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8. What is the ratio of 15 lbs. to 3 lbs. ? Of 21 lbs. Of 27 yds. to 9 yds.?

to 7 lbs? Of 35 bu. to 7 bu.?

9. What ratio is £1 to 10s.?

Note.-£1 is 20s. The question then is this: what is the ratio of 20s. to 10s.? Ans. 2.

10. What is the ratio of £2 to 5s.? Of £3 to 12s.? 11. What is the ratio of 2 yds. to 4 yds.? Of 21 yds. to 34 yds.?

308. Direct ratio is that which arises from dividing the antecedent by the consequent, as in Art. 305.

309. Inverse or reciprocal ratio, is the ratio of the reciprocals of two numbers. (Art. 280. Def. 11.) Thus, the direct ratio of 9 to 3, is 9: 3, or; the reciprocal ratio is, or÷}=}; (Art. 139;) that is, the consequent 3 is divided by the antecedent 9. Hence,

A reciprocal ratio is expressed by inverting the fraction which expresses the direct ratio; or when the notation is by points, by inverting the order of the terms. Thus, 8 is to 4, inversely, as 4 to 8.

310. Compound ratio is the ratio of the products of the corresponding terms of two or more simple ratios. Thus, The simple ratio of

And 66 " of

9:

8:

3 is 3;
4 is 2;

The ratio compounded of these is 72 : 12=6.

OBS. Compound ratio is of the same nature as any other ratio. The term is used to denote the origin of the ratio in particular cases.

311. From the definition of ratio and the mode of expressing it in the form of a fraction, it is obvious that the ratio of two numbers is the same as the value of a fraction whose numerator and denominator are respectively equal to the antecedent and consequent of the giv

QUEST.-308. What is direct or simple ratio? 309. What is inverse or reciprocal ratio? How is a reciprocal ratio expressed by a fraction? How by points? 310. What is compound ratio? Obs. Does it differ in its nature from other ratios? 311. What in effect is the ratio of two numbers?

en couplet; for each is the quotient of the numerator divided by the denominator. (Arts. 305, 110.)

Note.-Hence, from the principles of fractions already established, we may easily deduce the following truths respecting ratios.

312. To multiply the antecedent of a couplet by any number, multiplies the ratio by that number; and to divide the antecedent, divides the ratio: for, multiplying the numerator multiplies the value of the fraction by that number, and dividing the numerator divides the value. (Arts. 111, 112.) Thus, the ratio of The ratio of And 66

16: 4 is 4;

16×24 is 8, which equals 4×2; 162 4 is 2, 66

:

66

4÷2.

313. To multiply the consequent of a couplet by any number, divides the ratio by that number; and to divide the consequent multiplies the ratio: for, multiplying the denominator divides the value of the fraction by that number, and dividing the denominator multiplies the value. (Arts. 113, 114.)

Thus the ratio of 16: 4

The

And

66

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is 4;

16: 4×2 is 2, which equals 4÷2; 16:42 is 8, which equals 4×2.

314. To multiply or divide both the antecedent and consequent of a couplet by the same number, does not alter the ratio: for, multiplying or dividing both the numerator and denominator by the same number, does not alter the value of the fraction. (Art. 116.) is 3;

Thus, the ratio of

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12:4
12×2: 4x2 is 3;
12 24 2 is 3.

-

315. If the two numbers compared are equal, the ratio is a unit or 1: for, if the numerator and denomina

QUEST.-312. What is the effect of multiplying the antecedent of a couplet by any number? Of dividing the antecedent? How does this appear? 313. What is the effect of multiplying the consequent by any number? Of dividing the consequent? 314. What is the effect of multiplying or dividing both the antecedent and consequent by the same number?

tor are equal, the value of the fraction is a unit or 1. (Art. 117.) Thus the ratio of 6×2: 12 is 1; for the value of 11. (Art. 121.)

OBS. This is called a ratio of equality.

316. If the antecedent of a couplet is greater than the consequent, the ratio is greater than a unit: for, if the numerator is greater than the denominator, the value of the fraction is greater than 1. (Art. 117.) Thus the ratio of 12 4 is 3.

OBS. This is called a ratio of greater inequality.

317. If the antecedent is less than the consequent, the ratio is less than a unit: for, if the numerator is less than the denominator, the value of the fraction is less than 1. (Art. 117.) Thus, the ratio of 3: 6 is 3, or 1 ; for (Art. 120.)

OBS. This is called a ratio of less inequality.

11. What is the direct ratio of 3:9, expressed in the lowest terms? What the inverse ratio?

Ans. ; and ÷1=3. (Arts. 308, 309.) 12. What is the inverse ratio of 4 to 12? Of 9 to 24? Of 21 to 25 ? Of 40 to 56 ?

13. What is the direct ratio of 15s. to £2? 6d. to £1? Of £2, 10s. to £3, 5s.?

Of 6 to 18?

Of 13s.

14. What is the direct ratio of 6 inches to 3 feet? 15. What is the direct ratio of 15 oz. to 1 cwt.?

PROPORTION.

318. Proportion is an equality of ratios. Thus, the two ratios 6:3 and 4: 2 form a proportion; for =1, the ratio of each being 2.

QUEST.-315. When the two numbers compared are equal, what is the ratio? Obs. What is it called? 316. When the antecedent is greater than the consequent, what is the ratio? Obs. What is it called? 317. If the antecedent is less than the consequent, what is the ratio? Obs. What is it called? 318. What is proportion?

OBS. The terms of the two couplets, that is, the numbers of which the proportion is composed, are called proportionals.

319. Proportion may be expressed in two ways. First, by the sign of equality (=) placed between the two ratios.

Second, by four points or a double colon (: :) placed between the two ratios.

Thus, each of the expressions, 12: 6=4 : 2, and 12 6 4 2, is a proportion, one being equivalent to the other.

OBS. The latter expression is read, "the ratio of 12 to 6 equals the ratio of 4 to 2," or simply, " 12 is to 6 as 4 is to 2."

320. The number of terms in a proportion must at least be four, for the equality is between the ratios of two couplets, and each couplet must have an antecedent and a consequent. (Art. 306.) There may, however, be a proportion formed from three numbers, for one of the numbers may be repeated so as to form two terms. Thus the numbers 8, 4, and 2, are proportional; for the ratio of 8: 4 4:2. It will be seen that 4 is the consequent in the first couplet, and the antecedent in the last. It is therefore a mean proportional between 8 and 2.

OBS. 1. In this case, the number repeated is called the middle term or mean proportional between the other two numbers.

The last term is called a third proportional to the other two numbers. Thus 2 is a third proportional to 8 and 4.

2. Care must be taken not to confound proportion with ratio. (Arts. 305, 318.) In a simple ratio there are but two terms, an antecedent and a consequent; whereas in a proportion there must at least be four terms, or two couplets.

Again, one ratio may be greater or less than another; the ratio of 9 to 3 is greater than the ratio of 8 to 4, and less than that of 18 to 2. One proportion, on the other hand, cannot be greater or less than another; for equality does not admit of degrees.

QUEST. Obs. What are the numbers of which a proportion is composed, called? 319. In how many ways is proportion expressed? What is the first? The second? 320. How many terms must there be in a proportion? Why? Can a proportion be formed of three numbers? How? Will there be four terms in it? Obs. What is the number repeated called? What is the last term called in such a case? What is the difference between proportion and ratio?

321. The first and last terms of a proportion are called the extremes; the other two, the means.

OBS. Homologous terms are either the two antecedents, or the two consequents.

Analogous terms are the antecedent and consequent of the same couplet.

322. Direct proportion is an equality between two direct ratios. Thus 12:49:3 is a direct propor

tion.

OBS. In a direct proportion, the first term has the same ratio to the second, as the third has to the fourth.

323. Inverse or reciprocal proportion is an equality between a direct and a reciprocal ratio. Thus 8:4::: ; or 8 is to 4, reciprocally, as 3 is to 6.

OBS. In a reciprocal or inverse proportion the first term has the same ratio to the second, as the fourth has to the third.

=

Thus

324. If four numbers are in proportion, the product of the extremes is equal to the product of the means. Thus 8:46:3 is a proportion: for. (Art. 318.) Now 8x3=4×6.

Again, 12:6::: is a proportion. (Art. 323.)

And

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OBS. 1. The truth of this proposition may also be illustrated in the following manner:

We have seen that 2:3::6:9. (Art. 318.)

For obviously

3. (Art. 120.) Now,

QUEST.-321. Which terms are the extremes? Which the means? Obs. What are homologous terms? Analogous terms? 222. What is direct proportion? Obs. In direct proportion what ratio has the first term to the second? 323. What is inverse proportion? Obs. What ratio has the first term to the second in this case? 324. If four numbers are proportional, what is the product of the extremes equal to? Obs. If the product of the extremes is equal to the product of the means, what is true of the four numbers? If the products are not equal, what is true of the numbers?

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