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
| University of the State of New York. Examination dept - Examinations - 1895 - 436 pages
...imaginary. 6-7 Complete and prove the following theorem : if any number of quantities are in proportion **the sum of all the antecedents is to the sum of all the consequents** as ... 8-9 Write three terms of the expansion of [a -\- b~\ n and prove that it is true when n is any... | |
| George Washington Hull - Algebra - 1895 - 358 pages
...nq Multiplying, $P--JJ26n dq Whence, am :bn = cp: dq. THEOREM XVI. In a series of equal ratios, (he **sum of all the antecedents is to the sum of all the consequents** as any antecedent is to its consequent. Let a : 6 = с : d, And m:n=p:q. Then а с t Ь d' And ™=£.... | |
| James Bates Thomson - 1896 - 336 pages
...number of quantities are proportional, any antecedent is to Us consequent, as ¿he sum of all ¿he **antecedents is to the sum of all the consequents. Let a : b** :: e : d :: e : f, etc. Then a : b :: a+e+e : Ь + d+f, etc. For (Th. i), ad = be And, " af=be Also,... | |
| Webster Wells - Algebra - 1897 - 522 pages
...Dividing (I) by (2), —b = ~dWhence, a + b: a — o = c + d:c — d. 315. 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 by § 306, ad = be, and af= be. Also, ab = ba. Adding, a (b + d +f) = b(a + c + e).... | |
| Webster Wells - Algebra - 1897 - 422 pages
...-J--- a — о с — a Whence, a + b : a — b = с + d : с — d. 315. 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** = с : d = e : f. Then by § 306, ad = bс, and af= be. Also, ab = ba. Adding, a(b + d+f) = b(a + c... | |
| Webster Wells - Algebra - 1897 - 386 pages
...Dividing (1) by (2), f±£ = Whence, a + b: a — b = c + d: c — d. 315. 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 by § 306, ad = be, and a/= be. Also, ab = ba. Adding, a(b + d+f) = b(a + c +... | |
| Webster Wells - Algebra - 1904 - 384 pages
...Dividing (1) by (2), ±= Whence, a + b: a — b = c + d: с — d. 315. 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 by § 306, ad = be, and a/= be. Also, ab = ba. Adding, a (6 + d +/) = b (a +... | |
| George Washington Hull - Geometry - 1897 - 408 pages
...are proportional. §111 §111 Ax. 2 PROPOSITION XV. THEOREM. QED 133. 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. Let a: b— c : d= e:f. Let r •» the common ratio, Then... | |
| Fletcher Durell, Edward Rutledge Robbins - Algebra - 1897 - 482 pages
...+ d: c — d. 286. VI. Composition of Several Equal Batios ; that is, 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. а с eg eLet each of the equal ratios equal r. mu а с... | |
| James Howard Gore - Geometry - 1898 - 232 pages
...like powers of the terms are in proportion. PROPOSITION IX. THEOREM. 209. In a series of equal ratios, **any antecedent is to its consequent as the sum of...of all the consequents. Let a : b = c : d = e : f.** To prove a + c + e:b + d +/= a : b = c : d = e : f. Let r be the value of the equal ratios, that is,... | |
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