5. What is the ratio of 21 gallons to f of a hogshead ? Ans. 1. 6. What is the ratio of } of t of $ 2 : of $0.50 ? Ans. 7. What is the inverse ratio of 24:6? Ans. À 8. What part of 36 is 4 ? Ans. š. 9. What part of a farm of 94A. 2R. 16rd. is 11 A. 3R. ? 10. Which is the greater, the ratio of 17 to 9, or of 39 to 19 ? Ans. 39 : 19. 11. By how much does the ratio of 36 X 4 X 3:12 X 16 X 2 exceed that of 60 • (3 x 5): 20 X 2=8? Ans. 13. 12. What is the inverse ratio of .02 : 2.503 ? 13. Which is the greater, the ratio of į of } : } of , or that of 5:4? 14. The height of Bunker Hill Monument is 220 feet, and that of the great pyramid, Egypt, 500 feet ; what is the ratio of the height of the former to that of the latter ? 15. A certain farm contains 180 acres, and the township of which it forms a part is 36 square miles in extent. What is the ratio of the latter to the former ? 16. Find approximate values for the ratio of 4900 to 11283. Ans. 1, 4, 1},4, 7795; &c. 17. The ratio of the circumference of a circle to its diameter is 3.141592. Required approximate values for this ratio. Ans. 3, 2, 138, 13, &c., or 3, 34, 310, 311, &c. Ans. 15. ANALYSIS BY RATIO. 331. Operations by analysis may often be much abridged by ratio. Thus, frequently, it is more convenient to multiply or divide by the ratio a number bears to a unit of the same kind, than to multiply or divide by the number itself. This form of analysis is much used by business men ; and, like that by aliquot parts (Art. 114), is sometimes called Practice. EXAMPLES 1. What cost 14 tons 15cwt. 3qr. 20lb. of iron, at $ 60 a ton ? Ans. $ 887.85. OPERATION $ 60.00 = cost of 1 ton. 14 66 $ 840.00 14 tons. (10cwt.: 1 ton 3); } of $ 60 - 30.00 = 10cwt. (5cwt.: 10ewt.= }); } of $30= 15.00 5cwt. (2qr. : 5€wt. 1o); tb of $15 = 1.50= 2qr. (lqr. : 2cwt. 1); } of $ 1.50 = 0.75 Iqr. (15lb. : 3qr. = }); of $ 2.25 0.45 151b. (5lb. : 151b. }); } of $ 0.45 = 0.15 = 5lb. Ans. $ 887.85 = 15T. 3qr. 20lb. Since 1 ton costs $ 60, 14 tons will cost 14 times $ 60, or $ 840. 15cwt. 10cwt. + 50wt. Since the ratio of 100wt. to 1 ton or 20cwt. 1, 10cwt. will cost { as much as 1 ton, or $ 30 ; and as the ratio of 5cwt. to 10cwt. = 1, 5cwt. will cost } as much as 10cwt. or $ 15. 3qr. 2qr. + Iqr. Since the ratio of 2qr. to 5cwt. or 20qr. 1, 2qr. will cost 15 as much as 5cwt., or $ 1.50; and as the ratio of 1qr. to 2qr. = 1, lqr. will cost as much as 2qr., or $ 0.75. 20lb. = 15lb. + 516. Since the ratio of 15lb. to 3qr. or 75lb. }, 15lb. will cost } as much as 3qr., or $ 0.45; and as the ratio of 5lb. to 15lb. Ž, 5lb. will cost } as much as 15lb., or $ 0.15. The cost of the several parts equals the cost of the whole, or $ 887.85, Ans. 2. What is the value of 17 acres 3 roods 35 rods of land, at $ 80 per acre ? Ans. $ 1437.50. 3. What cost 16cwt. 3qr. 10lb. of guano, at $ 2.50 per cwt. ? 4. What cost 27cwt. lqr. 201b. of coffee, at $ 14 per cwt. ? Ans. $ 384.30. 5. If 1 yard of cloth cost $5.60, what will 7yd. 3qr. 2na. cost ? Ans. $ 44.10. 6. What cost 7 tons 13cwt. 2qr. of hay, at $ 20 per ton ? 7. What cost 99bu. lpk. 4qt. of wheat, at $1.92 per bushel ? Ans. $ 190.80. OPERATION. 66 $ 1.92 = cost of lbu. 100 $ 192.00 100bu. (2pk. : 1bu. = 1); 4 of $ 1.92 = 0.96 ) 2pk. (4qt. : 2pk. = 1); 1 of $0.96 = 0.24 $ 4qt. Ans. $190.80 99bu. 1pk. 4qt. 66 The quantity being nearly 100 bushels, we find the cost of 100 bushels by annexing two ciphers to $ 1.92, the cost of i bushel, and obtain $ 192, from which we subtract the cost of 2pk. 4qt., the difference of quantity between that given and 100 bushels; the cost of 2pk. $ 0.96; and that of 4qt. $ 0.24 ; $ 192 $ 0.96 + $ 0.24 $ 190.80 Aus. 8. What cost 19yd. 3qr. 2na. of cloth, at $ 4.40 per yard ? Ans. $ 87.45. 9. How much must be paid for 24A. 3R. 20p. of land, at $ 32 per acre ? 10. How much must be paid for 1991b. 12oz. of butter, at $ 0.30 per Ans. $ 59.925. 11. What cost 714 yards of broadcloth, at 15s.6d. per yard? Ans. 553£. 7s. 12. How much must be paid for the services of a man 2y. 9mo. 15da., at $ 450 yer year? Ans. $ 1256.25. 13. If 1 acre of land cost $ 80.50, what will 25 acres 2 roods 35 rods cost ? Ans. $ 2070.35+. 14. What cost 4981b. of tea, at 2s. 6d. per lb. ? 15. If lcwt. 2qr. 121b. of alum can be purchased for $ 4.05, how much can be purchased for $28.35 ? Ans. 11cwt. lqr. 91b. OPERATION. . $ 28.35 $ 4.05 = 7; 1cwt. 2qr. 12lb. x 7 = = llcwt. lqr. 91b. Ans. Since the ratio of $ 28.35 to $ 4.05 7, $ 28.35 will purchase 7 times as much as $4.05. By multiplying what the latter will purchase by the ratio, we have the answer required. 16. If ilgal. 3qt. Ipt. of molasses cost $ 5.83%, what will 35gal. 2qt. 1 pt. cost ? Ans. $ 17.514. 17. If 24 yd. 3qr. of cloth cost $ 49.50, what will 12yd. Iqr. 2na. cost ? 18. If 17bu. 2 pk. 4qt. of oats be paid for 14bu. 3pk. of salt, what quantity of oats must be paid for 73bu. 3pk. of salt? Ans. 88bu. Opk. 4qt. 19. If $ 9.75 will purchase 1T. 2cwt. 2qr. 15lb. of coal, how much will $ 3.25 purchase? 20. If a train of cars move at the average velocity of 27m. 3fur. 20rd. per lh. 20m., how far will it move in 4h. ? Ans. 82m. 2fur. 20rd. PROPORTION. 332. A PROPORTION is an equality of ratios. Any four numbers are in proportion, when the ratio of the first to the second is the same as that of the third to the fourth. Thus, the ratios 9 : 3 and 6: 2, being equal to each other, when written, 9:3 6:2, or , form a proportiori. Proportion is written with the sign of equality (=), or, as is more common, with four dots (::), between the ratios. Thus, 9:3 6: 2, or 9: 3 :: 6 : 2, expresses a proportion, and is read, The ratio of 9 to 3 is equal to the ratio of 6 to 2, or 9 is to 3 as 6 is to 2. 333. The terms of a proportion are the four numbers which form the proportion. These numbers are also called proportionals. The first and third terms, or proportionals, are called antecedents, the second and fourth are called consequents ; the first and last are called the extremes, the second and third the means; the first and second compose the first couplet, the third and fourth compose the second ; and when the ratio of the first of three terms is to the second as the ratio of the second is to the third, the second term is called a mean proportional to the other two terms. 334. A direct proportion is an equality between two direct ratios; an inverse or reciprocal proportion is an equality between a direct and an inverse or reciprocal ratio. Thus, the numbers 4, 2, 6, 3 are, as they stand, in direct proportion, denoting 4:2:: 6:3; but in the order 4, 2, 3, 6, are in inverse proportion, denoting that 4:2::}:, or the direct ratio of 4 to 2 is equal to the inverse ratio of 3 to 6. NOTE. — The term proportion, used alone, always means direct proportion. 335. In any proportion, if the antecedents or consequents, or both, are divided, or multiplied, by the same number, they are still proportionals. Thus, dividing the antecedents of the proportion 4:8::10 : 20 by 2, we have 2:8:: 5 : 20; dividing the consequents by 2, we have 4:4::10:10; and dividing both the antecedents and consequents by 2, we have 2:4:: 5:10; each of which results is a proportion, since if we divide the second term of each by the first, and the fourth by the third, the two quotients will be equal. The effect is the same when the terms are multiplied by the same number. 336. In every proportion the product of the two extremes is equal to the product of the two means. Thus, the proportion 16:8:: 20:10 may be expressed 16 fó. Now, if we reduce these fractions to a common denominator, we have 600 19600 ; but in this operation we multiplied together the two extremes of the proportion, 16 and 10, and the two means, 8 and 20; thus, 16 X 10 = 8 X 20. Hence, 1. If the extremes and one of the means are given, the other mean may be found by dividing the product of the extremes by the given mean ; or, 2. If the means and one of the extremes are given, the other extreme may be found by dividing the product of the means by the given extreme. SIMPLE PROPORTION. 337. Simple Proportion is an equality between two simple ratios. NOTE. — Simple Proportion is sometimes called the Rule of Three, and formerly was termed by arithmeticians the Golden Rule. 338. The object of that part of simple proportion which is usually included in arithmetics, is to find a fourth proportional to three given numbers, or, in other words, to find the fourth term of a proportion, when the other three terms are given. Ex. 1. If a man travel 243 miles in 9 days, how far will he travel in 24 days ? Ans. 648 miles. Since 9 days have the same ratio to 24 days as 243 miles, 9 da. : 2 4 da. :: 2 4 3 m. : the distance of travel in 9 days, 2 4 have to the distance of travel 9 7 2 in 24 days, we have the first three terms of a proportion 486 given, namely, the two means 9 ) 5832 and one of the extremes, from which to find the required exAns. 6 4 8 m. Extreme. treme. Now, to arrange the given numbers in the order of OPERATION. Extreme. Mean. Mean. Extreme. m. |