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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. "
An Elementary Geometry - Page 30
by William Frothingham Bradbury - 1872 - 110 pages
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University Algebra

Webster Wells - Algebra - 1879 - 468 pages
...each other. Thus, if a:b = 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...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 +/)...
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The Elements of Plane and Solid Geometry: With Chapters on Mensuration and ...

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)...
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A Treatise on Algebra

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 —...
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New Elementary Algebra: Containing the Rudiments of the Science for Schools ...

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 +...
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New Elementary Algebra: Designed for the Use of High Schools and ..., Book 1

Benjamin Greenleaf - Algebra - 1879 - 350 pages
...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 : b : : a-\-c -\- e : b -\- d-\- f. For, by Theo. L, ad = be, and af=...
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An Elementary Geometry: Plane, Solid and Spherical

William Frothingham Bradbury - Geometry - 1880 - 260 pages
...c : d ac that is j = 5 Hence 6 = #> C" C B that is a n : b" = c n : d* ._ -~> . ^ THEOREM IX. 23i If any number of quantities are proportional, any...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) THEOREM X. 24i If there are...
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New Practical Algebra: Adapted to the Improved Methods of Instruction in ...

James Bates Thomson - Algebra - 1880 - 324 pages
...v=|. That is, a : Ъ — e : d Again, 12 : 4 = 6 : 2, and 9:3 = 6:2 л 12 : 4 = g : 3 THEOREM X. Wlien any number of quantities are proportional, any antecedent...antecedents is to the sum of all the consequents. Adding (Ax, 2), ab + ad + af = ba + be + be Factoring, a <b + d +/) = b (a + e + e) Hence, (Th. 3),...
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A University Algebra

Edward Olney - Algebra - 1880 - 354 pages
...Ъ—dl У. СОЕ. — 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 : A, etc., a...
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The student's algebra

James Mackean - 1881 - 510 pages
...Mixing. PROP. XIV. — When several quantities are in continued proportion, any one of the antecedents is to its consequent as the sum of all the antecedents is to the sum of all the consequents. a ma + ne + pe Theorem IX., l=mb + nd+pf, and if mnp-1, a a+c+e . . then т = f,i _r~?; ... a:o::a...
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The Complete Algebra ...

Edward Olney - Algebra - 1881 - 504 pages
...,ec.) : (b + d+f+h + k + ,etc.) ::a:,or с : 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. _ aa -, -, а с , , Solution, v — т or ab = ba, - = ^01...
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