CHAPTER XI RATIO AND PROPORTION 108. Ratio. The quotient of one number divided by another of the same kind is called their ratio. Thus, the ratio of 12 inches to 6 inches is the fraction 12, or 4. The ratio of 2 feet to 3 feet is the fraction. The ratio of 10 cents to $1 is, or. Note that in every case a ratio is simply a fraction of the kind studied in arithmetic. The first number, or dividend, is called the antecedent; the second number, or divisor, is called the consequent. Thus, in the ratio, the antecedent is 6 and the consequent is 7. EXERCISES 1. What is the ratio of 5 quarts to 8 quarts? of 5 quarts to 10 quarts? 2. What is the ratio of 18 inches to 3 inches? of 18 inches to 1 foot? 3. What is the ratio of a foot to a yard? of a yard to an inch? 4. A stick was divided into two parts one of which contained 2 units and the other 7 units. What was the ratio of the two parts? 5. State (as a fraction in simplest form) the value of each of the following ratios. 6. Find the value of each of the following ratios. (a) to . (b) & to 8. (c) 1 to 23. 7. Give (as a fraction) the simplest form for each of the following ratios. (a) 2 a to 2 b. (b) 3 a2 to 6 b2. (c) a2-b2 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. 22, p. 22) will be 2 πr. Then the circumference (see Ex. Therefore, the ratio of the circumference Ans. to the radius will be 2 πr/r, or 2 π. 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 the square will be 2 r. Now proceed as in the FIG. 42. solution of Ex. 10, using Ex. 25, p. 22, and remembering that the area of the square here will be (2 r)2, or 4 r2.] The circle in Fig. 42 is said to be inscribed in the square; the square is circumscribed about the circle. FIG. 43. 12. Find the ratio of the volume of any cube to that of the sphere that will just fit inside it. [HINT. Let r be the radius of the sphere and use Ex. 28, p. 23.] 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 the cylinder to the volume of the sphere is simply. FIG. 44. [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. Likewise, we may write =, and }=}}, and 1 2 ; hence these are all true proportions. But 1 = 3 -6 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:b:: c: d. c:d, a:b::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. (b) 3:4-6:8. (c) 1:2:2:-4. :::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. |