The method of solution is indicated below: Let it be required to find a limiting value of x from Multiplying both members by 12 (principle 2°), we have, - 4x + 48 < 3x — 36; transposing and reducing (principle 1°), we have, -7x-84; changing the signs of both members (principle 3°), we have, 7x84; dividing both members by 7 (principle 2°), we have, CHAPTER X. RATIO, PROPORTION, AND SERIES. I. RATIO, AND PROPORTION. Explanation. 155. We are said to measure a quantity when we find how many times it contains a quantity of the same kind, taken as a standard; the latter quantity is called the unit of measure. As the unit is assumed to be a quantity whose value is known before the measure is made, we call it the antecedent; because the value of the quantity to be measured is found in terms of this antecedent, we call it a consequent. Mathematically speaking, the measurement is performed by dividing the consequent by the antecedent; the result is an abstract number which we call a ratio. This ratio, prefixed to the unit employed, is the expression for the value of the quantity to be measured; hence, we have the following Definition. 156. The ratio of one quantity to another is the result obtained by dividing the second quantity by the b first. Thus, the ratio of a to b is equal to in which a' b is the quantity to be measured or the consequent, and a is the unit or antecedent. Different Methods of Expressing a Ratio. 157. The ratio of a to b may be expressed by the b symbol or, it may be written a:b; in the latter a case the sign :, stands for is contained in. In both cases a is the unit, or antecedent, and b is the consequent. The antecedent and consequent are called terms of the ratio, the antecedent being the first term and the consequent being the second term. Definitions. 158. A proportion is an expression of equality between two ratios. A proportion may be written in two ways. Thus, if the ratio of a to b is equal to the ratio of c to d we may indicate this equality by either of the following expressions: Either of these expressions indicates that the ratio of a to b is equal to the ratio of c to d. The former may be read a is contained in b as many times as c is contained in d; the latter may be read a is to b, as c is to d. We may reverse these readings without error. 159. There are four terms in every proportion which have received different names with respect to each other. The first and third are antecedents; the second and fourth are consequents. The first and fourth are extremes; the second and third are means. The first and second form the first couplet; the third and fourth form the second couplet. The fourth term, is called a fourth proportional to the other three. When the second term is equal to the third, it is said to be a mean proportional between the other two. In this case, there are but three different terms in the proportion, and the last term is said to be a third proportional to the other two. In the proportion, d b = or abc:d, a a and c are antecedents, b and d consequents, a and b d extremes, b and c means ; or ab, is the first d c' a' couplet, or cd, is the second couplet, and d is a fourth proportional to a, b, and c. Also, in the proportion, b b is a mean proportional between a and c, and c is a third proportional to a and b. 160. Quantities are in proportion, by alternation, when antecedent is compared with antecedent, and consequent with consequent. 161. Quantities are in proportion, by inversion, when antecedents are made consequents, and consequents are made antecedents. 162. Quantities are in proportion, by composition, when the sum of antecedent and consequent is compared with either antecedent or consequent. 163. Quantities are in proportion, by division, when the difference of antecedent and consequent is compared with either antecedent or consequent. 164. Two varying quantities are reciprocally, or inversely proportional, when one is increased as many times as the other is diminished. In this case, their product is a fixed quantity, as xy = m. 165. Equimultiples of two quantities, are the results obtained by multiplying both by the same quantity. Thus, ma, and mb, are equimultiples of a and b, whatever may be the value of m. Principles of Proportion. 166. We shall demonstrate some of the most important principles of proportions, adopting both methods of writing proportions. |