II. RATIOS. NOTE.-In treating of Ratios and Proportion we shall first confine the discussion to commensurable quantities, and then extend the demonstrations to incommensurable quantities. 1. Ratio is the relation which one quantity bears to another in respect of magnitude, the comparison being made by considering what multiple, part, or parts, one is of the other. Thus, in comparing 6 with 3 we observe that it has a certain magnitude with respect to 3, which it contains twice ; again, in comparing it with 2 we see that it has a different relative magnitude, for it contains 2 three times; or, 6 is greater when compared with 2 than it is when compared with 3. The ratio of a to 6 is usually expressed by two points placed between them thus, a : b; and the former, a, is called the antecedent of the ratio, the latter, b, the consequent. 2. Since in the ratio a: b the comparison is made in regard to quantuplicity, the ratio may evidently be expressed by what is necessary to multiply b by to obtain a. But this, multiplier is the fraction 7 a Then the ratio a:b is measured by the fraction and for shortness õ' a a we may say that the ratio of a to b is equal to or is 7, or, in general, 6 A ratio is measured by the fraction which has for its numerator the antecedent of the ratio, and for its denominator the consequent of the ratio. 3. Hence, we may say that the ratio of a to.b is equal to the ratio of c to d, when = 3 4. If the terms of a ratio be multiplied or divided by the same quantity, the ratio is not altered. a ma For = 7 mbo REMARK.-ma and mb are called equimultiples of a and b. 5. In general, we can operate on ratios by the rules for operating on fractions. III. PROPORTION. Four quantities are said to be proportionals when the first is the same multiple, part, or parts of the second that the third is of the fourth ; that is, when the four quantities a, b, c, d, are a с called proportionals. This is usually expressed by saying a is to b as c is to d, and is thus represented a : 6 :: c:d, or thus, a : b=cid. The terms a and d are called the extremes, and b and c the means. THEOREM I. If four quantities are proportionals, the product of the extremes is equal to the product of the means. Let a, b, c, d be the four quantities; then, since they are proportionals, ä; and, by multiplying both sides of the equation by bd, b we have ad = bc. a C mean. NOTE.—It is evident that since the product of two magnitudes has no meaning, the multipliers, at least, must be abstract numbers. COROLLARY 1. Hence, if the first to the second as the second is to the third, the product of the extremes is equal to the square of the Thus, if a : b=b:(, then ac = - 6%. Cor. 2. Any three terms in a proportion being given, the fourth may be determined from the equation ad = bc. bc ad ad bc For d 6 7 d Hence, we have the Single Rule of Three in Arithmetic. = a = a C THEOREM II. If the product of two quantities be equal to the product of two others, the four are proportionals; the factors of either product being taken for the means and the factors of the other for the extremes. 6 Let xy = ab; then, dividing by ay, thus, or x : a=b:y. у a y' THEOREM III. If a:b=c:d and c:d=e:f, then also a:b=e:f. e a Because 2 = and = ", therefore 3 = or a : 6 =e: f. THEOREM IV. If four quantities be proportionals, they are proportionals when taken inversely. If a : b=c: d, then b:a=d: C. =g; divide unity by each of these equal quantities, thus, THEOREM V. If four quantities be proportionals, they are proportionals when taken alternately. If a :b=c:d, then a :c=b: d. 6 6 For c thus, a с a 7=; multiply by с or, a :(=b: d. REMARK 1.-Theorems IV. and V. are immediate consequences of Theorem II. For we can change the order of the terms in any way which still renders the product of the extremes equal to the product of the means. KEMARK 2.-Unless the four quantities are of the same kind, the alternation of the terms cannot take place ; because this operation supposes the first to be some multiple, part, or parts of the third. One line may have to another line the same ratio as one weight has to another weight, but there is no relation with respect to magnitude between a line and a weight. In such cases, however, if the four quantities be represented by numbers, or by other quantities which are all of the same kind, the alternation may take place, and the conclusions drawn from it will be just. THEOREM VI. When four quantities are proportionals, the first together with the second is to the second, as the third together with the fourth is to the fourth. If a : b=c:d, then a + b :b=c+d:d. For 6 = a; and, adding 1 to both sides, or, a +6:6=c+ d : d. This operation is called componendo, and the quantities are said to be in proportion by composition. THEOREM VII. Also, the excess of the first above the second is to the second, as the excess of the third above the fourth is to the fourth. a с a C a or, 6:6=(-d : d. This operation is called dividendo, and the quantities are said to be proportionals by division. 6 d · Cor. Since 7 = gives (Theorem IV.), then Theorems VI. and VII. give a + b:a=c+d: a - b:a=c-d:6, or, by inversion, a: a+b=c:c+d b=1:6 d. That is, the first is to the sum of the first and second, as the third is to the sum of the third and fourth. Again, the first is to its excess above the second, as the third is to its excess above the fourth. This last operation is called convertendo. THEOREM VIII. When four quantities are proportionals, the sum of the first and second is to their difference, as the sum of the third and fourth is to their differ a: a ence. If a : 6=c:d, then a +b: a – b=c+d:c-d. For (Theorem VI.) a +6 C+ d 6 d When any number of quantities are proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents. Let a:b=c:d=e:f; then a: b= a +cte: 6+ d + f. Because ad = bc; because af = be; also, ab = ba, 6 d' 6 hence, ab + ad + af = ba + bc + be, that is, a(b + d +f)=b(a +c+e). Hence (Theorem II.), a :b=a+cte: 6 + d +f; and similarly when more quantities are taken. THEOREM X. When four quantities are proportionals, if the first and second be multiplied or divided by any quantity, as also the third and fourth, the resulting quantities will be proportionals. Let a : b=c:d; then ma : mb = nc: nd. If the first and third be multiplied or divided by any quantity, and also the second and fourth, the resulting quantities will be proportionals. Let a :b=c:d, then ma : nb = mc : nd. MC For i = , therefore, ma mc ma nd or, ma : nb = mc : nd. |