8:4 and 12:6 each being equal to 2, form a proportion. 381. Proportion is indicated in two ways: 1. By a double colon placed between the two ratios. 2. By the sign of equality placed between the two ratios. Thus, 2:54:10 2:5= : 4:10. 382. Since each ratio consists of two terms, every proportion must consist of at least four terms. Each ratio is called a couplet and each term a proportional. 383. The Antecedents of a proportion are the first and third terms, that is, the antecedents of its ratios. The Consequents of a proportion are the second and fourth terms, or the consequents of its ratios. 384. The Extremes are the first and fourth terms. The Means are the second and third terms. 385. Three numbers may be in proportion when the first is to the second as the second is to the third. Thus, the numbers 3, 9, and 27 are in proportion since 3:9::9:27, the ratio of each couplet being }. In such a proportion the second term is said to be a mean proportional between the other two. 386. In every proportion the product of the extremes is equal to the product of the means. Thus, in the proportion 3:5::6:10 we have 3 × 10 − 5 × 6. 387. Four numbers that are proportional in the direct order are proportional by inversion, and also by alternation, or by inverting the means. Thus, the proportion 2:3::6:9, by inversion becomes 3:2:: 9:6, and by alternation 2:6 :: :3:9. 388. From the preceding principles and illustrations, it follows that any three terms of a proportion being given, the fourth may readily be found by the following rule: RULE. Or, -I. Divide the product of the extremes by one of the means, and the quotient will be the other mean. II. Divide the product of the means by one of the extremes, and the quotient will be the other extreme. EXAMPLES. Find the term not given in each of the following proportions: 1. 48:20:( ):50. 2. 42:70::3:( ). 3. ( ): 300::20:100. 4. 1:( ):7: 84. 5. 48 yd.:( )::$67.25: $201.75. 6. 3 lb. 12 oz. :( ):: $3.50: $10.50. Ans. 120. Ans. 6. Ans. 12. Ans. 144 yd. Ans. 11 lb. 4 oz. 7. ( ): $38.25::8 bu. 2 pk. :76 bu. 2 pk. 8. 41:381():761. Ans. $4.25. |
