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of the same kind in respect of magnitude, that is, the one being so many times or parts of a time greater or less than the other. Ratio is thus merely a short name for relative or comparative magnitude, and is expressed by the quotient of the former of the two numbers or quantities divided by the latter. 5 is the ratio or measure of the relative magnitude of 10 to 2, because 10:2=5, that is, 10 contains 2 five times,- it is five times as great as 2. Again, the fraction ? is the ratio of 7 to 9: 7 is Iths of 9, or 7 divided by 9 is 1 : the ratio in this case cannot be reduced to a whole number. 13 denotes the ratio of 11 to 8, for 11:-8==13.
So far as abstract numbers are concerned, it appears, therefore, that ratio has the same meaning as quotient. If quantities be compared, they must, as above stated, be of the same kind and denomination, otherwise no intelligible comparison can be made between them. It is impossible to form any idea of the relation which may exist in point of magnitude between such dissimilar quantities, for instance, as 6 pounds weight and 10 pounds sterling ; lbs. must be compared with lbs, and £s with £s; the denomination of the quantities must also be the same before one can be properly expressed as a fraction of the other and their ratio be determined. Shillings bear no ratio to pence, or bushels to gallons, but when each pair of quantities is reduced to a common denomination, its ratio becomes assignable by division, as in the case of abstract numbers. Thus, the ratio of 12 cwt. to 4 cwt.—of 12 feet to 4 feet,--of 12 hours to 4 hours, &c., &c., is denoted by the number 3, which forms the quotient of 12; 4, and expresses the magnitude of the former of any of these quantities as compared with that of the latter, irrespective of the nature or designation of the quantities themselves. It is essential to correct notions on the subject of ratio to keep in mind this truth,—that so long as two quantities admit of being compared in respect of magnitude, the ratio of one to the other, or the quotient of the first divided by the second, is always an abstract number, whole or fractional, which signifies how often the second is contained in the first, and has no reference whatever to the character or species of the quantities,—whether they both be pounds weight, pounds sterling, yards, gallons, or other concrete units. The term ratio, accordingly, when applied to the mutual relation of quantities of the same kind, has a more restricted meaning than the term quotient, which is used to denote not only how many times the divisor is contained in the dividend, but also as a name for the result that arises from taking any part, such as the half, fourth, seventh, &c., of a given quantity. 2 bushels, for instance, is the quotient of 10 bushels divided by the number 5,-it is the fifth part of 10 bushels ; but the quotient of 10 bushels divided by 5 bushels is the ratio or abstract number 2.
As a fraction represents the divison of its numerator by its denominator (see page 66), and as the ratio between two numbers or like quantities is the quotient of the first of these numbers, &c., divided by the second, it follows that every ratio may properly be expressed in the form of a fraction. This point, indeed, as has just been stated, is a necessary consequence of the definition of a ratio. But it is more generally the practice to designate a ratio by placing dots between the numbers or quantities compared ; thus, 7:9 implies the ratio of 7 to 9, and, therefore the division of 7 by 9, as otherwise represented by the sign + placed between the dividend and divisor (7 - 9)*
When any two fractions or ratios are equal, the equality constitutes what is called a proportion. Thus, since = %, or, by using the sign of division, 3:7=6 · 14, these four numbers are proportionals, or in proportion, the
* Au explanation of the signs used to denote ratio and proportion was givon at page 65.
first divided by the second, being equal to the third divided by the fourth. If we take away the line between the two points in the sign of division, it becomes : and if we retain merely the extremities of the lines in the sign of equality, it becomes ::, hence, 3 ; 7= 6; 14 may be written 3 : 7 :: 6 : 14, which is read, 3 is to 7 as 6 is to 14, or, as 3 is to 7 so is 6 to 14. This probably is the origin of the proportional points.* Sometimes, the sign of equality is employed instead of the four dots ::, and the proportion written, 3:7 = 6 : 14.
The first term of a ratio, is called an antecedent, and the second term, its consequent. So, in a proportion or an equality of two ratios, the terms of the first and second ratios are respectively antecedents and consequents to each other. Hence, the numerator of every fraction when regarded as a ratio, is an antecedent and the denominator, its consequent. Of the four numbers which constitute a proportion, the first and fourth terms are called the extremes, and the second and third, the means, that is, the first antecedent and second consequent are the extremes ; the first consequent and second antecedent, the means ; thus, in,
9 : 11 :: 45 : 55 9 and 55 are the extremes : 11 and 45 the means.
In a proportion, the product of the extremes is always equal to the product of the means.
9 : 5 :: 18 : 10
Therefore, 10 x 9=5 x 18 To prove that this is a general property of proportionals, and not true merely of the numbers in the instance above cited, let the two ratios which form the proportion be represented as fractions, then we have
5 x 18
Let these fractions be reduced to a common denominator as follows:
5x10 = 5x10 Here, the denominators being equal, it is seen that the numerators also are equal, for 9 x 10 or 90=5 x 18 or 90, but 9 x 10 is the product of the ex. tremes, and 5 x 18 the product of the means. The conclusion in this case is perfectly general ; as the process which leads to it has, it is obvious, no necessary connection with the particular numbers above employed.+
It follows from this proposition, that when the product of any two numbers equals the product of two others, the four numbers are proportionals; the « extremes" being the factors of one of such products, and the “ means " the factors of the other. Thus, since 4 x 9=12 x 3, if 4 and 9 be made the first and fourth terms, and 12 and 3 the second and third terms respectively of a proportional statement, it will be found on applying the test of equality of ratios, that the four numbers so arranged are proportionals, –
4 : 12 :: 3 : 9
Let these fractions be brought to a common denominator, and the equality of the ratios or proportionality of the numbers will at once be apparent. • See Ritchie's Principles of Geometry.
Officers who wish fully to understand the principle of the most important part of all arithmetic --that on which the rule of three is founded, -would do well to repeat the form of demonstration given in the text, on several sets of equal ratios or fractions.
Again, since the product of two or more numbers is the same in whatever order the factors are multiplied together, we have
(1.) 4 : 12 :: 3 : 9 or 4 =
9 : 3 :: 12 : 4 or j = Generally, when four numbers are proportional, the
(1.) 1st : 2nd :: 3rd : 4th
4th : 2nd :: 3rd : 1st
4th : 3rd :: 2nd : 1st It is plain that four additional proportions may be formed by placing the second ratio first in each of the preceding statements.
Thus, as in (1.) the 1st : : 2nd :: 3rd : 4th, then by merely changing the order of the ratios, we have, 3rd : 4th :: 1st : 2nd, and similarly for the others of the series.
The rule in this case, expressed generally, is, that four numbers which are proportional in any given order, will also be proportional when that order is varied, so that the same two terms go together either as extremes or means. If that condition be not observed, there will be no proportion between the numbers ; thus,
4 : 12 :: 3 : 9
whence, 12 : 9 :: 4 : 3 in which 12 and 3 are made extremes instead of means ; but it is not true that,
12 : 3 :: 9 : 4 for by this arrangement, the extremes consist of numbers which were respectively a mean and an extreme in the original proportion, and it is seen that the 1st x 4ih is not equal to the 2nd x 3rd.
As the product of the extremes equals the product of the means, it is possible, when any three terms of a proportion are given, to find the fourth by calculation, which merely amounts to saying, that when a dividend and divisor are given we know how to obtain the quotient. If the product of the extremes be divided by one of the means, the quotient will be the other mean; and if the product of the means be divided by one of the extremes, the quotient will be the other extreme. Suppose that the first term of the proportion, 4 : 12 :: 3 : 9 were missing, and we wished to supply it ; as we know that the product of the means should equal that of the extremes, we have 12 x 3 or 36=9x the absent term or the other extreme; and the question becomes, by what number must 9 be multiplied 80 that the product shall be 36 ? Evidently, by the quotient of 36 divided by 9, that is, 4.
Again, if the second term (12) were wanting, and the other three given, we should have the product of the two extremes and one mean from which to find the other mean; that is, we should have 36 given as dividend and 3 as divisor. The method of deducing the third and fourth terms is equally obvious. In every proportion, therefore, the 1st term—2nd x3rd ; the 2nd=1stxtth; the 3rd=1st x4th; and the 4th="
th_2nd x 3rd 4th of these formulæ, the last is the most useful and the most frequently applied, as it
is that by which we are enabled to resolve questions in the Rule of Three. Owing to its importance it may be well to show how the truth of the conclusion, that the 4th term of a proportion is equal to the product of the 2nd and 3rd divided by the 1st, can be established by independent reasoning based on the nature of ratios, no less than by the process of deriving it from the identity between the product of the means and that of the extremes. Let all but the fourth term of a proportion be given as follows,
2 : 3 :: 6 : required term or using the fractional form,
3 = required denominator. Now, whatever part or fraction 2 is of 3, the same fraction must 6 be of the required denominator. 2 is grds of 3 (page 66), therefore, 6 must also be rds of the denominator sought. As 6 is the įrds of a certain number, the half of 6 or 3, is 1 of it, and 3 times 3 or 9, is the whole of it, accordingly if we put 9 for the 4th term, we see that it satisfies both the tests by which the proportionality of four numbers is determined, for 2 x 9=3 X 6, and =%. As another example, let the statement be.
5 : 7 :: 11 : required term. Here, 11 must be the same fraction of the unknown term that 5 is of 7, that is, it must be şths of it. But if 11 be the çths of a number, ? is the 4th of it, and 7 times y or 4 is the number required. The complete proportion will therefore be
5 : 7 :: 11 : 4 Proof, 5x4=77=7x117
also, =*= 5 In each of these examples, the result has in reality been arrived at by the process of multiplying the second and third terms together and dividing the product by the first, as the formulæ directs.
It should be remarked that in strictness, the only proper mode of determining the fourth term of a proportion from the other three, is to multiply the third by the ratio of the second to the first. But this evidently amounts to the same as the multiplication of the third into the second, and the division of such product by the first :
11 x }(the ratio of 7 to 5) = 11x7 To supply the fourth term of a proportion is to find what is called a fourth proportional to the other three : thus, 20 is a fourth proportional to 6, 8, and 15, for 6 : 8 :: 15 : 20. When the second and third terms are the same, the fourth term is said to be a third proportional to the first and second, and is found by multiplying the second term into itself-squaring the second--and then dividing by the first :
12 x 12 8 : 12 :: 12 : 8 = 18 It is indifferent, however, whether we call 18 a third proportional to 8 and 12 are fourth proportional to 8, 12, and 12. In the former case, the middle term is said to be a mean proportional between the first and third ; 12 is a mean
proportional between 8 and 18, for 8 : 12 :: 12 : 18. As the square of the second is equal to the product of the first by the third-as 12 x 128 x 18 - it is plain that when any two numbers are given, a mean proportional between them may be found, by multiplying the numbers together and taking the square root of their product.* In the present instance 8 x 18 = 144, and the square root of 144 is 12.
The computation of mean proportionals is occasionally needed in the practice of gauging.
A ratio is not altered in value if both its terms be multiplied or divided by the same number,
2 : 3 as 4 : 6, or as 6 : 9, or as 8 : 12, &c., &c.,
and 8 : 12 as 2 : 3; or : *, &c. This property is proved in the same manner as in the case of fractions, which may be used to represent the ratios, (see page 67).
4= = =, &c., where the numerator and denominator of the first fraction are multiplied successively by 2, 3, 4, &c. It is in virtue of this property, that a proportion or the equality of two ratios is established, for in every proportion the third term is a certain number of times or parts of a time, the first and the fourth is an equal number of times or parts of a time the second that is,
6 : 8 :: 15 : 20
and 20 (the 4th term) is 24 times 8 (the 2nd term). Having given any ratio, therefore, we may form an infinite number of ratios which shall be equal to it, simply by multiplying or dividing both its terms by the same integer or fraction.
A proportion is not affected when its first and second terms, its third and fourth, its first and third, or its second and fourth terms, are multiplied or divided by the same number, thus,
6 : 8 :: 15 : 20 And as = 15, it is plain that both terms of or 18 may be multiplied or divided by the same number without affecting the equality of the fractions, and as the second and third terms may be made to exchange places, altering the proportion to 6 : 15 :: 8 : 20, or 1st : 3rd :: 2nd : 4th (page 108), it is evident that both terms of more may be similarly treated and the propor. tion still be preserved. It is not allowable, however, to multiply or divide the 1st and 4th or the 2nd and 3rd terms by the same number, for is not equal to that is, the ratio of the 1st to the 4th is not the same as that of the 2nd to the 3rd.
There are several other properties of proportional numbers, but the foregoing comprise all that are necessary or useful in the applications of arithmetic.
RULE OF THREE._This well-known rule is, in effect, identical with the method of finding a fourth proportional to three given numbers. Every question in the Rule of Three is solved by aid of the principle, that the answer or required fourth term is equal to the product of the second and third terms divided by the first, and is of the same denomination—when the question is properly stated as the third term.
* See EVOLUTION at the end of this chapter.