The quantity bring nearly 100 bushels, we find the cost of 100 bushels by annexing two ciphers to $ 1.92, the cost of 1 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 -f$0.24 = $190.80 Ans. 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 1b.? 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. $ 2070.35-4-. 14. What cost 4981b. of tea, at 2s. 6d. per 1b.? 15. If lcwt. 2qr. 121b. of alum can be purchased for $ 4.05, how much can be purchased for $ 28.35? Ans. llcwt. lqr. 91b. OPERATION. . $28.35-4-$4.05 =7; 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 11 gal. 3qt. lpt. of molasses cost $5.83£, what will 35gal. 2qt. lpt. cost? Ans. $ 17.51A. 17. If 24yd. 3qr. of cloth cost $ 49.50, what will 12yd. lqr. 2na. cost? 18. If 17bu. 2pk. 4qt. of oats be paid for 14bu. 3pk. of salt, what quantity of oats must be paid for 7obu. 3pk. of salt? Ans. 88bu. Opk. 4qt. 19. If $9.75 will purchase IT. 2cwt. 2qr. 151b. 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, E) : 3 = 6 : 2, or § — form a proportion. 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. 333i 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 J^ — fg. Now, if we reduce these fractions to a common denominator, we have Jg6^1- = V6(r , uut m tn's 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. 337i 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 Ans. 648 miles. Since 9 days have the same ratio to 24 days as 243 miles, the distance of travel in 9 days, have to the distance of travel in 24 days, we have the first three terms of a proportion given, namely, the two means and one of the extremes, from which to find the required extreme. Now, to arrange the given numbers in the order of he travel in 24 days? OPERATION. Extreme. Mean. 9 da. : 2 4 da.i Ans. 6 4 8 m. Extreme. a proportion, or state the question, we make the 243 miles the third term, because it is of the same kind as the required fourth term, and as from the nature of the question the latter must be greater than the third term, we make the greater of the other two numbers the second term, and the less the first; and, then, the product of the means divided by the given extreme gives the required extreme (Art. 336). By Analysis. — If a man travel 243 miles in 9 days, he will in 1 day travel \ of 243 miles = 27 miles; then, if he travel 27 miles in 1 day, in 24 days he will travel 24 times 27 miles = 648 miles, the answer, as before. By Ratio. — 9 : 24 = = f; 243 miles -t- f = 648 miles, Ans. 2. If 15 yards of cloth cost $ 48.90, what will 5 yards cost? Ans. $16.30. Operation. .We statg tue question by making 15:5::$4 8.9 0:$— $ 48.90 the third term, because it is of the same kind as the required term. ,5 X 4 8.9 0 icon Then, since the answer must be less — = 1 6.3 0. tnan $ 48.90, because 5 yards will cost g less than 15 yards, we make 5 yards, 3 Ans. $ lo.oO. tne iess 0f the two numbers, the second term, and 15 yards the first; and proceed as in the first example, except that we abridge the work by canj cellation. By Analysis. — If 15 yards cost $ 48.90, 1 yard will cost fa oi $48.90 = $ 3 26; then, if 1 yard cost $3.26, 5 yards will cost 5 times $ 3.26 == $ 16.30. Rule. — Write the given number that is of the same kind as the required fourth term, or answer, for the third term of the proportion. Of the other two numbers write the larger for the second term, and the less for the first, when the answer should exceed the third term; but write the less for the second term, and the larger for the first, when the answer should be less than the third term. Multiply the second and third terms lot/ether, and divide their product by the first; or divide the third term by the ratio of the first term to the second. Note 1. — When the first and second terms are of different denominations, they must be reduced to the same denomination; and when the third term is a compound number, it must be reduced to the lowest denomination mentioned in it. The answer will be of the same denomination as the third term. Note 2. — To shorten the operations, factors common to the dividend and divisor may be cancelled. Note 3. — The pupil should perform these questions by analysis, as well as by proportion, and introduce cancellation when it will abbreviate the operation. Examples. 3. It. 16 acres of land cost $720, what will 197 acres cost? Ans. $ 8865. 4. If $ 8865 buy 197 acres, how many acres may be bought for $ 720? 5. What will 84hhd. of molasses cost, if 15hhd. can be purchased for $ 175.95? Ans. $ 985.32. 6. If $ 100 gain $ 6 in 12 months, how much would it gain in 40 months? Ans. $ 20. 7. If a certain vessel has provisions sufficient to last a crew of 10 men 45 days, how long would the provisions last if the vessel were to ship 5 new hands? Ans. 30 days. 8. If 7 and 9 were 12, what, on the same supposition, would 8 and 4 be? 9. If 9 men can perform a certain piece of labor in 17 days, how long would it take 3 men to do it? Ans. 51 days. 10. If 3 men can perform a piece of labor in 51 days, how many must be added to the number to perform the labor in 17 days? Ans. 6. 11. A rectangular piece of land containing an acre is 5£ rods in breadth. What is its length? Ans. 29T'T rods. 12. If $100 gain $ 6 in a year, how much will $850 gain? 13. If $ 100 gain $ 6 in a year, how much would be sufficient to gain $ 32 in a year? Ans. $ 533.33J. 14. If 20 gallons of water weigh 1671b., what will 180 gallons weigh? Ans. 15031b. 15. If a staff 3 feet long cast a shadow of 2 feet, how high is that steeple whose shadow is 75 feet? Ans. 112£ feet. 16. If 5|cwt. be carried 36 miles for $4.75, how far might it be carried for $ 160? Ans. 1212T§ miles. 17. If 100 workmen can perform a piece of work in 12 days, how many men are sufficient to perform the work in 8 days? Ans. 150. 18. If TV of a yard cost ^ of a dollar, what will f of a yard cost?" Ans. $ 0.48. 19. What must be paid for 21A. 3R. 20p. of land, if 36A. 3R. cost $ 1260? Ans. $ 750. |