and the sum of the squares of its figures added to the remainder, the order of the figures will be inverted. What is the number? Ans. 37. 4. Find two numbers such that their sum, their product, and the difference of their squares shall be equal to one another. Ans. 5 and 5. 5. There are two pieces of cloth of different lengths; and the sum of the squares of the number of yards in each is 145; and one half the product of their lengths plus the square of the length of the shorter is 100. What is the length of each? Ans. Shorter, 8; longer, 9 yards. 6. Find two numbers such that the greater shall be to the less as the less is to 24, and the difference of their squares shall be 33. 7. The area of a rectangular field is 1575 square rods; and if the length and breadth were each lessened 5 rods, its area would be 1200 square rods. What are the length and breadth? 8. Find two numbers such that their sum shall be to 6 as 9 is to the greater, and the sum of their squares shall be 45. Ans. 9 and 3, or 6 and 3. 9. The fore wheels of a carriage make 2 revolutions more than the hind wheels in going 90 yards; but if the circumference of each wheel is increased 3 feet, the carriage must pass over 132 yards in order that the fore wheels may make 2 revolutions more than the hind wheels. What is the circumference of each wheel? Ans. Fore wheels, 13 feet; hind wheels, 15 feet. 10. Find two numbers such that five times the square of the greater plus three times their product shall be 104, and three times the square of the less minus their product shall be 4. SECTION XXI. RATIO AND PROPORTION. 192. RATIO is the relation of one quantity to another of the same kind; or, it is the quotient which arises from dividing one quantity by another of the same kind. Ratio is indicated by writing the two quantities after one another with two dots between, or by expressing the division in the form of a fraction. Thus, the ratio of a to b is written, a : b, or; read, a is to b, or a divided by b. a 193. The TERMS of a ratio are the quantities compared, whether simple or compound. The first term of a ratio is called the antecedent, and the other the consequent; and the two terms together are called a couplet. 194. An INVERSE, or RECIPROCAL RATIO, of any two quantities is the ratio of their reciprocals. Thus, the direct ratio of a to b is a : b, i. e. and the inverse ratio of a to b is 1 b 1 1 a b a α bi b 195. PROPORTION is an equality of ratios. Four quantities are proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth. The equality of two ratios is indicated by the sign of equality (=) or by four dots (: :). equals c to d, or a is to b as c is to d, or a divided by b equals c divided by d. 196. In a proportion the antecedents and consequents of the two ratios are respectively the antecedents and consequents of the proportion. The first and fourth terms are called the extremes, and the second and third the means. 197. When three quantities are in proportion, e. g. a: b - b: c, the second is called a mean proportional between the other two; and the third, a third proportional to the first and second. 198. A proportion is transformed by ALTERNATION when antecedent is compared with antecedent, and consequent with consequent. 199. A proportion is transformed by INVERSION when the antecedents are made consequents, and the consequents antecedents. 200. A proportion is transformed by COMPOSITION when in each couplet the sum of the antecedent and consequent is compared with the antecedent or with the consequent. 201. A proportion is transformed by DIVISION when in each couplet the difference of the antecedent and consequent is compared with the antecedent or with the consequent. THEOREM I. 202. In a proportion the product of the extremes is equal to the product of the means. THEOREM II. 203. If the product of two quantities is equal to the prod uct of two others, the factors of either product may be made the extremes, and the factors of the other the means of a proportion. 204. If four quantities are in proportion, they will be in proportion by alternation. 205. If four quantities are in proportion, they will be in 206. If three quantities are in proportion, the product of the extremes is equal to the square of the mean. Let a:b=b: c By Theorem I. ac = b2 THEOREM VI. 207. If four quantities are in proportion, they will be in proportion by composition. N |