4. Find a fourth proportional to 2, 5, and 4; 35, 20, and 14. 5. Write out the proofs for the following, stating the theorem in full in each case: (_a_) The product of the extremes equals etc. (_b_) If the product of two numbers equals the product of two other numbers, either pair etc. (_c_) Alternation. (_d_) Inversion. (_e_) Composition. (_f_) Division. (_g_) Composition and division. (_h_) In a series of equal ratios, the sum of the antecedents is to the sum of the consequents etc. (_i_) Like powers or like roots of the terms of a proportion etc. 6. If x : m :: 13 : 7, write all the possible proportions that can be derived from it. [See (5) above.] 7. Given rs = 161m; write the eight proportions that may be derived from it, and quote your authority. 8. (_a_) What theorem allows you to change any proportion into an equation? (_b_) What theorem allows you to change any equation into a proportion? 9. If xy = rg, what is the ratio of x to g? of y to r? of y to g? 10. Find two numbers such that their sum, difference, and the sum of their squares are in the ratio 5 : 3 : 51. (_Yale._) ~Reference:~ The chapter on Ratio and Proportion in any algebra. An easy and powerful method of proving four expressions in proportion is illustrated by the following example: Given a : b = c : d; prove that 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2. Let a/b = r. Therefore a = br. Also c/d = r. Therefore c = dr. Substitute the value of a in the first ratio, and c in the second: Then (3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3b^3r^3 + 5b^3r)/(3b^3r^3 - 5b^3r) = [b^3r(3r^2 + 5)]/[b^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5). Also (3c^3 + 5cd^2)/(3c^3 - 5cd^2) = (3d^3r^3 + 5d^3r)/(3d^3r^3 - 5d^3r) = [d^3r(3r^2 + 5)]/[d^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5). Therefore (3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3c^3 + 5cd^2)/(3c^3 - 5cd^2). Axiom 1. Or, 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2. If a : b = c : d, prove: 1. a^2 + b^2 : a^2 = c^2 + d^2 : c^2. 2. a^2 + 3b^2 : a^2 - 3b^2 = c^2 + 3d^2 : c^2 - 3d^2. 3. a^2 + 2b^2 : 2b^2 = ac + 2bd : 2bd. 4. 2a + 3c : 2a - 3c = 8b + 12d : 8b - 12d. 5. a^2 - ab + b^2 : (a^3 - b^3)/a = c^2 - cd + d^2 : (c^3 - d^3)/c. 6. The second of three numbers is a mean proportional between the other two. The third number exceeds the sum of the other two by 20; and the sum of the first and third exceeds three times the second by 4. Find the numbers. 7. Three numbers are proportional to 5, 7, and 9; and their sum is 14. Find the numbers. (_College Entrance Board._) 8. A triangular field has the sides 15, 18, and 27 rods, respectively. Find the dimensions of a similar field having 4 times the area.