Either system, rigidly adhered to, is correct; but the English, being considered the more simple and natural, is adopted in this work.

240. An inverse or reciprocal ratio of any two quantities is the ratio of their reciprocals (106); thus, the direct ratio of 6 to 3 is 6:3 = 6 ÷ 3, and the reciprocal ratio of 6 to 3 is J:}=} ÷ }=} × 1 = 13÷6=3:6; .. any direct ratio by the English method is a reciprocal ratio by the French, and vice versa.

241. If the antecedent equals the consequent, the ratio is a unit, and is called a ratio of equality; thus, 5 : 5 = 1, is a ratio of equality.

242. If the antecedent is greater than the consequent, the ratio is more than a unit, and is called a ratio of greater inequality; thus, 12: 4 = 3, is a ratio of greater inequality.

243. If the antecedent is less than the consequent, the ratio is less than a unit, and is called a ratio of less inequality; thus, 2:10= , is a ratio of less inequality.

244. The antecedent and consequent being a dividend and divisor, it follows that any operations on them will affect the value of the ratio just as like operations on the dividend and divisor will affect the quotient; or as like operations on the numerator and denominator of a fraction will affect the value of the fraction; .*.,

(a) If the antecedent be multiplied by any number, the ratio is multiplied by the same (59, a); thus,

12:3 4, but 2 X 12:3 = 2 X 4; and
26, but 2 X 2: 62 X §.

(b) If the antecedent be divided, the ratio is divided (59, b);

(c) If the consequent be multiplied, the ratio is divided (59, c); thus,

(d) If the consequent be divided, the ratio is multiplied (59, d);

thus,

18: 6 = 3, but 18: 6-2 2 X 3; and

2:10= , but 2:10÷ 2 =

2 X 1.

(e) If the terms of the ratio are both multiplied or both divided by the same number, the ratio is not changed (60, Cor. and 61, Cor.); thus,

12:3= 4, and 5 × 12 : 5 × 3 = 4; also,
5, and 20 2:4÷ 2 = 5.

20:4

245. The ratio of two fractions that have a common denominator is the same as the ratio of their numerators; thus, 20:25 6: 3, since multiplying both terms by 20 does not alter the ratio.

=

246. The direct ratio of two fractions that have a common numerator is the inverse ratio of their denominators; thus, §: 2 = 1:12 = 12:72 12:6; for, first, we divide the terms by 5 (244, e), then reduce them to a common denominator, and, finally multiply them by 72 (244, e).

247. The antecedent, consequent and ratio are so related to each other, that, if either two of them be given, the other may be found; thus, in 12 : 3 = 4, we have

antecedent consequent ratio,

248. When there is but one antecedent and one consequent, the ratio is said to be simple; thus, 15: 5 = 3, is a simple ratio.

249. When the corresponding terms of two or more simple ratios are multiplied together, the resulting ratio is said to be compound;

A compound ratio is always equal to the product of the simple ratios of which it is compounded.

NOTE..

A compound ratio is not different in its nature from a simple ratio, but it is called compound merely to denote its origin.

(a) If each of the terms of a ratio be squared (94, b, Note 1), the compound ratio so formed is called a duplicate ratio, and is equal to the square of the simple ratio; thus, 62: 2232, i. e. 36: 4 9, is the duplicate of 6: 2 3.

(b) If each term be cubed (94, b, Note 1), the result is called a triplicate ratio, and is equal to the cube of the simple ratio; thus, 48: 23 23, i. e. 64: 88, is the triplicate ratio of 4:2= 2.

(c) A similar result will be obtained by raising the terms of a ratio to any like powers, and also by taking any like roots (94, b, Note 1).

(d) If the square root of each term be taken, the resulting ratio is called the sub-duplicate ratio; if the cube root, the subtriplicate ratio; etc.

(e) A double or duple ratio is twice a given ratio, and is obtained by multiplying the antecedent or by dividing the consequent by 2 (244, a and d); a triple ratio is three times a ratio, and is obtained by multiplying the antecedent or dividing the consequent by 3; etc. Let not the pupil confound duple, triple, quadruple, etc., with duplicate, triplicate, quadruplicate, etc.

(f) The half, third, fourth, etc. of a ratio are called the subduple, sub-triple, sub-quadruple ratio, etc.

What operations upon the terms of a ratio will produce the sub-duple, sub-triple, sub-quadruple ratio, etc.?

§ 26. PROPORTION.

250. PROPORTION is an equality of ratios.

251. At least 2 ratios and.. 4 terms are required to form a proportion.

252. The proportionality of the four numbers, 8, 4, 6 and 3, may be indicated thus,

8:46:3,

which is read, 8 is to 4 as 6 is to 3; or, as 8 is to 4 so is 6 to 3.

Any 4 numbers are proportional, and may be written and read in like manner, if the quotient of the 1st divided by the 2d is equal to the quotient of the 3d divided by the 4th.

253. The 1st and 4th terms are called extremes, and the 2d and 3d, means. The 1st and 3d are the antecedents of the two ratios and the 2d and 4th are the consequents.

254. In a proportion the product of the extremes is equal to the product of the means; thus, in 8: 4 :: 6:3, we have 8 × 3 = 4 X 6; for, from the definition of proportion, we have of , and, if each member of this equation (7, Note) be multiplied by the product of the denominators, we have 8 × 3 = 4 X 6.

This is one of the easiest tests of proportionality.

255. Any changes in the order or magnitude of the terms of a proportion which do not affect the EQUALITY of the ratios will not destroy the proportionality. These changes are very numerous; some of them will be noticed in the Supplement.

256. Since the product of the extremes is equal to the product of the means, any one term may be found when the other three are given; for the product of the extremes divided by either mean will give the other mean, and the product of the means divided by either extreme will give the other extreme.

257. When three numbers are in proportion, as, e. g., 46: 69, the 2d is called a mean proportional between the 1st and 3d, and the 3d, a third proportional to the 1st and 2d.

(a) A mean proportional between two numbers may be found by multiplying the two given numbers together and then resolving the product into two equal factors; thus, the mean proportional to 2 and 8 is 4, for 2 X 8 164 X 4 ; .. 2 : 4 ::

4:8.

(b) A third proportional to two numbers may be found by dividing the square of the 2d by the 1st. The third proportional to 5 and 10 is 20; for 1025 20;.. 5: 10 :: 10: 20.

258. In all examples in Simple Proportion there are three numbers given to find a fourth; .. Proportion is often called the Rule of Three.

Two of the three given numbers must be of the same kind, and the other is of the same kind as the answer.

Ex. 1. If 3 men build 6 rods of wall in a day, how many rods will 5 men build?

This example may be analyzed as follows:- -If 3 men build 6 rods, 1 man will build of 6 rods, i. e. 2 rods; and if one man build 2 rods, 5 men will build 5 times 2 rods, i. e. 10 rods, Ans.; but, to solve it by proportion, we say that the given number of rods has the same ratio to the required number of rods that 3 men have to 5 men: thus,

3 men : 5 men :: 6 rods: required number of rods. Now, since the means and 1st extreme are given, we find the 2d extreme by dividing the product of the means by the given extreme (256); thus,

6 X 530 and 30 ÷ 3 = 10 Ans. as before.

Every example in Simple Proportion is solved in like manner. Hence,

RULE.

Write that given number which is of the same kind as the required answer, for the third term; consider whether the