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. 4.

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 18° to 45' 30"?

Ans. 11. 3

14. What is the ratio of

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

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 31 and the ratio 7, what is the antecedent? Ans. 13 13. 19. If the antecedent be of and the consequent .75, what is the ratio?

20. If the conséquent be $6.123 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 proportional.

423. Proportion is indicated in three ways:

1st. By a double colon placed between the two ratios; thus, 3:4::9:12 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, 3:4 = 9:12 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, 12 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 3, or 11.

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 1 = 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,

624 × 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:

3. 41 yd. (?): $9.75 $29.25.

4. (?) 21 A. 3 R. 20 P.:: $1260: $750.

:

Ans. 65. Ans. 197 A.

Ans. 13 yd.

CAUSE AND EFFECT.

429. Every question in proportion may be considered as a comparison of two causes and two effects. Thus, if 3 dollars as a cause will buy 12 pounds as an effect, 6 dollars as a cause will buy 24 pounds as an effect. Or, if 5 horses as a cause consume 10 tons as an effect, 15 horses as a cause will consume 30 tons as an effect.

Causes and effects in proportion are of two kinds-simple and compound.

430. A Simple Cause or Effect contains but one element; as price, quantity, cost, time, distance, or any single factor used as a term in proportion.

431. A Compound Cause or Effect is the product of two or more elements; as the number of workmen taken in connection with the time employed, length taken in connection with breadth and depth, capital considered with reference to the time employed, etc.

432. Since like causes will always be connected with like effects, every question in proportion must give one of the following

statements:

1st Cause

: 2d Cause

= 1st Effect : 2d Effect.

1st Effect : 2d Effect = 1st Cause : 2d Cause.

in which the two causes or the two effects forming one couplet, must be like numbers and of the same denomination.

Considering all the terms of a proportion as abstract numbers,

we may say that

1st Cause : 1st Effect

But as ratio is the result of comparing two numbers or things of the same kind, (417), the first form is regarded as the most natural and philosophical.

SIMPLE PROPORTION.

433. Simple Proportion is an equality of two simple ratios, and consists of four terms.

Questions in simple proportion involve only simple causes and simple effects.

FIRST METHOD.

1. If $8 will buy 36 yards of velvet, how many yards may be bought for $12?

ANALYSIS. The required term in this example is an effect; and the statement is, $8 is to $12 as 36 yards is to (?), or how many yards. Dividing 12 x 36, the product of the means, by 8, the given extreme, we have (?)

2. If 6 horses will draw 10 tons, how many horses will draw

RULE. I. Arrange the terms in the statement so that the causes shall compose one couplet, and the effects the other, putting (?) in the place of the required term.

II. If the required term be an extreme, divide the product of the means by the given extreme; if the required term be a mean, vide the product of the extremes by the given mean.

NOTES. 1. If the terms of any couplet be of different denominations, they must be reduced to the same unit value.

2. If the odd term be a compound number, it must be reduced either to its lowest unit, or to a fraction or a decimal of its highest unit.

3. If the divisor and dividend contain one or more factors common to both, they should be canceled. If any of the terms of a proportion contain mixed