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364. Ratio is the comparison with each other of two numbers of the same kind. It is of two kinds-arithmetical and geometrical.

365. Arithmetical Ratio is the difference of the two numbers.

366. Geometrical Ratio is the quotient of one number divided by the other.

367. When we use the word ratio alone, it implies geometrical ratio, and is expressed by the quotient arising from dividing one number by the other. Thus, the ratio of 4 to 8 is 2, of 10 to 5 is, etc.

368. Ratio is indicated in two ways.

1st. By placing two points between the numbers compared, writing the divisor before and the dividend after the points. Thus, the ratio of 5 to 7 is written 5:7; the ratio of 9 to 4 is written 9: 4.

2d. In the form of a fraction; thus, the ratio of 9 to 3 is; the ratio of 4 to 6 is .

369. The Terms are the two numbers compared.

370. The Antecedent is the first term.

371. The Consequent is the second term.

372. No comparison of two numbers can be fully explained but by instituting another comparison; thus, the

It is thought best to omit the questions at the bottom of the pages in the remaining part of this work, leaving the teacher to use such as may be deemed appropriate.

comparison or relation of 4 to 8 cannot be fully expressed by 2, nor of 8 to 4 by 1. If the question were asked, what relation 4 bears to 8, or 8 to 4, in respect to magnitude, the answer 2, or 1, would not be complete nor correct. But if we make unity the standard of comparison, and use it as one of the terms in illustrating the relation of the two numbers, and say that the ratio or relation of 4 to 8 is the same as 1 to 2, or the ratio of 8 to 4 is the same as 1 to 1, unity in both cases being the standard of comparison, then the whole meaning is conveyed.

373. A Direct Ratio arises from dividing the consequent by the antecedent.

374. An Inverse or Reciprocal Ratio is obtained by dividing the antecedent by the consequent. Thus, the direct ratio of 5 to 15 is 15 = 3; and the inverse ratio of 5 to 15 is.

375. A Simple Ratio consists of a single couplet; as 3:12.

376. A Compound Ratio is the product of two or more simple ratios. Thus, the compound ratio formed from the simple ratios of 3:6 and 8:2 is x = 3 x 8:6 x 2 = }=}

377. In comparing numbers with each other, they must be of the same kind, and of the same denomination.

378. The ratio of two fractions is obtained by dividing the second by the first; or by reducing them to a common denominator, when they are to each other as their numerators. Thus, the ratio of is ÷ 18 = 2, which is & 7% the same as the ratio of the numerator 3 to the numerator 6 of the equivalent fractions and.

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Since the antecedent is a divisor and the consequent a dividend, any change in either or both terms will be governed by the general principles of division, (87.) We have only to substitute the terms antecedent, consequent, and ratio, for divisor, dividend, and quotient, and these principles become

GENERAL PRINCIPLES OF RATIO.

PRIN. I. Multiplying the consequent multiplies the ratio; dividing the consequent divides the ratio.

PRIN. II. Multiplying the antecedent divides the ratio; dividing the antecedent multiplies the ratio.

PRIN. III. Multiplying or dividing both antecedent and consequent by the same number does not alter the ratio. These three principles may be embraced in one

GENERAL LAW.

A change in the CONSEQUENT produces a LIKE change in the ratio; but a change in the ANTECEDENT produces an OPPOSITE change in the ratio.

379. Since the ratio of two numbers is equal to the consequent divided by the antecedent, it follows, that

1. The antecedent is equal to the consequent divided by the ratio; and that,

2. The consequent is equal to the antecedent multiplied by the ratio.

EXAMPLES FOR PRACTICE.

1. What part of 9 is 3 ?

; or, 9:3 as 1:1, that is, 9 has the same ratio to 3

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12. What is the ratio of 3 gal. to 2 qt. 1 pt? Ans. .

13. What is the ratio of 6.3s. to 8s. 6d. ?

14. What is the ratio of 5.6 to .56?

Ans. 1.
Ans.

15. What is the ratio of 19 lbs. 5 oz. 8 pwts. to 25 lbs. 11 oz. 4 pwts.?

16. What is the inverse ratio of 12 to 16?

17. What is the inverse ration of & to ‡ ?

Ans. 11.

Ans. 3. Ans.

Ans.

18. What is the inverse ratio of 5 to 171? 19. If the consequent be 16 and the ratio 24, what is the antecedent? Ans. 7.

20. If the antecedent be 14.5 and the ratio 3, what is the consequent ? Ans. 43.5. 21. If the consequent be and the ratio, what is the antecedent? Ans. 1.

22. If the antecedent be and the ratio, what is the consequent ? Ans. o.

PROPORTION.

380. Proportion is an equality of ratios. Thus, the ratios 6 4 and 12: 8, each being equal to , form a proportion. 381. Proportion is indicated in two ways:

1st. By a double colon placed between the two ratios; thus, 2:5: 4:10.

2d. By the sign of equality placed between the two ratios; thus, 2:5 = 4:10.

382. Since each ratio consists of two terms, every proportion must consist of at least four terms.

383. The Extremes are the first and fourth terms. 384. 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 3.

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 3:56:10 we have 3 × 10 − 5 × 6.

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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: 6:39.

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. I. Divide the product of the extremes by one of the means, and the quotient will be the other mean. Or,

II. Divide the product of the means by one of the extremes, and the quotient will be the other extreme.

EXAMPLES FOR PRACTICE.

Find the term not given in each of the following proportions:

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SIMPLE PROPORTION.

389. Simple Proportion is an equality of two simple ratios, and consists of four terms, any three of which being given, the fourth may readily be found.

390. Every question in simple proportion involves the principle of cause and effect.

391. Causes may be regarded as action, of whatever kind, the producer, the consumer, men, animals, time, distance, weight, goods bought or sold, money at interest, etc.

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