CHAPTER VII. Practical Application of the Rule of Proportion. 130. The rule of proportion is divided into simple and compound. Simple proportion is the equality of the ratio of two quantities, to that of two other quantities. (see § 65.) Compound proportion is the equality of the ratio of two quantities to another ratio, the antecedent and consequent of which are respectively the products of the antecedents and consequents of two or more ratios. (see § 82, chap. IV.) Simple Proportion. 131. The object of that part of simple proportion which is usually taught in courses of arithmetic, is to find the number which has the same ratio to one of three given numbers, that there is between the other two; or to find a fourth proportional to three given numbers. Example 1. If I can purchase 4 yards of cloth for $35.50, what quantity ought I to get, at the same rate, for $106.50? This can be ascertained by simple proportion: for the quantities purchased at a given rate must be directly as the prices paid; therefore 4 yards the quantity purchased for 835-50 must be less than the quantity purchased for $106.50, in the same ratio in which the former sum of money is less than the latter, or in the ratio 3550 cents to 10650 cents, or of the abstract numbers 3550 10650. Therefore, 3550 : 10650 4 yards to the quantity sought; which fourth proportional is found (§ 68) by taking the product of the second and third terms and dividing that product by the first: thus, 10650 X 4 3550 42600 12 yards. 2. If I pay $106-50 for 12 yards of cloth, what must I pay, at the same rate, for 4 yards? Here the thing sought being a sum of money, the given sum of money, $106.50 must be the third term of the analogy. And as the answer must be a less sum of money, the two given quantities of cloth must be stated in a greater inequality; that is, as 12 to 4. So that, as 12: 4 :: 10650 cents to the sum sought. Therefore 10650×4÷÷÷12=3550 cents, or $35.50, the answer. This question is formed from the last, and the result obtained is sufficient to prove the accuracy of the work in both solutions. 3. If a mason can build a wall in 6 days, working 7 hours a day, how many hours a day must he work in order to build it in 5 days? It is plain that he must work a greater number of hours each day, and therefore the fourth term of the analogy must be greater than the third term, 7 hours; and hence the first two terms must be stated in a ratio of less inequality; thus 5: 6: 7 hours to the number of hours sought. The answer, therefore is 4, or 8 hours; that is, 8 hours and 24 minutes. The truth of this may be proved by forming another question in which this answer shall be one of the given terms, and any one of the former given terms shall be the term sought. Thus : 4. If a mason, working 8 hours and 24 minutes a day, build a wall in 5 days, how many hours a day must he work in order to build it in 6 days? Here it is plain that he must work a less number of hours each day; and therefore the fourth term of the analogy must be less than the third term, 83 hours: and hence the first two terms must be stated in a ratio of greater inequality, that is, as 6: 5:: 82 hours to the number of hours sought. The answer, therefore, is 8×5÷6, or 42 =7. hours.* *We may, in like manner, form two other questions from Ex. 3.Thus if a mason working 8 hours and 24 minutes a day, build a walk in 5 days, in how many days shall he build it, working 7 hours a day? Or, 2dly, if he build it in 6 days, working 7 hours a day, in how many days shall he build it, working each day 8 hours and 24 minutes? And thus whenever a question has been solved by the rule of proportion, the student may be profitably exercised in forming three other questions adapted to prove the truth of his answer, since we can find any one of the four terms of an analogy from having given the three others. It is proper to observe that the first two examples are performed by what is usually called Direct Proportion, and the last two examples by Inverse Proportion; or, in other words, by what is commouly, though improperly, called, "The Rule of Three direct and inverse." This distinction is perfectly useless; and, like all useless distinctions, it is calculated only to perplex the learner, and to render a simple subject complicated. The preceding four examples may serve to illustrate the following general rule for solving all questions in simple proportion, whether direct or in verse. Three numbers being given, to find a fourth Pro portional. 132. RULE.- Arrange the three given terms in the same line, in succession, placing the one which is of the same kind with the required term the third in order; and if, by the nature of the question, the required term is to be greater than the third term, put the greater of the other two terms in the second place; otherwise put the less in that place. Then, if the first two terms be not of the same simple denomination, reduce them to the same denomination, usually the lowest mentioned in either. Find the product of the second and third terms, and divide it by the first. The quotient is the fourth proportional in the same denomination as the third term, and may be reduced to a higher denomination if necessary. * * In the method of arranging the terms which is delivered in almost all the books on Arithmetic, and which is very generally employed in practice, the term which is of the same kind with the answer is put in the second place. This method is entirely subversive of the principles of proportion, and is calculated to prevent the learner from acquiring just views of this subject, as in it a ratio is, in many cases, instituted between quantities entirely different in kind. The method above explained (taken from Thomson's Arithmetic, and which has also been adopted by Walker and some other late writers) is founded on principles strictly mathematical; and besides being very simple and easy in practice, it possesses the advantage of training the mind to accurate thinking, and of preparing it for the subsequent study of Mathematics. It also precludes the necessity of what is generally called the Rule of Three Inverse; as all the questions usually solved under that head, may be solved by the rule above delivered. When the first and second terms are in the same denomi nation, they are evidently in the same ratio as the numbers which express them; and therefore they are to be reduced to the same denomination, if they be not so already. Ex. 1. If 12 yards of cloth cost 33 dollars, what would 8 yards cost at the same rate? The answer to this question must evidently be in money, and therefore $33, the money given in the question, must occupy the third place and, as 8 yards will cost less than 12 yards, 8 yards, the less of As 12 8: 33: 8 12)264 $22 these two terms, must be put in the second place. Hence, the terms will be arranged, and the operation performed, as in the margin; and it appears that, at the rate specified in the question, 8 yards would cost $ 22.* 2. How many men would perform in 168 days a piece of work which 108 men can perform in 266 days? As 168: 266 :: 108: Here, it is plain that the answer to this question must be men; and therefore the given number of men must be put in the third place; and as the answer must be a greater number of men, in order to perform the work in less time than 266 days, we must put 266, the greater of the other two terms, in the second place hence the terms will be arranged and the operation performed, as in the margin; and the answer is found to be 171 men.t 266 648 648 216 168)28728(171 168 1192 1176 168 168 *In this example, it is evident that as often as 8 yards are contained in 12 yards, so often would the answer be contained in $33. Hence the quantities are proportionals, and the reason of the process is evident from what goes before. The reason of the operation will also be plain if it be considered, that the price of one yard would be found by dividing $33 by 12, and † It is obvious that the terms must be thus arranged: since 266×108 would be the number of days in which the work would be performed by Exercises.-1. If 57cwt. of sugar cost $ 432, what would 95cwt. cost at the same rate? Ans. $720. 2. If 385yds. of linen cost $315, how much might be bought for $90 ? Ans. 110yds. 3. If the yearly rent of a farm containing 182 acres be $273, what is the rent of a part of it containing 43 acres? Ans. $64.50. 4. If 275 reams of paper cost $990, what would 990 reams cost? Ans. $ 3564. 5. If 96 men reap 40 acres of grain in a week, how many men would reap 65 acres in the same time? Ans. 156. 6. If 84 sheep can be grazed in a field for 12 days, how long might 112 sheep have been grazed equally well in the same field? Ans. 9 days. 7. If the shilling loaf weighs 36 ounces when flour is $ 4 per barrel, how much must it weigh when flour is $6 per barrel? Ans. 24oz. 8. If a person lent me $270 for 8 months; in return for his kindness how much ought I to lend him for 18 months? Ans. $120. 9. How many men must be employed to finish a canal in 12 days, which 5 men could perform in 36 days? Ans. 15. 10. If a person travel 12 hours a day, and finish his jour ney in three weeks, how long should the same journey take him if he travelled only 9 hours a day?' Ans. 4 weeks. 11. If 24 pioneers can make a trench in 12 days, what length of time would the same work employ 9 men? Ans. 32 days. 12. Suppose 50 men build a house in 60 days, how many men would build the same in 100 days ? Ans. 30. 13. If a besieged garrison have 4 months' provisions, at the rate of 18 ounces per day for each man, how long will that the price of eight yards would be found by multiplying the quotient by 8; and the result will evidently be the same that would be obtained by multiplying $33 by 8, and dividing the product by 12. The method. according to the rule is in general preferable, however, as it has the advantage of freeing the operation as much as possible from fractional quantities. one man; and if this be divided by 168, the quotient must be the numher of days required; consequently, 266 and 108 must be made the second and third terms that their product may be taken in the operation. This question, as well as many others in this article, belongs to Inverse Proportion. |