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
| James Wallace MacDonald - Geometry - 1894 - 76 pages
...powers or like roots are in proportion. Proposition XI. A Theorem. 127. 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. COROLLARY. The sura of any number of the antecedents is... | |
| James Wallace MacDonald - Geometry - 1889 - 80 pages
...consequents. Proposition X. A Theorem. Proposition XI. A Theorem. 127. 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. COROLLARY. The sum of any number of the antecedents is... | |
| Charles Davies - Algebra - 1889 - 330 pages
...+ d+f+h+ &c. _ b Ac. a •/ Honce, the following principle : 10. In any continued proportion, tlie **sum of all the antecedents is to the sum of all the consequents,** as any antecedent is to the corresponding consequent. Let us assume the two equations, bd . fh - =... | |
| Joseph Ray - Algebra - 1894 - 422 pages
...15 : 135 : : 8 : 72. 278. Propositioa XII. — In any number of proportions having the same ratio, **any antecedent is to its consequent as the sum of...antecedents is to the sum of all the consequents. Let a : b** : : С : d : : m : n, etc. Then, a : b : : a+c+wi : o+ d+n. Since a : b : : с : d, we have be— .ad... | |
| Webster Wells - Algebra - 1890 - 604 pages
...(1) by (2), a — oc — a Whence, a + 6 : a — b = c + d:c — d. 390. 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 Art. 381, ad = be, and af= be. Also, ab = ba. Adding, a(6 + d+/) = 6(a + c... | |
| George William Jones - Algebra - 1892 - 300 pages
...and like roots of the terms of a proportion are proportional. THEOR. 9. In a continued proportion, **the sum of all the antecedents is to the sum of all the consequents** as any antecedent is to its consequent. For, let a : b - с : d= e :/= - - then va/a = b/b, с/а -... | |
| Webster Wells - Geometry - 1894 - 400 pages
...by the second, we have a -f- bc -\- d PROPOSITION VIII. THEOREM. 239. 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:b=a+c+e:b+d+f. Let r denote the value of each of the given ratios. Then, r = ;hy = rWhence,... | |
| Webster Wells - Geometry - 1894 - 256 pages
...the second, we have a -\- b _ c -f- d PROPOSITION VIII. THEOREM. 239. 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** if. To prove a:b = a-\-c-\-e:b-\-d-\-f. Let r denote the value of each of the given ratios. lli-, Whence,... | |
| Webster Wells - Geometry - 1894 - 398 pages
...by the second, we have a -\- bc -f- d PROPOSITION VIII. THEOKEM. 239. 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. To prove a\b — a-\-c-\-e:b-\-d-\-f. Let r denote the value of each of the given... | |
| Webster Wells - Geometry - 1894 - 400 pages
...VIII. THEOREM. 239. In a series of equal ratios, any antecedent is to its consequent as the sum of nil **the antecedents 'is to the sum of all the consequents. Let a : b** = e : d = e :f. To prove a : b = a -f- c -f- e : b -\- d -f- f. Let r denote the value of each of the... | |
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