 | Edward Olney - Algebra - 1878 - 516 pages
...: (6 + d+/+ ^ + fc+,ete.) : : a : b, or c : d, or e : f, etc. That is, in a series of equal ratios, the sum of all the antecedents is to the sum of all the consequents, as any antecedent is to its consequent SOLUTION. =- = r or a& = ba, oo ac , , — = -j or ad = be,... | |
 | Edward Olney - 1878 - 360 pages
...Ъ— dt 72. Сов. — 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 : : с : d : : e :f: :g :n, etc., a +... | |
 | Benjamin Greenleaf - Algebra - 1879 - 350 pages
...5=7Therefore, by Art. 38, Ax. 7, j = ^, or. a : b : : c : d. THEOREM X. 321. If any number of quantities are proportional, any antecedent is to its consequent...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, ab-\-ud-\-af—... | |
 | Benjamin Greenleaf - Algebra - 1879 - 376 pages
...Therefore, by Art. 38, As^ 7, -r = -,, or. a : b : : c : d. THEOREM X. 324 1 If any number of quantities are proportional, any antecedent is to its consequent...antecedents is to the sum of all the consequents. Let « : 4 : : c : d : : e : f; then g:t: : m -\- c -\- e : l-\-d-\- f. For, by Theo. I., ad = bc, and... | |
 | Webster Wells - Algebra - 1879 - 468 pages
...e:f and c:d — e:f ae с е then, -=- and -d = -f Therefore, - = od 351. If any number of quantities are proportional, any antecedent is to its consequent,...antecedents is to the sum of all the consequents. Thus, if a : b = c: d = e :f then (Art. 343), ad = bc and af=be also, ab = ab Adding, a (b + d +/)... | |
 | Isaac Sharpless - Geometry - 1879 - 282 pages
...E : F. TTK AC A EC ^f AE Proposition 16. Theorem. — If any number of quantities be in proportion, any antecedent is to its consequent, as the sum of...antecedents is to the sum of all the consequents. If A : B : : C : D : : E . F, etc., then A : B :: A+C+E,etc. : B+D + F,etc. Let A = mB, then (IV. 6)... | |
 | Elias Loomis - Algebra - 1879 - 398 pages
...n, ma _mc •rib ~ nd1 or ma :nb::mc: nd. 309. If any number of quantities are proportional, any one 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, since a : b : : c : d, ad=bc; (1.) and, since a:b::e:f, af=be; (2.) also ab —... | |
 | Horatio Nelson Robinson - Algebra - 1879 - 332 pages
...PROPOSITION XIH 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), <z:f = c:d (B), a: b=m: n. . . . (C), &c. = &c. We are to prove that a: b = (a + c +... | |
 | Benjamin Greenleaf - Algebra - 1879 - 350 pages
...Therefore, by Art 38, Ax. 7, f = ¿, or. a : b : : с : d. THEOREM X. 324. If any number of quantities are proportional, any antecedent is to its consequent as the sum of аи the antecedents is to the sum of all the consequents. Let a : b : : с : d : : e : f; then a :... | |
 | William Frothingham Bradbury - Geometry - 1880 - 260 pages
...6» = #> C" C B that is a n : b" = c n : d* ._ -~> . ^ THEOREM IX. 23i If any number of quantities are proportional, any antecedent. is to its consequent...sum of all the consequents. Let a :b = c : d =e :f Now ab — ab (A) and by (12) ad = bc (B) and also af = be (C) Adding (A), (B), (C) < Hence, by (14)... | |
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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. 
