From (1), by Theorem III. a+b:c+d=b: d Cor. It follows that either couplet of a proportion may be multiplied or divided by any quantity, and the resulting quantities will be in proportion. And since by Theorem III. if a: b = ma: mb, a: ma=b: mb, or ma: a mb: b, it follows that both consequents, or both antecedents, may be multiplied or divided by any quantity, and the resulting quantities will be in proportion. THEOREM XI 212. If four quantities are in proportion, like powers or like roots of these quantities will be in proportion. Since n may be either integral or fractional, the theorem is proved. THEOREM XII. 213. If any number of quantities are proportional, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. 214. If there are two sets of quantities in proportion, their products, or quotients, term by term, will be in proportion. PROBLEMS IN PROPORTION. 215. By means of the principles just demonstrated, a proportion may often be very much simplified before making the product of the means equal to the product of the extremes; and a proportion which could not otherwise be reduced by the ordinary rules of Algebra may often be so simplified as to produce a simple equation. 1. The cube of the smaller of two numbers multiplied by four times the greater is 96; and the sum of their cubes is to the difference of their cubes as 210: 114. What are the numbers? From (2), by Theo. X., Cor. x + y3 : x3 — y3 — 35 : 19 2. The product of two numbers is 78; and the difference of their cubes is to the cube of their difference as 283: 49. What are the numbers? SOLUTION. Let x = the greater and y = = the less. 283:49 (2) Then xy 78 (1) x3 — y3: x3 — 3 x2 y + 3 x y3 — y3· From (2), by division, 3xy-3xy: (xy)3 234:49 Dividing 1st couplet by xy, = Dividing antecedents by 78, Extracting the square root, Whence, X 3. The sum of the cubes of two numbers is to the cube of their sum as 13: 25; and 4 is a mean proportional between them. What are the numbers? 4. The difference of two numbers is 10; and their product is to the sum of their squares as 6: 37. the numbers? What are From (1) and (3) we find x = 12 and y = 2. 5. The product of two numbers is 136; and the dif ference of their squares is to the square of their difference as 25 9. What are the numbers? Ans. 8 and 17. 6. As two boys were talking of their ages, they discovered that the product of the numbers representing their ages in years was 320, and the sum of the cubes of these same numbers was to the cube of their sum as 7:27. What was the age of each? Ans. Younger, 16; elder, 20 years. 7. As two companies of soldiers were returning from the war, it was found that the number in the first multiplied by that in the second was 486, and the sum of the squares of their numbers was to the square of the sum as 13: 25. How many soldiers were there in each company? Ans. In 1st, 27; in 2d, 18. 8. The difference of two numbers is to the less as 100 is to the greater; and the same difference is to the greater as 4 is to the less. What are the numbers? NOTE. Multiply the two proportions together. (Theorem XIII.) SECTION XXII. PROGRESSION. 216. A PROGRESSION is a series in which the terms increase or decrease according to some fixed law. 217. The TERMS of a series are the several quantities, whether simple or compound, that form the series. The first and last terms are called the extremes, and the others the means. ARITHMETICAL PROGRESSION. 218. An ARITHMETICAL PROGRESSION is a series in which each term, except the first, is derived from the preceding by the addition of a constant quantity called the common difference. 219. When the common difference is positive, the series is called an ascending series, or an ascending progression; when the common difference is negative, a descending series. Thus, a, a + d, a + 2d, a + 3d, &c. is an ascending arithmetical series in which the common difference is d; and a, a- -d, a 2d, a 3d, &c. is a descending arithmetical series in which the common difference is d. 220. In Arithmetical Progression there are five elements, any three of which being given, the other two can be found: 1. The first term. |