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 - 1906 - 484 pages
...О (л Яi— ПГ~ т TI "VCl Vс In like manner, -1— = — — 344. In a series of equal ratios, **any antecedent is to its consequent as the sum of...the sum of all the consequents. Let a : b = c: d = e** :/. Then by § 331, ad — be, and af= be. Also, a6 = ba. Adding, а(b + d +/) = b(a + с + e). Whence,... | |
| Isaac Newton Failor - Geometry - 1906 - 440 pages
...13 = x — y : 3, find the ratio of x to y. PROPOSITION IX. THEOREM 336 In a continued proportion, **any antecedent is to its consequent as the sum of...antecedents is to the sum of all the consequents.** PROOF ab = ba, Iden. ad = be, § 328 and of = be. . § 328 Adding, ab + ad + af= ba + be + be ; Ax.... | |
| Isaac Newton Failor - Geometry - 1906 - 431 pages
...Ifx+y:l3 = x — y:3, find the ratio of x to y. PROPOSITION IX. THEOREM 336 In a continued proportion, **any antecedent is to its consequent as the sum of...antecedents is to the sum of all the consequents.** PROOF ab = ba, I den. ad = be, § 328 and af = be. § 328 Adding, ab + ad -f <tf= ba + &c + be ; Ax.... | |
| Webster Wells - Algebra - 1906 - 550 pages
...-ï— = -^— • л/о л/d 344. //ia series of equal ratios, any antecedent is to its consequent an **the sum of all the antecedents is to the sum of all the consequents. Let** a:b = c:d = e:f. Then by § 331, ad = be, and of= be. Also, ab = ba. Adding, a(b + d +/) = b(a + с... | |
| Edward Rutledge Robbins - Geometry, Plane - 1906 - 268 pages
...a : b = c : r. ) Proof : am = 6c.and ar = be (?) (290). 301. THEOREM. 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: Set each given ratio = m; thus, acea = m; - = m; - =... | |
| Edward Rutledge Robbins - Geometry - 1907 - 428 pages
...a : b = c : r. \ Proof : am = be and ar = bc (?) (290). 301. THEOREM. 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. „. acea Glven: I-2-/-T To Prove: f±£±i+|= f = 1 , etc.... | |
| Webster Wells - Algebra - 1908 - 262 pages
...proportion. Let the proportions be ?=v and - = ?• 6 a / ft frl. 149. In a series of equal ratios, **any antecedent is to its consequent as the sum of...antecedents is to the sum of all the consequents. Let,** a:b = c:d = e:f. Then, - ---/ b b+d+f EXERCISE 59 The following problems lead both to integral and... | |
| Webster Wells - Algebra - 1908 - 456 pages
...then, — — — . bd b" d" In like manner, — = ^- • П,, ",~i 152. In a series of equal ratios, **any antecedent is to its consequent as the sum of...antecedents is to the sum of all the consequents.** e). :b + d+f. (§141) In like manner, the theorem may be proved for any number of equal ratios. 153.... | |
| Webster Wells - Geometry, Plane - 1908 - 206 pages
...- a2 _ 2 a + 6 _ x — va;2 — a2 2 a — 6 PROP. VIII. THEOREM 224. 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. We have ba = ab. And from (1), be = ad, and be =... | |
| Webster Wells - Geometry - 1908 - 336 pages
...division to the following: x + Vx2 - n" _ 2 q + 6 PROP. VIII. THEOREM 224. 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. !_!_!. a) To Prove a + c + e = g. b+d+fb Proof. We have ba... | |
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