(9) A detachment was marching in regular column with 5 men more in depth than in front; but upon the enemy coming in sight, the front was increased by 845 men; and by this movement the detachment was drawn up in 5 lines. Required the number of men. Ans. 4550. (10) What two numbers are those, the difference of which multiplied by the greater produces 40, and by the less 15? Ans. 8 and 3. (11) Find two numbers, the sum of which is 15, and the sum of their squares 113. Ans. 7 and 8. (12) A and B can do a piece of work in m days, A and C in n days, B and C in r days: what portion of the work can each do in a day? Ans. The portions which A, B, and C respectively can 125. An Inequality is an expression consisting of two members, with the sign> or < between them. 126. Both members of an inequality may be increased or diminished by any quantity, multiplied or divided by any quantity which does not change their signs, raised to any power, or any root of each extracted, and the same kind of inequality will evidently still subsist. (Art. 40.) 127. But if a factor be introduced which changes the sign of each member, the inequality will be reversed; for, if A> B, then, by transposition, -B>-A, or -A <-B. EXAMPLES. a b (1) To find whether one fraction be >,, or < α whence it is obvious that is >, =, or < b) is necessarily a positive quantity, whether a be greater or less than b, a2 2ab+b2 is > 0 and.. by adding 2ab to both sides, we have that is, the sum of any quantity, except unit, and its reciprocal, is > 2. EXAMPLES FOR PRACTICE. (1) Which is the greater, √2+√7 or v 3 + √5? 128. Ratio is the relation which one quantity bears to another, with respect to magnitude, the comparison being made by considering what multiple, part or parts, one is of the other. 129. The quantities to be compared must evidently be of the same kind; for although we can compare the abstract numbers 3 and 9, yet no comparison can exist between 3 pounds and 9 yards. They must also be referred to the same unit; thus, in order to consider what part 5s. is of £3, we reduce these quantities to the same denomination, as 5s. and 60s. or £1 and £3. 130. If a and b be two magnitudes, their ratio will be represented by the fraction it is also written thus, a: b, α b and read thus, a to b; and the former term is called the antecedent of the ratio, and the latter the consequent. 131. If the terms of a ratio be multiplied or divided by the same quantity, the ratio is not altered. α ma For = (Art. 68), or a : b = ma : mb. 132. A ratio is said to be a ratio of greater or less inequality, according as the antecedent is greater or less than the consequent. 133. A ratio of greater inequality is diminished, and of less inequality increased, by adding any quantity to both its terms. Let x be added to the terms of the ratio then will a b a that is, according as the original ratio is one of greater or less inequality. b 134. Hence, conversely, a ratio of greater inequality is increased, and of less inequality diminished, by subtracting from its terms a quantity less than either of them. 135. If the antecedents of any ratios be multiplied together, and also the consequents, the resulting ratio is called their sum, or the ratio compounded of them. duplicate, triplicate, &c. ratio of a: b, or its double, treble, &c. ratio. Also, the ratios α a &c. are termed the sub-duplicate, sub-triplicate, &c. ratios of a : b, or the half, third, &c. ratios of a: b. The indices, shewing what multiple or part of the simple. ratio is taken, are called the measures of the ratios. 137. If the consequent of the preceding ratio be the antecedent of the succeeding one, and any number of such ratios be taken, the ratio which results from their composition, is that of the first antecedent to the last consequent. Let ab, bc, c d be the ratios, the compound ratio is abc: bed, or, dividing by bc, a: d. 138. A ratio of greater inequality, compounded with another, increases it; and a ratio of less inequality diminishes it. For, let the ratio x:y be compounded with the ratio a;b; then, the resulting ratio ax: by is> or <a: b, according α is> or < that is, according as x is > or < y. PROPORTION. 139. Proportion consists in the equality of ratios. Thus, a C if = the four quantities a, b, c, d constitute a propor b tion, and are called proportionals, the first being the same multiple, part or parts, of the second, that the third is of the fourth. This proportion is also written thus, a bed, but commonly thus, a: b:: c: d, and read thus, the ratio of a to b is equal to that of c to d, or thus, a is to b as c to d. The terms a and d are called the extremes, and b and c the |