 If any number of quantities are proportional, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let a : b = c : d = e :f Now ab = ab (1) and by Theorem I.  An Elementary Geometry - Page 30by William Frothingham Bradbury - 1872 - 110 pagesFull view - About this book
 | Horatio Nelson Robinson - Algebra - 1874 - 338 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 - Geometry - 1874 - 206 pages
...115. If any number of magnitiides are proportional, any antecedent is to its consequent as the sitm of all the antecedents is to the sum of all the consequents. Let A : B : : C : D : : E : F; then will A : B: : A + C + E : B \-D-\-F. For, from the given proportion, we have AXD = BXC, and AXF... | |
 | Benjamin Greenleaf - Algebra - 1875 - 338 pages
...= -f, , ce and 5=?. Therefore, by Art. 38, Ax. 1, | = ^, or, a : b : : c : d, THEOREM X. 324. -//'' any number of quantities are proportional, any antecedent...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;... | |
 | Benjamin Greenleaf - Geometry - 1875 - 204 pages
...remaining terms will be in proportion. THEOREM X. 115. If any number of magnitudes are proportional, awy antecedent is to its consequent as the sum of all...all the consequents. Let A : B : : C : D : : E : F; then will A : B: :A+C+E: B-\-D + F. For, from the given proportion, we have By adding AXB to the sum... | |
 | 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 = «/•... | |
 | Benjamin Greenleaf - 1876 - 332 pages
...d : d ice and - = -. Therefore, by Art, 38, Ax. 7, -r = 3, or, a : b : : c : d. THEOREM X. f 324« If any number of quantities are proportional, any...all the consequents. Let a : b : : c : d : : e : f; then a : b : : a-\-c-\-e:b-\-d-\-f. For, by Theo. I., ad=.bc, and af=. be ; also, ab = ba. Adding,... | |
 | 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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