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
| 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 +/)... | |
| 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
...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=... | |
| 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... | |
| 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),... | |
| 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... | |
| 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... | |
| Edward Olney - Algebra - 1881 - 504 pages
...,eíc.) : (b + d+f+h + k + ,etc.) ::a:ö,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... | |
| |