PROPORTION; OR THE RULE OF THREE. T 94. 1. If a piece of cloth, 4 yards long, cost 12 dollars, what will be the cost of a piece of the same cloth 7 yards long? Had this piece contained twice the number of yards of the first piece, it is evident the price would have been twice as much; had it contained 3 times the number of yards, the price would have been 3 times as much; or had it contained only half the number of yards, the price would have been only half as much; that is, the cost of 7 yards wai be such part of 12 dollars as 7 yards is part of 4 yards. 7 yards is of 4 yards; consequently, the price of 7 yards must be of the price of 4 yards, or of 12 dollars. of 12 dollars, that is, 12 × 7 = 421 dollars, Answer. 2. If a horse travel 30 miles in 6 hours, how many miles will he travel in 11 hours, at that rate? 11 hours is of 6 hours, that is, 11 hours is 1 time 6 hours, and of another time; consequently, he will travel, in 11 hours, 1 time 30 miles, and of another time, that is, the ratio between the distances will be equal to the ratio between the times. of 30 miles, that is, 30 X = 230 55 miles. If, then, no error has been committed, 55 miles must be of 30 miles. This is actually the case; for z = V. Ans. 55 miles. Quantities which have the same ratio between them are raid to be proportional. Thus, these four quantities, hours. hours. miles. miles. written in this order, being such, that the second contains the first as many times as the fourth contains the third, that is, the ratio between the third and fourth being equal to the ratio between the first and second, form what is called a proportion. It follows, therefore, that proportion is a combination of two equal ratios. Ratio exists between two numbers; but proportion requires at least three. To denote that there is a proportion between the numbers , 11, 30, and 55, they are written thus : which is read, 6 is to 11 as 30 is to 55; that is, 6 is the same part of 11, that 30 is of 55; or, 6 is contained in 11 as many times as 30 is contained in 55; or, lastly, the ratio or relation of 11 to 6 is the same as that of 55 to 30. T 95. The first term of a ratio, or relation, is called the antecedent, and the second the consequent. In a proportion there are two antecedents, and two consequents, viz. the antecedent of the first ratio, and that of the second; the consequent of the first ratio, and that of the second. In the proportion 6 11 :: 30: 55, the antecedents are 6, 30; the consequents, 11, 55. : The consequent, as we have already seen, is taken for the numerator, and the antecedent for the denominator of the fraction, which expresses the ratio or relation. Thus, the first ratio is, the second ; and that these two ratios are equal, we know, because the fractions are equal. The two fractions 1 and 3 being equal, it follows that, by reducing them to a common denominator, the numerator of the one will become equal to the numerator of the other, and, consequently, that 11 multiplied by 30 will give the same product as 55 multiplied by 6. This is actually the case; for 11 X 30: 330, and 55 X 6 = 330. Hence it follows, If four numbers be in proportion, the product of the first and last, or of the two extremes, is equal to the product of the second and third, or of the two means. Hence it will be easy, having three terms in a proportion given, to find the fourth. Take the last example. Knowing that the distances travelled are in proportion to the times or hours occupied in travelling, we write the proportion thus: hours. hours. miles. miles. Now, since the product of the extremes is equal to the product of the means, we multiply together the two means, 11 and 30, which makes 330, and, dividing this product by the known extreme, 6, we obtain for the result 55, that is, $5 miles, which is the other extreme, or term, sought. 3. At $54 for 9 barrels of flour, how many barrels may be purchased for $186 ? In this question, the unknown quantity is the number of barrels bought for $186, which ought to contain the 9 barrels as many times as $186 contains $54; we thus get the following proportion: A Any three terms of a proportion being given, the operation by which we find the fourth is called the Rule of Three. just solution of the question will sometimes require, that the order of the terms of a proportion be changed. This may be done, provided the terms be so placed, that the product of the extremes shall be equal to that of the means. 4. If 3 men perform a certain piece of work in 10 days, how long will it take 6 men to do the same? The number of days in which 6 men will do the work being the term sought, the known term of the same kind, viz. 10 days, is made the third term. The two remaining terms are 3 men and 6 men, the ratio of which is . But the more* men there are employed in the work, the less time will be required to do it; consequently, the days will be less in * The rule of three has sometimes been divided into direct and inverse, a distinction which is totally useless. It may not however be amiss to explain, in this place, in what this distinction consists. The Rule of Three Direct is when more requires more, or less requires less, as in this example:-If 3 men dig a trench 48 feet long in a certain tine, how many feet will 12 men dig in the same time? Here it is obvious, that the more men there are employed, the more work will be done; and therefore, in this instance, more requires more. Again:-If 6 men dig 48 feet in a given time, how much will 3 men dig in the same time? Here less requires less, for the less men there are employed, the less work will be done. The Rule of Three Inverse is when more requires less, or less requires more, as in this example:-If 6 men dig a certain quantity of trench in 14 hours, how many hours will it require 12 men to dig the same quantity? Here more requires less; that is, 12 men being more than 6, will require less time. Again-If 6 men perform a piece of work in 7 days, how long will 3 men be in performing the same work? Here less requires more; for the number of men, being less, will require more time. ૨ proportion as the number of men is greater. There is still a proportion in this case, but the order of the terms is inverted; for the number of men in the second set, being two times that in the first, will require only one half the time. The first number of days, therefore, ought to contain the second as many times as the second number of men contains the first. This order of the terms being the reverse of that assigned to them in announcing the question, we say, that the number of men is in the inverse ratio of the number of days. With a view, therefore, to the just solution of the question, we reverse the order of the two first terms, (in doing which we invert the ratio,) and, instead of writing the proportion, 3 men: 6 men, (§,) we write it, 6 men : 3 men, (,) that is, men. men. days. days. Note. We invert the ratio when we reverse the order of the terms in the proportion, because then the antecedent takes the place of the consequent, and the consequent that of the antecedent; consequently, the terms of the fraction which express the ratio are inverted; hence the ratio is inverted. Thus, the ratio expressed by = 2, being inverted, is Having stated the proportion as above, we divide the product of the means, (10 × 3 = 30,) by the known extreme, 6, which gives 5, that is, 5 days, for the other extreme, or term sought. Ans. 5 days. From the examples and illustrations now given we deduce the following general RULE. Of the three given numbers, make that the third term which is of the same kind with the answer sought. Then consider, from the nature of the question, whether the answer will be greater or less than this term. If the answer is to be greater, place the greater of the two remaining numbers for the second term, and the less number for the first term; but if it is to be less, place the less of the two remaining numbers for the second term, and the greater for the first; and, in either case, multiply the second and third terms together, and divide the product by the first for the answer, which will always be of the same denomination as the third term. Note 1. If the first and second terms contain different denominations, they must both be reduced to the same denomination; and if the third term be a compound number, it either must be reduced to integers of the lowest denomination, or the low denominations must be reduced to a fraction of the highest denomination contained in it. Note 2. The same rule is applicable, whether the given quantities be integral, fractional, or decimal. EXAMPLES FOR PRACTICE. 5. If 6 horses consume 21 bushels of oats in 3 weeks, how many bushels will serve 20 horses the same time? Ans. 70 bushels. 6. The above question reversed. If 20 horses consume 70 bushels of oats in 3 weeks, how many bushels will serve 6 horses the same time? Ans. 21 bushels. 7. If 365 men consume 75 barrels of provisions in 9 months, how much will 500 men consume in the same time? 8. If 500 men consume 102 months, how much will 365 men time? Ans. 102 barrels. barrels of provisions in 9 consume in the same Ans. 75 barrels. 9. A goldsmith sold a tankard for 10 £. 12 s., at the rate of 5 s. 4 d. per ounce; I demand the weight of it. Ans. 39 oz. 15 pwt. 10. If the moon move 13° 10′ 35′′ in 1 day, in what time does it perform one revolution? Ans. 27 days, 7 h. 43 m. 11. If a person, whose rent is $145, pay $12'63 parish taxes, how much should a person pay whose rent is $378? Ans. $32'925. 12. If I buy 7 lbs. of sugar for 75 cents, how many pounds can I buy for $6? Ans. 56 lbs. 13. If 2 lbs. of sugar cost 25 cents, what will 100 lbs. of coffee cost, if 8 lbs. of sugar are worth 5 lbs. of coffee? Ans. $20. 14. If I give $6 for the use of $100 for 12 months, what must I give for the use of $357'82 the same time? Ans. $21'469. 15. There is a cistern which has 4 pipes; the first will fill it in 10 minutes, the second in 20 minutes, the third in |