Find the ratio in lowest terms of 14. 1 mi. to 120 rd. 15. 3 gal. to 2 qt. 1 pt. 16. 8 yd. to 9 inches. 17. $1.68 to $.24. 18. 1 wk. 3 da. 12 hr. to 9 wk. 19. 10 A. 60 sq. rd. to 6 A. 110 sq. rd. 20. 25 bu. 2 pk. 6 qt. to 40 bu. 4.5 pk. 21. 7 cwt. 40 lb. to 5 T. 8 cwt. 22. Find the reciprocal of the ratio of 75 to 15. 23. Find the reciprocal of the ratio of 2 qt. 1 pt. to 4 gal. 1 qt. 1 pt. PROPORTION. 365. A proportion is an equation in which each member is a ratio; or it is an equality of ratios. The equality of two ratios may be indicated by the sign =, or by the double colon (::). Thus, we may indicate that the ratio of 8 to 4 is equal to that of 6 to 3, in any of the following ways: 6 8 8÷4= 6 ÷ 3. This proportion, in any of its forms, is read, The ratio of 8 to 4 is equal to the ratio of 6 to 3, or, 8 is to 4 as 6 is to 3. 366. Since each ratio consists of two terms, every proportion must consist of at least four terms. Each ratio is called a couplet, and each term is called a proportional. 367. The antecedents of a proportion are the first and third terms; the consequents are the second and fourth terms. 368. The extremes are the first and fourth terms; the means are the second and third terms. In the proportion 8: 4 :: 10: 5, 8, 4, 10, and 5 are the proportionals; 8:4 is the first couplet, 10 : 5 the second couplet; 8 and 10 are the antecedents, 4 and 5 are the consequents; 8 and 5 are the extremes, 4 and 10 are the means. When the two means are the same number, that number is said to be a mean proportional between the two extremes. 369. A simple proportion is an expression of equality between two simple ratios. Thus, 4:8= 5:11, and 8: 12 10 15 are simple proportions. 370. An expression of equality between two ratios, one or both of which are compound, is sometimes called a compound proportion. Thus, 3:4 = 14:28 is a proportion, composed of a compound and a simple ratio, and may be read, 3 x 6 is to 4 x 9 as 14 is to 28, equivalent to the simple proportion 18: 36 = 14: 28. 371. The solution of problems in proportion depends upon the following PRINCIPLES.-I. The product of the extremes of a proportion is equal to the product of the means. II. The product of the extremes, divided by either mean, will give the other mean. III. The product of the means divided by either extreme will give the other extreme. 372. Find the term omitted, and represented by x, in each 373. The proper arrangement of the terms in the form of a proportion is called the statement of the question. 1. If 4 tons of coal cost $24, what is the cost of 12 tons ? 4 tons, and $x, the cost of 12 tons, form the other couplet. 2. If 8 men earn $320 in 8 da., what will 12 men earn in The value of the required term in this example depends upon two conditions: 1st, the number of men at work; 2d, the number of days they work. Consider each condition separately, and arrange the terms of the same unit value in couplets. Then find the required term by PRIN. III. 3. If 96 cords of wood cost $240, what will 40 cords cost? 4. If 20 horses consume 36 tons of hay in 9 mo., how many tons will 12 horses consume in 18 months? RULE.-1. Of the given numbers, make that the third term which is of the same kind as the answer, or number sought. 2. Arrange the remaining numbers, two of a kind, into couplets, in such a manner that the terms shall have the same ratio to each other as the third term has to the required term. 3. Divide the product of the means by the given extreme, and the quotient is the fourth or required term. 1. If the terms of any couplet are of different denominations, they must be reduced to the same denomination. 2. If the odd term is a compound number, it must be reduced to its lowest unit. 3. If the divisor and dividend contain factors common to both, cancel them. 5. If 12 gallons of wine cost $30, what will 63 gallons cost? 6. If 9 bu. of wheat make 2 bbl. of flour, how many barrels of flour will 100 bu. make? 7. If 6 bu. of oats cost $3, what will 94 bu. cost? 8. What will 87.5 yd. of cloth cost, if 14 yd. cost $.42? 9. If by selling $1500 worth of dry goods I gain $275.40, what amount must I sell to gain $1000? 10. What will 114 lb. of tea cost, if 3 lb. 12 oz. cost $3.50? 11. If a speculator in grain gain $26.32 by investing $325, how much would he gain by investing $2275? 12. If 16 horses consume 128 bu. of oats in 50 da., how many bushels will 5 horses consume in 90 da. ? 13. If 18 men can build 42 rd. of wall in 16 da., how many men can build 28 rd. in 8 da. ? yr. 14. If a man clears $750 by his business in 1 6 mo., how much would he gain in 3 yr. 9 mo. at the same rate? 15. If a certain business yield $350 net profits in 10 mo., in what time would the same business yield $1050 profits? 16. If $480 gain $84 interest in 30 mo., what sum will gain $21 in 15 mo. ? 17. If 24 yd. of cloth 1 yd. wide cost $3.37, what cost 361 yd., 11 yd. wide? 18. How much land worth $16.50 an acre should be given in exchange for 140 acres, worth $24.75 an acre? 19. If I gain $155.52 on $1728 in 1 yr. 6 mo., how much will I gain on $750 in 4 yr. 6 mo.? 20. If 450 tiles, each 12 in. square, will pave a cellar, how many tiles that are 9 in. by 8 in. will pave the same? CAUSE AND EFFECT. 374. The terms of a proportion have not only the relations of magnitude, but also the relations of cause and effect. 375. Causes, in computation, are things that produce something as, men at work, money lent, horses, time, etc. 376. Effects are the results of causes; as, work done, interest drawn, cost, distance traveled, etc. 377. Every problem in proportion may be considered as a comparison of two causes and two effects. Thus, if 4 tons as a cause will bring, when sold, $24 as an effect, 12 tons as a cause will bring $72 as an effect. Or, if 6 horses as a cause draw 10 tons as an effect, 9 horses as a cause will draw 15 tons as an effect. 378. Since like causes produce like effects, the ratio of two like causes equals the ratio of two like effects produced by these causes. Hence, 1st cause 2d cause 1st effect: 2d effect. 1. If 8 men earn $32 in one week, what will 15 men earn at the same rate, in the same time? couplet of the proportion. The required term is found by PRIN. III. |