482. 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. 186, 187.)

Thus, the ratio of

The ratio of

And 66

16: 4 is 4;

16×24 is 8, which equals 4×2;

16÷2:4 is 2, which equals 4÷2.

OBS. With a given consequent the greater the antecedent, the greater the ratio; and on the other hand, the greater the ratio, the greater the antecelent. (Art. 187. Obs.)

483. 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. 188, 189.)

Thus, the ratio of 16:4 is 4;

The
And

66

66

16:4X2 is 2, which equals 4÷2;
16:42 is 8, which equals 4x2.

OBS. With a given antecedent, the greater the consequent, the less the ratio; and the greater the ratio, the less the consequent. (Art. 189. Obs.)

484. 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. 191.) Thus, the ratio of

12:4 is 3;

12X2: 4X2 is 3;

12 2:42 is 3.

485. If the two numbers compared are equal, the ratio is a unit or 1, and is called a ratio of equality. Thus, the ratio of 6×2:12 is 1; for the value of 12=1. (Art. 196.)

QUEST.-482. What is the effect of multiplying the antecedent of a complet by any number? Of dividing the antecedent? 483. What is the effect of multiplying the consequent by any number? Of dividing the consequent? Why? 484. What is the effect of mul tiplying or div ding both the antecedent and consequent by the same number? Why? 485. When the two numbers compared are equal, what is the ratio? What is it called?

486. If the antecedent of a couplet is greater than the consequent, the ratio is greater than a unit, and is called a ratio of greater inequality. Thus, the ratio of 12: 4 is 3; for the value of 23. (Art. 196.)

487. If the antecedent is less than the consequent, the ratio is less than a unit, and is called a ratio of less inequality. Thus. the ratio of 3: 6 is, or ; for 3. (Art. 195.)

OBS. 1. The direct ratio of two fractions which have a common numeralar, is the same as the reciprocal ratio of their denominators. Thus, the ratio of is the same as :, or 8: 4.

2. The ratio of two fractions which have a common denominator, is the same as the ratio of their numerators. Thus, the ratio of : 4 is the same as that of 8:4, viz: 2. Hence,

488. The ratio of any two fractions may be expressed in whole numbers, by reducing them to a common denominator, and then using the numerators for the terms of the ratio. (Art. 484.) Thus, the ratio of to is the same as

33. What is the direct ratio of 4 to 12, terms?

2, or 6: 2.

expressed in the lowest
Ans..

Ans. 1÷=3.
Of 9 to 63?

Of 250 to 32?
Of 28 to 7?

34. What is the inverse ratio of 4 to 12? 35. What is the direct ratio of 64 to 8? 36. What is the direct ratio of 84 to 21? 37. What is the inverse ratio of 4 to 16? 38. What is the inverse ratio of 42 to 6? 39. Which is the greater, the ratio of 63 to 9, or that of 72 to 8? 40. Which is the greater, the ratio of 86 to 240, or that of 45 to 72 ?

Of 8 to 72?

41. Which is the greater, the ratio of 120 to 85, or that of 240 to 170?

42. Which is the greater, the ratio of 624 to 416, or that of 936 to 560 ?

43. Is the ratio of 5X6 to 24, a ratio of greater, or less inequality?

QUEST.-486. When the antecedent is greater than the consequent, what is the ratio oalled? 487. If the antecedent is less than the consequent, what is the ratio called 7 488. How may the ratio of two fractions be expressed in whole numbers 7

44. Is the ratio of 6×9 to 7×8, a ratio of greater, or less inequality?

45. Is the ratio of 2×4×16 to 4×32 a ratio of greater, or less inequality?

46. What is the ratio compounded of the ratios of 5 to 3, and 12 to 4 ?

47. What is the ratio compounded of 8: 10, and 20 : 16 ? 48. What is the ratio compounded of 3 : 8, and 10:5? 49. What is the ratio compounded of 18: 20, and 30: 40? 50. What is the ratio compounded of 35: 40, and 60:75, and 21 to 19?

51. What is the ratio compounded of 60: 40, and 12:24, and 25:30?

489. In a series of ratios, if the consequent of each preceding couplet is the antecedent of the following one, the ratio of the first antecedent to the last consequent, is equal to that compounded of all the intervening ratios.

Thus, in the series of ratios 3:4

4:7

7:16

the ratio of 3 to 16, is equal to that which is compounded of the ratios of 3: 4, of 4:7, and 7:16; for, the compound 3×4X7 3

ratio is

4×7×16 ̄16, or 3:16.

490. If to or from the terms of any couplet, two other numbers having the same ratio be added or subtracted, the sums or remainders will also have the same ratio. (Thomson's Legendre, B. III., Prop. 1, 2.) Thus, the ratio of 12 : 3 is the same as that of 20:5. And the ratio of the sum of the antecedents 12-+20 to the sum of the consequents 3+5, is the same as the ratio of either couplet. That is,

12+20 12 20
3+5
3 5

So also the ratio of the difference of the antecedents, to the difference of the consequents, is the same. That is,

20-12 12 20 20-12:5—3::12:3=20:5, or 5—3—3

91. If in several couplets the ratios are equal, the sum of he antecedents has the same ratio to the sum of all the conseEs, which any one of the antecedents has to its consequent.

erefore the ratio of (12+15+18): (4+5+6)=3.

. 1. A ratio of greater inequality is diminished by adding the same numboth terms. Thus, the ratio of 8:2, is 4; and the ratio of 8+4:2+

A ratio of less inequality is increased by adding the same number to both ms. Thus, the ratio of 2:8 is 4, and the ratio of 2+16:8+16 is 1.

PROPORTION.

92. PROPORTION is an equality of ratios. Thus, the two 6:3 and 4: 2 form a proportion; for, the ratio of each

2.

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

93. Proportion may be expressed in two ways.

rst, by the sign of equality (=) placed between the two

.

cond, by four points (: :) placed between the two ratios.

us, each of the expressions, 12:6=4:2, and 12:6::4:2, roportion, one being equivalent to the other. The latter exion is read, “the ratio of 12 to 6 equals the ratio of 4 to 2," mply, "12 is to 6 as 4 is to 2."

. The sign (::) is said to be derived from the sign of equality, the four being merely the extremities of the lines.

94. The number of terms in a proportion must at least be for the equality is between the ratios of two couplets, and couplet must have an antecedent and a consequent. (Art. 476.) ere may, however, be a proportion formed from three numfor one of the numbers may be repeated so as to form two

T.-492. What is Proportion? 493. How many ways is proportion expressed? s the first? The second? 494. How many terms must there be in a proportion ? Can a proportion be formed of three numbers? How?

terms. Thus, the numbers 8, 4, and 2, are proportional; for the ratio of 8:4=4:2. It will be seen that 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. 474, 492.) 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 raio 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.

495. The first and last terms of a proportion are called the cxtremes; 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.

496. Direct proportion is an equality between two direct ratios. Thus, 12:4:: 9:3 is a direct proportion.

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

497. 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.

498. If four numbers are proportional, the product of the extremes is equal to the product of the means.

Thus, 8:4:: 6:3 is

a proportion; for &=3, (Art. 492,) and 8×3=4×6.

QUEST.-Obs. What is the number repeated called? What is the last term cal'ed in such a case? What is the difference between proportion and ratio? 495. Which terms are the extremes? Which the means? Obs. What are homologous terms? Analogous terins? 496. What is direct proportion? Obs. In direct proportion what ratio has the first term to the second? 497: What is inverse proportion? Obs. What ratio has the first term to the second in this case? 498. If four numbers are proportional, what is the prxduct of the extremes equal to ?