ART. 238. PROPORTION is an equality of ratios. Thus the ratios 9: 3 and 12 : 4 are equal, and when united form a proportion. Proportion is usually expressed by four dots between the two ratios; thus, the proportion in the preceding example is written 9:3:12:4, and is read, 9 is to 3 as 12 to 4. The numbers which form a proportion are called proportionals. The first and third are called antecedents, the second and fourth are called consequents; also, the first and last are called extremes, and the remaining two the means. ART. 239. Any four numbers are said to be proportional to each other when the first contains the second as many times as the third contains the fourth; or when the second contains the first as many times as the fourth contains the third. Thus, 9 has the same ratio to 3 that 12 has to 4, because 9 contains 3 as many times as 12 contains 4. ART. 240. If the antecedents or consequents of a proportion, or both, are divided 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 consequents and antecedents by 2, we have 2: 4 :: 5:10; each of which 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. = = 126. 639 ART. 241. The product of the extremes of a proportion is equal to the product of the means. Thus, the proportion 14: 7:: 18: 9 may be expressed fractionally, 8. Now, if we reduce these fractions to a common denominator, we have 126 but in this operation we multiplied together the two extremes of the proportion, 14 and 9, and the two means, 18 and 7; thus, 14 X9=18 X 7. QUESTIONS. - Art. 238. What is proportion? How is proportion expressed? What are the numbers called that form a proportion? Which are the extremes? Which the means? Art. 239. When are numbers said to be in proportion to each other? - Art. 240. What is the effect of dividing the antecedents or consequents of a proportion? Of multiplying them? - Art. 241. How does the product of the extremes compare with that of the means? ART. 242. 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. Thus, if the extremes are 3 and 24, and the given mean 6, the other mean is 12; because 24 × 3 72; and 72÷6 = 12. ART. 243. 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. Thus, if the means are 8 and 16, and the given extreme 4, the other extreme is 32; because 16 X 8128; and 128 ÷ 4 = 32. SIMPLE PROPORTION. ART. 244. SIMPLE PROPORTION is an expression of the equality between two simple ratios. NOTE.Simple Proportion is sometimes called the Rule of Three. ART. 245. Method of stating and solving questions in Simple Proportion. Ex. 1. If 7lb. of sugar cost 56 cents, what will 36lb. cost? Ans. $2.88. Extreme. OPERATION. Mean. Mean. 7 lb. 3 6 lb. :: 5 6 cts. 36 336 168 7)20.16 Since 7lb. have the same ratio to 361b. as 56 cents, the cost of the former, have to the cost of the latter, we have the first three terms of a proportion given, namely, one of the extremes and the two means. Now, to ascertain which of these terms are the means, and which the extreme, we arrange them in the order of a proportion, or state the question, by making 56 cents the third term, because it is of the same kind, and has the same proportion to the required answer, or fourth term, as the first has to the second. From the nature of the question, since the answer will be more than 56 cents, or the third term, the second term must be greater than the first; we make 36lb. the second term, and 71b. the first, and then proceed as in Art. 246. $2.88 Extreme. QUESTIONS. Art. 242. If the extremes and one of the means are given, how can the other mean be found? - Art. 243. When the means and one of the extremes are given, how can the other extreme be found?- Art. 244. What is simple proportion? How many terms are given in questions in simple proportion? BY ANALYSIS. If 7lb. cost 56 cents, 1 pound will cost of 56 cents, which is 8 cents. Then, if 1lb. cost 8 cents, 361b. will cost 36 times as much; that is, 36 times 8 cents, which are $2.88, Ans. as before. OPERATION. bar. bar. $. 76:12:: 456 Ex. 2. If 76 barrels of flour cost $456, what will 12 barrels cost? Ans. $72. We state this question by making $456 the third term, because it is of the same kind of the required answer. Then, since the answer must be less than $456, because 12 barrels will cost less than 76 barrels, we make 12 barrels, the smaller of the other two terms, the second term, and 76 barrels the first term, and proceed as before. 76)5472 ($7 2 12 532 152 BY ANALYSIS. 152 $456, which is $6. of If 76 barrels cost $456, 1 barrel will cost 12 times as much; that is, $72, Ans. as before. Ex. 3. If 3 men can dig a well in 20 days, how long will it take 12 men? OPERATION. men. men. days. 12:3:: 20 3 12)60 BY ANALYSIS. 5 days. Ans. 5 days. Since the required answer is days, we make 20 days the third term. And as 12 men will dig the well in less time than 3 men, the answer must be less than 20 days. Therefore we make 3 men the second term, and 12 men the first, and proceed as in the other examples. -If 3 men dig the well in 20 days, it will take one man 3 times as long, that is, 60 days. Again, we say, If one man dig the well in 60 days, 12 men would dig it in 12 of 60 days, that is, 5 days, Ans. as before. From the preceding examples we deduce the following 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 together, and divide their product by the first. QUESTIONS. What is meant by stating the question? Which of the terms given in the example do you make the third? Why? Which the second? Why? Which the first? Why? After the question is stated, how do you obtain the answer? NOTE 1. When the first and second terms consist 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 denomThe answer will be the same denomination as the ination mentioned in it. 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. Ex. 4. If 16 bushels of wheat are worth $24, what are 96 bushels worth? OPERATION BY CANCELLATION. bu. bu. $. 16: 96:24 6 96×24 = $144 BY ANALYSIS AND CANCELLATION. 6 $24 × 96 = 16 $144 Ans. $144. We first state the question as directed in the rule, and then write the second and third terms above a horizontal line, with the sign of multiplication between them, for a dividend, and the first term below the line for a divisor, and cancel the common factors. By this method of analysis we first place the $24, which is of the same kind of the required answer, above a line for a dividend; and then say, Since $24 is the price of 16 bushels, 1 bushel will cost of $24, and express the division by placing the 16 below the line for a divisor. Now, since we have an expression for the price of 1 bushel, we next express the multiplication of it by 96 bushels, the price of which is required, and then cancel as before. EXAMPLES FOR PRACTICE. 5. What cost 9 gallons of molasses, if 63 gallons cost $14.49? Ans. $2.07. 6. What cost 97 acres of land, if 19 acres can be obtained for Ans. $1721.75. $337.25? 7. If a man travel 319 miles in 11 days, how far will he travel Ans. 1363 miles. in 47 days? 8. When $120 are paid for 15 barrels of mackerel, what Ans. $632. will be the cost of 79 barrels? QUESTIONS. What is the rule for simple proportion? How should the pupil perform the questions? How do you state the question and arrange the terms for cancellation? What do you cancel? How do you arrange the terms for cancellation by analysis? 9. If 9 horses eat a load of hay in 12 days, how many horses would it require to eat the hay in 3 days? Ans. 36 horses. 10. When $5.88 are paid for 7 gallons of oil, what cost 27 gallons? Ans. $22.68. 11. When $10.80 are paid for 91b. of tea, what cost 147lb Ans. $176.40. 12. What cost 27 tons of coal, when 9 tons can be purchased for $85.95? Ans. $257.85. 13. If 15 tons of lead cost $105, what cost 765 tons? Ans. $5355.00. 14. If 16hhd. of molasses cost $320, what cost 176hhd.? Ans. $3520.00. 15. If 15cwt. 3qr. 17lb. of sugar cost $124.67, what cost 76cwt. 2qr. 19lb.? Ans. $600.56+. 16. If the cars on the Boston and Portland Railroad go one mile in 2 minutes and 8 seconds, how long will they be in passing from Haverhill to Boston, the distance being 32 miles? Ans. 1h. 8min. 16sec. 17. If a man travels 3m. 7fur. 18rd. in one hour, how far will he travel in 9h. 45min. 19sec.? Ans. 38m. 2fur. 32rd+ 18. A fox is 96 rods before a greyhound, and while the fox is running 15 rods the greyhound will run 21 rods; how far will the dog run before he can catch the fox? Ans. 336 rods. 19. If 5 men can reap a field in 12 hours, how long would it take them if 4 men were added to their number? Ans. 63 hours. 20. Ten men engage to build a house in 63 days, but 3 of their number being taken sick, how long will it take the rest to complete the house? Ans. 90 days. 21. If a 4 cent loaf weighs 5oz. when flour is $5 per barrel, what should it weigh when flour is $7.50 per barrel? Ans. 34oz. 22. If 7 men can mow a field in 10 days when the days are 14 hours long, how long would it take the same men to mow the field when the days are 13 hours long? Ans. 1019 days. 23. If 291b. of butter will purchase 40lb. of cheese, how many pounds of butter will buy 791b. of cheese? Ans. 574 lb. 24. If of a yard cost of a dollar, what will $94 cost? yd. yd. $. of a yard Ans. $0.767. 1X }} X { = }}}=$0.76, Ans. |