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 - Geometry - 1898 - 264 pages
...From(l), o_ = c- (§ 237) ac and o^-ft^Cj-d. ac PROP. VIII. THEOREM. 240. In a series of equal ratios, **the sum, of all the antecedents is to the sum of all the consequents** as any antecedent 18 to its consequent. Given a:b = c:d=e:f. (1) To Prove a + c + e:b + d +/= a : b.... | |
| Arthur A. Dodd, B. Thomas Chace - Geometry - 1898 - 468 pages
...: W~C'*— CD : C' D' . Now substitute these values in your first equations. By proportion, §198, **the sum of all the antecedents is to the sum of all the consequents** as any antecedent is to its consequent. Can you write an equation so that the sum of the AS in the... | |
| Webster Wells - Algebra - 1899 - 444 pages
...c--d a — о с — a 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 — bc, and а/= ЬeAlso, ab = ba. Adding, a (b + d +/) = b (и + с... | |
| James Howard Gore - Geometry - 1899 - 266 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...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 :/. Let r be the value of the equal... | |
| James Howard Gore - Geometry - 1899 - 266 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...antecedents is to the sum of all the consequents.** To prove a + c + e:b + d +/= a:b = c:d = Let r be the value of the equal ratios, that is, acje From... | |
| Webster Wells - Geometry - 1899 - 424 pages
...bc — d .: a + b: a — 6 = c + d:c — d. PROP. VIII. THEOREM. 240. 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. Given a:b = c:d = e:f. (1) To Prove a + c + e :b + d+f= a:... | |
| George Egbert Fisher - 1900 - 444 pages
...From a : b = b : c, we have, by Art. 8, &2 = ac ; whence b = -^/(ac). 19. 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** щ : dl = щ : d¡ = n3 : d3 = ••• = v, ^ = V,^V,^ = V}.... Ctl -,2 , «3 Then, n1 = vd¡, nj... | |
| George Egbert Fisher - 1901 - 320 pages
...a : b = b : c, .we have, by Art. 8, b2 = ас ; whence b = ^/(ac). 19. 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** ni : dl = n2: <Z2 = n3 d3=—=v, * = v,b = v,b = v,.... di a, d3 Then, n1 = vdu n¡ = vd¡, n3 = vdt,... | |
| Thomas Franklin Holgate - Geometry - 1901 - 462 pages
...h.-.R, or - = ^-^, that P:R = hm: kn. ' kn R kn' 240. THEOREM. If any number of ratios are equal, then **the sum of all the antecedents is to the sum of all the consequents** as any one antecedent is to its consequent. Let a^ : 6! = 02 : 62 = a3 : 63 = •••. It is required... | |
| James Harrington Boyd - Algebra - 1901 - 812 pages
...?^*. [{491] ac By dividing (1) by (2), 2-±| = e-±± 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 is to its consequent. Let the 'ratios be (1) j- = J- = ± = = r. (2) a = Ar,... | |
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