1. Define ratio, proportion, mean proportional, third proportional, fourth proportional. 2. Find a mean proportional between 4 and 16; 18 and 50; and 3. Find a third proportional to 4 and 7; 5 and 10; and 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 write all the possible proportions that can be derived from it. [See (5) above.] 7. Given 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 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. Given prove that Let Also Substitute the value of a in the first ratio, and c in the second: Then Also Axiom 1. Or, If prove: 1. 2. 3. 4. 5. 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. |