...: and if the consequents be equal, the ratio is the greater which has the greater antecedent. 165. A ratio of greater inequality is diminished, and a...of less inequality is increased, by adding the same quantity to both its terms. Let a : b be a ratio of inequality, and suppose a> to be added to both...

...and since — —. is less than £, the ratio of a : a + b a + bb is less than that of a : b. III. A ratio of greater inequality is diminished, and a...of less inequality is increased, by adding the same quantity to both of its terms. Let - represent any ratio, and add n to each of its terms, then these...

...fraction increased, by the addition of the same quantity to both its terms (Art. 39), it follows that a ratio of greater inequality is diminished, and a ratio of less inequality increased, by adding the same quantity to both its terms. Conversely, a ratio of greater inequality...

...of these is the greatest, we infer that the ratio of 2 to 7 is less than the ratio of 3 to 8. (207.) A ratio of greater inequality is diminished, and a...of less inequality is increased, by adding the same quantity to both terms. i>i&-r' 2 2 + 1 3 To prove the proposition generally, let j- represent any...

...: 9 ; and, since — :-=• is less than T , the ratio of a : a-4-b ajo o is less than a : !>. 236. A ratio of greater inequality is diminished, and a...of less inequality is increased by adding the same quantity to both terms. Let j- represent any ratio, and add n to each of the terms, then these two...

...of these is the greatest, we infer that the ratio of 2 to 7 is less than the ratio of 3 to 8. (207.) A ratio of greater inequality is diminished, and a...of less inequality is increased, by adding the same quantity to both terms. 3 3+1 4 Thus, 2>2+l'0r3; 2 2+1 3 3<3TT' °r 4" To prove the proposition generally,...

...equality, according аз the antecedent is greater than, less than, or equal to, the consequent. 376. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by adding any quantity to both terms of the ratio. Let the ratio be j- , and let a new ratio be formed by adding...

...according as the antecedent is equal to, greater than, or less than, the consequent. THEOREM XV. A Eatio of greater Inequality is diminished, and a Ratio of less Inequality is increased, by adding the same Quantity to both its Terms, provided that the terms of the given ratio and the quantity added are all...

...or more ratios may be compared by comparing the values of the vulgar fractions which represent them. A ratio of greater inequality is diminished, and a...of less inequality is increased, by adding the same quantity to both its terms. If the terms of a ratio be multiplied or divided by the same quantity,...

...equality, according as the antecedent is greater than, less.thsai, or equal to the consequent. 353. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by adding any number to both terms of the ratio. Let the ratio be j , and let a new ratio be formed by adding...