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 - 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 - 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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