...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 +/)...

...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)...

...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 —...

...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 +...

...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=...

...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...

...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),...

...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...

...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...

...,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...