Dropping 2xy from both members, dividing by a, and transposing, we have. 2√xy=a-3 √xy=2 (7) This value of ay put in equation (3), gives x+y=5 (8) From equations (7) and (8), we find x and y, as taught in Art. 101. 3. The sum of the first and third of four numbers in geometrical progression is 20, and the sum of the second and fourth is 60. What are the numbers? Ans. 2, 6, 18, 54. 4. Divide the number 210 into three parts, so that the last shall exceed the first by 90, and the parts be in geometrical progression. Ans. 30, 60, and 120. 5. The sum of four numbers in geometrical progression is 30; and the last term divided by the sum of the mean terms is 1. What are the numbers? Ans. 2, 4, 8, and 16. 6. The sum of the first and third of four numbers in geometrical progression is 148, and the sum of the second and fourth is 888. What are the numbers? Ans. 4, 24, 144, and 864. 7. The continued product of three numbers in geometrical progression is 216, and the sum of the squares of the extremes is 328. What are the numbers? Ans. 2, 6, 18. 8. The sum of three numbers in geometrical progression is 13, and the sum of the extremes being multiplied by the mean, the product is 30. What are the numbers? Ans. 1, 3, and 9. 9. There are three numbers in geometrical progression whose product is 64, and the sum of their cubes is 584. What are the numbers? Ans. 2, 4, and 8. Let x2, xy, y2 represent the three numbers. From (2) subtract three times (1), and we have x®-2x3y3+y=392=196•2 (5) We give the minus sign to a3, because y must be greater than a from the position it occupies in our notation, and 23—ya or y3—3, when squared, will produce the same power. Subtracting (6) from (4), and 10. There are three numbers in geometrical progression, the sum of the first and last is 52, and the square of the mean is 100. What are the numbers? Ans. 2, 10, 50. 11. There are three numbers in geometrical progression, their sum is 31, and the sum of the squares of the first and last is 626. What are the numbers? Ans. 1, 5, 25. 12. It is required to find three numbers in geometrical progression, such that their sum shall be 14, and the sum of their squares 84. Ans. 2, 4, and 8. 13. There are four numbers in geometrical progression, the second of which is less than the fourth by 24; and the sum of the extremes is to the sum of the means, as 7 to 3. What are the numbers? Ans. 1, 3, 9, and 27. 14. The sum of four numbers in geometrical progression is equal to the common ratio +1, and the first term is. What are the numbers? Ans. To, T, 1%, 77. PROPORTION. (ART. 117.) Two magnitudes of the same kind can be compared with one another, and the numerical relation between them determined. The manner of determining this relation, is to divide one by the other, and the quotient is called the ratio between the two numbers. When two quantities have the same ratio as two other quantities, the four quantities may constitute a proportion. Therefore, proportion is the equality of ratios. The last is the modern method, and means that the ratio of a to b is equal to the ratio of c to d. b a d с If a is taken as the unit of measure between a and b, then is the numerical ratio between these two magnitudes. If c is taken for the unit of measure between c and d, then is the numerical ratio between these two magnitudes. The magnitudes a and b may be very different in kind from those of c and d; for instance, a and b may be bushels of wheat, and c and d sums of money. This manner of comparing magnitudes, by taking one of them as a whole (regardless of other units) is called geometrical proportion, and if there are more than two magnitudes having the same ratio, the magnitudes are said to be in geometrical progression. Two magnitudes compared by ratio are called a couplet. Thus, a : b is a couplet, and c:d is another couplet. The first magnitude of a couplet is called the antecedent, the second the consequent. A ratio can exist between two magnitudes; but a proportion requires four-two antecedents and two consequents having the same ratio. All operations in proportion rest on this fundamental equation; and to prove a principle or an operation true, we directly, or remotely compare the principle or the operation to this equation, and if we find a correspondence, the principle or the operation is true-otherwise, false. PROPOSITION I. In every proportion, the product of the extremes is equal to the product of the means. Multiply both members of this equation by ac, and as the product of equal factors are equal (Ax. 3), That is, the product of c and b, the means, is equal to the product of a and d, the extremes. SCHOLIUM.-Divide both members of this equation by a, This equation shows, that the fourth term of any proportion may be found from the first three, by the following RULE.-Multiply the second and third terms of the proportion together, and divide that product by the first term. This is a part of the well known rule of three, in Arithmetic. PROPOSITION II. Conversely. If the product of two quantities is equal to the product of two others, then two of them may be taken for the means, and the other two for the extremes of a proportion. Divide both members of this equation by any one of the four factors, say c, then we have Divide this last equation by another of the factors, say a, This is the fundamental equation for proportion, and gives a:b-c:d Now, as the principle is established, we may proceed more summarily, and take the two factors in one member for the extremes, To fill up the means, we must take the factor which has the same name as a to stand before the equality, and the other factor to stand after the equality will be of the same name as d, and the proportion will be complete. If the quantities are all numerals, it is immaterial which factor stands first in the means. Thus, a:b=c:d are proportions equally true in numeriOr, a:cb:d cal values. |