 In a series of equal ratios, any antecedent is to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Let a: 6 = c: d = e :/. Then, by Art.  A Treatise on Algebra - Page 220by Elias Loomis - 1868 - 384 pagesFull view - About this book
 | William Frothingham Bradbury - Geometry - 1873 - 132 pages
...n rf" that is a" : 6" = c" : d n THEOREM IX. 23. If any number of quantities are proportional, any antecedent is to its consequent as the sum of all...antecedents is to the sum of all the consequents. Let a :b = e : d=.e :f Now ab = ab (A) and by (12) ad —be (B) and also af=be (C) Adding (A), (B), (C) a (b... | |
 | Edward Olney - Algebra - 1873 - 354 pages
...: b—dl У 2. COR. — If there be a series of equal ratios in the form of a continued proportion, the sum of all the antecedents is to the sum of all the consequents, as any one antecedent is to its consequent. DEM. — If a : b : : e : d : : e :f: : g : h, etc., a... | |
 | Horatio Nelson Robinson - Algebra - 1874 - 340 pages
...PROPOSITION Xin. 275. If any number of proportionals have the same ratio, any one of the antecedents will be to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let a : b = a : b (A) Also, a : b = с : d (B) a : b =m : n (С) &c. = &c. We are to prove that a : b = (a +... | |
 | Benjamin Greenleaf - Algebra - 1875 - 338 pages
...1, | = ^, or, a : b : : c : d, THEOREM X. 324. -//'' any number of quantities are proportional, any antecedent is to its consequent as the sum of all...antecedents is to the sum of all the consequents. Let a:b::c:d::e:f; then a : b : : a -f- c -|- e : b -\- d -|- f. For, by Theo. I., ad = bc, and af=be;... | |
 | Horatio Nelson Robinson - Algebra - 1875 - 340 pages
...Х1П. 275. If any number of proportionals have the same ratio, any one of the antecedents will be to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let a:b=a:b.... (A), a:f=c:d.... (B), / a : b = m: n. . . . (C), &c. = &c. We are to prove that a: b= (a... | |
 | William Frothingham Bradbury - 1875 - 280 pages
...5" = c" : <f THEOREM XII. 213. If any number of quantities are proportional, any antecedent is to us consequent as the sum of all the antecedents is to the sum of all the consequents. Let a : 6 = с : d=e : f Now ab = ab (1) and by Theorem I. ad = be (2) and also af=be (3) Adding(l),(2),(3),... | |
 | Horatio Nelson Robinson - Algebra - 1875 - 430 pages
...If there be a proportion, consisting of three or more equal ratios, then either antecedent will be to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Suppose a:b=c:d — e:f = g:h=, etc. Then by comparing the ratio, a : b, first with itself, and afterward... | |
 | William Guy Peck - Algebra - 1875 - 348 pages
...: b + d+f+h + &c. :: a:b; (11) hence, the following principle : 10°. In any continued proportion, the sum of all the antecedents is to the sum of all the consequents, as any antecedent is to the corresponding consequent. ь d " bc = ad. a — c' b a = ê' " be = «/•... | |
 | William Guy Peck - Conic sections - 1876 - 412 pages
...be multiplied or divided by the same quantity. PROPOSITION VIII. THEOREM. In a continued proportion, the sum of all the antecedents is to the sum of all the consequents as any antecedent is to the corresponding consequent. Assume the continued proportion, z 7 /• * df... | |
 | Richard Wormell - 1876 - 268 pages
...B + F. -F; .-. A + E : В + F = E : F = С: D. THEOREM LXX. If there be any number of equal ratios, the sum of all the antecedents is to the sum of all the consequents as either antecedent is to its consequent. Let A : В = С : D = E : F. By Theorem LXIX., A + E:B +... | |
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