The ratio of the reciprocals of two numbers is called the reciprocal, or inverse, ratio of those numbers. Thus, the reciprocal of 2 is ; of 3, ; and the reciprocal ratio of 2 to 3 is the ratio of to. The ratio of tois } ÷ } = } × } = §, or 3 : 2. Hence, 270. The Reciprocal or Inverse Ratio of any two terms is the same two terms with their positions interchanged. 271. Variation. One thing varies directly as another, when it increases as the other increases, and decreases as the other decreases. Thus, if the rate of motion be uniform, the distances moved over will vary directly as the time; that is, in twice the time, the distance moved over will be twice as great, and in three times the time, it will be three times as great, &c. One thing varies inversely as another, when it increases as the other decreases, and decreases as the other increases. Thus, the time of moving over a given space varies inversely as the velocity; that is, if the velocity is twice as great, the space will be moved over in one-half the time; if the velocity is three times as great the space will be moved over in one-third the time; &c. PROPORTION. 272. Proportion is equality of ratios. 273. A proportion is a statement of the equality of ratios. Thus, 66 = {, which may also be expressed 6 : 3= 8 4, or 6 3 8 4, is a proportion. The last is read 6 is to 3 as 8 is to 4." Any one of these expressions may be changed to either of the other two at pleasure. 274. The numbers which form a proportion are called proportionals, or terms of the proportion. 275. Three numbers are proportionals when the first is to the second as the second is to the third. Thus, 2: 4 :: 4 : 8. In this proportion the second term, 4, is called the mean proportional between the other two terms. 276. Four numbers are proportionals when the first is to the second as the third is to the fourth. Thus, 2: 4 :: 6:12. 277. The Extremes of a proportion are the 1st and 4th terms. 278. The Means are the 2d and 3d terms. 279. The Antecedents are the 1st and 3d terms. 280. The Consequents are the 2d and 4th terms. 281. In every proportion, the product of the extremes is equal to the product of the means. Thus, in the proportions 3 : 6 :: 5 : 10, 3 × 10 6 × 5; and 7: 2 :: 14: 4, 7×4 = 14 × 2. This Principle is called the test of the accuracy of the proportion. This Principle also affords. an easy method of finding any one term that may be wanting in a proportion. EXAMPLE 1.-What is the 4th term in 9: 18 :: 5 : ( ) ? SOLUTION. 18 x 5 9 = 10, Ans. EXPLANATION.-The product of the means is 5 × 1890. The product of the extremes must also be 90 (281). Since one of the extremes is 9, the other extreme must be 90÷9: = 10, Ans. EXAMPLE 2.—What is the 2d term in 3 : ( ) :: 4 : 12? EXPLANATION.-The product of the extremes is 3 × 12 = 36. SOLUTION. 3 x 12 4 = 9, Ans. The product of the means must also be 36 (281). Since one of the means is 4, the other mean must be 36 ÷ 4 = 9, Ans. RULE I. Divide the product of the means by the given extreme; the quotient is the other extreme. II. Divide the product of the extremes by the given mean ; the quotient is the other mean. Cancelling common factors from the antecedents or consequents, or either couplet, will not destroy the proportion; because the results will in each case stand the test. PROBLEMS. Find the unknown term in the following: 1. 134: (). 2. 4 2 10 :'( ). 3. 68 ( ): 12. 4. 9:5:( ): 10. 5. 3:( ): 8: 6. 6. 5:( ) :: 20: 12. In reference to the number of terms in their ratios, proportions are Simple or Compound. 282. A Simple Proportion is a statement of the equality of two simple ratios. Thus, 5 8: 10: 16, or = 18, is a simple proportion. 283. A Compound Proportion is a statement of the equality of a simple and a compound ratio, or of two compound ratios. Thus, 4 6, or 2 x 6: 2:3 SIMPLE PROPORTION. Simple Proportion is employed to find a fourth proportional, when three are given. Oral Exercises EXAMPLE.—If 9 lb. sugar cost 81, what cost 12 lb. ? SOLUTION.-If 9 lb. cost 814., 12 lb. will cost more in the ratio of 12 to 9, or 12 = as much. of 819 = 108, or $1.08. 108. Therefore, if 9 lb. of sugar cost 814, 12 lb. cost PROBLEMS. 1. If 6 yd. carpet cost $15, what do 10 yd. cost? 2. If a tree 20 ft. high casts a shadow 8 ft. long, how high is the tree whose shadow is 32 ft. long? 3. 7 bbl. molasses cost $63. What cost 14 bbl. ? 4. 2 lb. coffee last 6 persons 8 days. How long will it last 24 persons? 5. Paid $1.20 for 3 lb. butter. How many lb. can be bought for $3.60 ? 6. 5 horses eat a ton of hay in 8 months. How long would it take 2 horses to eat the same amount ? 7. If the boarding of 3 men for a week costs $21, how many men can be boarded the same length of time for $63 ? 8. If a 10 loaf of bread weighs 12 oz. when flour is $8 a bbl., how much should it weigh when flour is $16 a bbl.? 9. If 3 lb. sugar cost 33, what will 6 lb. cost? 10. If 4 bu. apples cost $3, what do 8 bu. cost? 11. If 5 T. iron cost $400, what will 8 T. cost? 12. If 6 bu. wheat cost $9, what do 5 bu. cost? 13. If 11 lb. coffee cost $1.65, what do 12 lb. cost? 14. If 12 lb. rice cost 72, what do 11 lb. cost? 15. If 15 bbl. beef contain 3000 lb., how many do 20 bbl. contain? Written Exercises EXAMPLE 1.-If 50 bu. wheat cost $130, what will 60 bu. cost at the same rate? |