6. Find the value of each of the following ratios. (a) i to š. (b) , to . (c) 11 to 25. 7. Give (as a fraction) the simplest form for each of the following ratios. (a) 2 a to 2 b. (6) 3 aż to 6 62. (c) a? — 62 to a+b. 8. State which is the antecedent and which the consequent in each of the parts of Exs. 5, 6, and 7. 9. What is the ratio of one side of a square to any other side? 10. What is the ratio of the circumference of any circle to its radius? SOLUTION. Call the radius r. Then the circumference (see Ex. 22, p. 22) will be 2 ar. Therefore, the ratio of the circumference to the radius will be 2 ir/r, or 27. Ans. 11. The figure shows a circle surrounded by a square which it just touches on all four sides. Find the ratio of the area of the circle to that of the square. [Hint. Let r be the radius. Then a side of FIG. 42, the square will be 2 r. Now proceed as in the solution of Ex. 10, using Ex. 25, p. 22, and remembering that the area of the square here will be (2 r)?, or 4 r2.] The circle in Fig. 42 is said to be inscribed in the square; the square is circumscribed about the circle. 12. Find the ratio of the volume of any cube to that of the sphere that will just fit inside it. (Hint. Letr be the radius of the sphere and use Ex. 28, p. 23.) Fig. 43. The sphere of Fig. 43 is said to be inscribed in the cube. 13. It is shown in geometry that the volume of a right circular cylinder is equal to the area of its base multiplied by its height. By means of this result show that when a sphere is completely surrounded by a cylinder in the manner shown in the figure (that is, the cylinder and sphere just touching each other above and below and on the side) then the ratio of the volume of Fig. 44. the cylinder to the volume of the sphere is simply . (Hint. Let r represent the radius of the sphere. Then the radius of the base of the cylinder will be r, and the height will be 2 r.] 14. Show that the circumferences of any two circles have the same ratio as their radii. [Hint. Let R be one radius and r the other.] 15. Show that the areas of any two circles have the same ratio as the squares of their radii. 16. In the figure are two circles, each surrounded by a square which it just touches on all four sides. Show that . the areas of the two circles have the same ratio as the areas of their surrounding squares. 17. What is the ratio of the areas of two squares if the side of one is double the side of the other? Answer the same question for two circles if the radius of one is double that of the other. 18. What is the ratio of the volumes of two cubes if the edge of the one is double that of the other? Answer the same question for two spheres if the radius of the one is double that of the other. 19. Mr. A's automobile travels at the rate of 25 miles an hour, while Mr. B's travels at 20 miles an hour. What is the ratio of the time it will take A to make any given journey as compared to the time it will take B? 109. Proportion. A proportion is an expression of equality between two ratios, or fractions. For example, since is the same as , we have the proportion s=4. Likewise, we may write {=}, and }={}, and hence these are all true proportions. But {= is not a true proportion since the two fractions here are unequal. Every proportion is thus seen to be an equality of the form a/b=c/d where a, b, c and d stand for numbers. These four numbers are called the terms of the proportion. The first and fourth numbers (that is, a and d) are called the extremes, while the second and third (b and c) are called the means. Besides writing a proportion in the form a/b=c/d, it may be written in the form a:b=c:d, or also in the form a:0::c:d. In all cases it is read “a is to b as c is to d,” and it means that the fraction a/b is equal to the fraction c/d. Note. Every proportion is thus a fractional equation of the kind studied in Chapter X. EXERCISES 1. Using the language of proportion, read each of the following proportions. (a) = 19. (c) -1:2::2:-4. (6) 3:4=6:8. (d) 1:1::2:1. 2. State what are the extremes and what the means in each part of Ex. 1. 3. State any proportions that you can make out of the following four quantities : 2 inches, 8 inches, 4 inches, 16 inches. (Hint. 2 inches is to 8 inches as ...] 4. State any proportions that you can make out of the following four quantities : 1 inch, 3 inches, 1 foot, 1 yard. [Hint. First express all the quantities in inches.] 5. State any proportions you can make out of the following four quantities : 1 pint, 1 quart, 1 gallon, 2 gallons. 6. Proceed as in Ex. 5 for the following quantities : 1 second, 1 minute, half an hour, a day and a half. 7. Proceed as in Ex. 5 for the following quantities : 1 cent, 1 dollar, 1 centimeter, 1 meter. [Hint. Compare money ratio with distance ratio.] 8. Proceed as in Ex. 5 for the following quantities : 8 ounces, 1 pound, 1 pint, 1 quart. 9. Proceed as in Ex. 5 for the following quantities : 25 miles an hour, 30 miles an hour, 10 gallons of gasoline, 12 gallons of gasoline. For further exercises on this topic, see Appendix, p. 306. 110. Algebraic Proportions. If we consider the algebraic fraction (a-6)/(ab) we see (upon dividing both numerator and denominator by ab) that it reduces to a/b. In other words, we have ab _ a ab2 6 This is an example of an algebraic proportion. Similarly 2 x2y = 2 • 4 xyz 2 z is an algebraic proportion, and it may be written also in the form 2 xły: 4 xyz=x: 2 z. Likewise, since we have (a? — 62) : (a−b)=(a+b):1. 111. Principle. Let a/b=c/d be any proportion. By multiplying both sides of this equality by bd (Axiom III, $ 9) we obtain Xbd= xbd, or ad=bc. This result may be stated in words as follows: PRINCIPLE. In any proportion the product of the means (see § 109) equals the product of the extremes. This principle is often useful in testing the correctness of a proportion. Thus, 6: 9=14:21 is a true proportion because the product of the means, which is 9X14, is equal to the product of the extremes, which is 6X21; but 6:9=8:15 is not a true proportion because 9X8 is not equal to 6X15. Similarly 23 : x2y=x:y because xy · X=X3 . y. |