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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

## College Algebra

James Harrington Boyd - Algebra - 1901 - 818 pages
...= e=± [1491] ac By dividing (1) by (2), J±| = ^ 493. THEOREM IX. — In a series of equal ratios the sum of all the antecedents is to the sum of all the consequents as any one antecedent w to its consequent. Let the ratios be (1) •£• = -£• = £ = ..... = r....

## Plane and Solid Geometry

James Howard Gore - Geometry - 1902 - 266 pages
...are in proportion. PROPOSITION IX. THEOREM. 209. In a series of equal ratios, any antecedent is to Us consequent as the sum of all the antecedents is to the sum of all the consequents. Let a:b = c:d=e:f. To prove a + c + e:b + d +f= a:b = c:d = e:f. Let r be the value of the equal ratios,...

## Mathematics

American School (Chicago, Ill.) - Engineering - 1903 - 426 pages
...j- = -jn That is, a" : 6" = cn : <f* NOTE. (Of course n may be either fractional or integral.) IX. If any number of quantities are proportional, any...antecedents is to the sum of all the consequents. Now, ab = ab (A) And,' ad = be (B) And also, af =. be (C) Adding ( A), (B), (C), a (b + d + /) = b...

## Mathematics, mechanics, heat

American School (Chicago, Ill.) - Engineering - 1903 - 390 pages
...THEOREH IX. 139. If any number of quantities are proportional, any antecedent is to its consei/uenl 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 (A) And by (128), ad = be (B) And also, af =. be (C) Adding (A), (B),...

## A College Algebra

Henry Burchard Fine - Algebra - 1904 - 612 pages
...(1) and (2), x = 0, 0, or - 7/3. Theorem. In a series of equal ratios any antecedent is to its 687 consequent as the sum of all the antecedents is to the sum of all the consequents. Thus, if ai : bi = O2 : 62 = О» : b»i then аi : bi = ai + a2 + о.t . Ьi + bt + b¡. For let r...

## A College Algebra

Henry Burchard Fine - Algebra - 1904 - 616 pages
...(1) and (2), x = 0, 0, or - 7/3. Theorem. In a series of equal ratios any antecedent is to its 687 consequent as the sum of all the antecedents is to the sum of all the consequents. Thus, if Oi : 61 = O2 : 62 = a3 : 63, then ai:b1 = a1 + a2 + a3:bl + bz + b3. For let r denote the...

## The Essentials of Algebra: For Secondary Schools

Robert Judson Aley, David Andrew Rothrock - Algebra - 1904 - 344 pages
...given proportion and reducing each member to a fractional form. THEOREM VI. 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. Proof. Let the equal ratios be ^!= (?=.#=<* = BDFH '"" Then...

## Solid Geometry

Fletcher Durell - Geometry, Solid - 1904 - 232 pages
...is to the second as the difference of the last two is to the last. 312. In a series of equal ratios, the sum of all the antecedents is to the sum of all the consequents as any one antecedent is to its consequent. 314. Like powers, or like roots, of the terms of a proportion...