The Progressive Higher Arithmetic

2d. In the form of a fraction. Thus, the ratio of 8 to 24 is written 24; the ratio of 7 to 5 is 5.

412. The Terms of a ratio are the two numbers compared.

The Antecedent is the first term; and

The Consequent is the second term.

The two terms of a ratio taken together are called a couplet. 413. A Simple Ratio consists of a single couplet; as 5 15. 414. A Compound Ratio is the product of two or more simple ratios. Thus, from the two simple ratios, 5 : 16 and 8: 2, we

5:16 8: 2

may form the compound ratio 5×8:16×2, or 16 ×

415. The Reciprocal of a ratio is 1 divided by the ratio; or, which is the same thing, it is the antecedent divided by the consequent. Thus, the ratio of 7 to 9 is 7:9 or 2, and its reciprocal is 7.

NOTE. The quotient of the second term divided by the first is sometimes called a Direct Ratio, and the quotient of the first term divided by the second, an Inverse or Reciprocal Ratio.

416. One quantity is said to vary directly as another, when the two increase or decrease together in the same ratio; and one quantity is said to vary inversely as another, when one increases in the same ratio as the other decreases. Thus time varies directly as wages; that is, the greater the time the greater the wages, and the less the time the less the wages. Again, velocity varies inversely as the time, the distance being fixed; that is, in traveling a given distance, the greater the velocity the less the time, and the less the velocity the greater the time.

417. Ratio can exist only between like numbers, or between two quantities of the same kind. But of two unlike numbers or quantities, one may vary either directly or inversely as the other. Thus, cost varies directly as quantity, in the purchase of goods; time varies inversely as velocity, in the descent of falling bodies. In all cases of this kind, the quantities, though unlike in kind, have a mutual dependence, or sustain to each other the relation of cause and effect.

418. In the comparison of like numbers we observe,

I. If the numbers are simple, whether abstract or concrete, their ratio may be found directly by division.

II. If the numbers are compound, they must first be reduced to the same unit or denomination.

III. If the numbers are fractional, and have a common denominator, the fractions will be to each other as their numerators; if they have not a common denominator, their ratio may be found either directly by division, or by reducing them to a common denominator and comparing their numerators.

419. 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, (117). 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 conse quent by the same number does not alter the ratio.

420. These three principles may be embraced in one

GENERAL LAW.

A change in the consequent by multiplication or division produ ces a LIKE change in the ratio; but a change in the antecedent produces an OPPOSITE change in the ratio.

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

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

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

EXAMPLES FOR PRACTICE.

1. What part of 28 is 7?

1 has to .

; or, 28: 7 as 1 : ; that is, 28 has the same ratio to 7 that

Ans. .

Ans. .

5. What is the ratio of 7 6. What is the ratio of 7

7. What is the ratio of

to 26?

to 21? to?

8. What is the ratio of 1 mi. to 3 fur.?

9. What is the ratio of 1 wk. 3 da. 12 h. to 9 wk.? Ans. 6. 10. What is the ratio of 10 A. 1 R. 20 P. to 6 A. 2 R. 30 P.? 11. What is the ratio of 25 bu. 2 pk. 6 qt. to 40 bu. 4.5 pk.? 12. What is the ratio of 183° to 45' 30"?

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15. Find the reciprocal of the ratio of 42 to 28.

Ans. 11.

16. Find the reciprocal of the ratio of 3 qt. to 43 gal.

17. If the antecedent be 15 and the ratio, what is the consequent?

Ans. 12.

18. If the consequent be 34 and the ratio 7, what is the antecedent? Ans. 1.

19. If the antecedent be of and the consequent .75, what is the ratio?

20. If the consequent be $6.12 and the ratio 25, what is the antecedent? Ans. $.245.

21. If the ratio be and the antecedent, what is the consequent?

22. If the antecedent be 13 A. 3 R. 25 P. and the ratio, what is the consequent ? Ans. 6 A. 2 R. 10 P.

PROPORTION.

422. Proportion is an equality of ratios. Thus, the ratios 5:10 and 6: 12, each being equal to 2, form a proportion.

NOTE.When four numbers form a proportion, they are said to be propor

tional.

423. Proportion is indicated in three ways:

1st. By a double colon placed between the two ratios; thus, 3:49:12 expresses the proportion between the numbers 3, 4, 9, and 12, and is read, 3 is to 4 as 9 is to 12.

2d. By the sign of equality placed between two ratios; thus, 3:49:12 expresses proportion, and may be read as above, or, the ratio of 3 to 4 equals the ratio of 9 to 12.

3d. By employing the second method of indicating ratio; thus, indicates proportion, and may be read as either of the above forms.

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

The Extremes are the first and fourth terms; and

The Means are the second and third terms.

425. Three numbers are proportional when the first is to the second as the second is to the third. Thus, the numbers 4, 6, and 9 are proportional, since 4:6 6:9, the ratio of each couplet being, or 14.

426. When three numbers are proportional, the second term is called the Mean Proportional between the other two. 427. If we have any proportion, as

3: 154: 20,

Then, indicating this ratio by the second method, we have 15 = 20.

Reducing these fractions to a common denominator,

And since these two equal fractions have the same denominator, the numerator of the first, which is the product of the means, must be equal to the numerator of the second, which is the product of the extremes; or, 15 x 420 × 3. Hence,

I. In every proportion the product of the means equals the product of the extremes.

Again, take any three terms in proportion, as

4: 6-6:9

Then, since the product of the means equals the product of the extremes,

62 4 x 9. Hence,

II. The square of a mean proportional is equal to the product of the other two terms.

428. Since in every proportion the product of the means equals the product of the extremes, (427, I), it follows that, any three terms of a proportion being given, the fourth may 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.

The required term in an operation will be denoted by (?), which may be read "how many," or "how much."

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