This answer, it will be perceived, is the same as that found by analysis. A little observation will show that the original third term 60, has been multiplied by both the second terms, 35 and 6, and divided by both the first terms, 20 and 2. Both the multiplications were not, it is true, performed at the same time, nor both the divisions; but first came a multiplication by 35, and then a division by 20, and then another multiplication by 6, followed by another division by 2. But, by the rules for composite numbers, it is evident, that the effect would have been just the same, if we had multiplied at once, by the product of the two multipliers, and then divided by the product of the two divisors. Thus, 35×6×60 12600 =315. This expression may be simplified, by rejecting equal factors from both divisor and dividend, which is, in fact, reducing the Fraction to lower terms. It will then be The two proportions, above, may therefore be combined into one, and 60 made the common third term to both the ratios, of time and of men-Thus, 20 men : 35 men We see, that the ratio of 60 acres to the answer, is dependent on two other given complete ratios. The question, then, might be solved differently still, by expressing each of the complete given ratios fractionally, and multiplying the remaining term by them successively. Thus, 361, and 3. Therefore, 7×3×60 The ratio of the third term to the answer, is made up of several simple ratios, multiplied together, or compounded, and hence it is called a COMPOUND RATIO. Thus, in the last example, 60 has to the answer, a compound ratio of the times and the men. A proportion, in which compound ratios occur is called a COMPOUND PROPORTION.. A proportion consisting of Simple ratios, is called a SIMPLE PROPOR TION. 11. If 64 men can dig 27 cellars, which are 24 ft. long, 18 ft. wide, and 16 ft. deep, in 36 days, working 8 hours a day; how many men will it take to dig 48 cellars, which are 54 ft. long, and 30 ft. w.de, and 13 ft. deep, in 10 days, if they only work 6 hours in the day? We will exhibit in detail the manner of stating, and solving this question by proportion. There are a number of circumstances, af. fecting the answer. But we will first inquire how many men it would take, if nothing but the NUMBER of cellars were concerned. The question, then, would be, if 64 men can dig 27 cellars, how many men can dig 48 cellars in the same time? More cellars will require more men. By Simple Proportion 27 cellars: 48 cellars :: 64 men: *** men. Next, we will find how many it would take, if nothing but the LENGTH of the cellar, were concerned. The question, then is, if 64 men can dig cellars, 24 ft. long, in a certain time, how many men can dig similar cellars, 54 ft. long, in the same time? More length will require more men. By Simple Proportion 24 feet: 54 feet :: 64 men: *** men. Next, we will find how many, if nothing but the WIDTH were concerned. The question, then is, if 64 men dig cellars 18 ft. wide, how many men will dig similar cellars 30 ft. wide ? More width will require more men. By Simple Proportion 18 feet: 30 feet: : 64 men : *** men. Next, we will find how many, if nothing but DEPTH were concerned. The question, then is, if 64 men dig cellars 16 feet deep, how many men will dig similar cellars 13 ft. deep? Less depth will require less men. By Simple Proportion 16 feet: 13 feet :: 64 men : *** men. Next, we will find how many, if nothing but the NUMBER OF DAYS were concerned. The question, then is, if 64 men dig cellars in 36 days, how many men will dig similar cellars in 10 days? Less days will require more men. By Simple Proportion 10 days: 36 days :: 64 men : *** men. Next, we will find how many, if nothing but the HOURS IN A DAY were concerned. The question, then is, if 64 men dig cellars by working 8 hours a day, how many men will dig similar cellars, working 6 hours a day. Less hours will require more men. *** By Simple Proportion 6 hours: 8 hours :: 64 men : men. All which may be collected, as in the last example, into one statement, thus : 27 cellars: 48 cellars 24 ft. length: 54 ft. length 6 hours: 8 hours :: 64 men: *** men. Multiply together all the consequents, in the complete ratios, as before, and 64 by the product. 48×54×30×13×36×8×64-18, 632,540,160. This must be divided by the product of the antecedents. 27X24X18×16×10×6=11,197,440. Then 18632540160 =1,664 Ans. 11197440 This is the method by proportion. Its principal disadvantage consists in the prolixity of the operation, when there are many ratios to be compounded. The process becomes shorter if we express each ratio fractionally, reduce it to its lowest terms, and finally mul. tiply them all together with the third term 64. 27:48=4=1 | 16 : 13=1} 16×9×5×13×18×4, 9X4×3×16× 5 ×3 X64=431308=1,664 25920 This is sometimes called the method by Ratios. It is recommended for prac tice, in preference to any other, particularly with the following modifications. Let the third term, (or antecedent of the incomplete ratio,) be written down, with all the second terms, (or consequents of tho complete ratios,) as factors in the numerator of a Fraction, connected by the sign of Multiplication; and all the first terms, (or antecedents of the complete ratios,) in the same manner, for a denominator, thus, Now observe whether any term in the Numerator has a common factor with one in the denominator, and, if so, divide both by that factor, and place the quotients directly under or over the terms divid. ed. Thus, 27 and 54 may each be divided by 27. I accordingly divide and place the quotient, 1, under 27, and 2, over 54. In the same manner, I proceed with 24 and 48; 18 and 36; 16 and 64; and 10 and 30. I then see that the factor 6, in the denominator, and the quotient 3 over 30, may be divided by 3. I, therefore, divide and place the quotient 1, a step higher still, and the quotient 2, under 6, like the others. I then see a 2 over 54, and also a 2 under 6. I divide these, and put down the quotients, (1, in each case,) like the 1 over 30. The whole Denominator is now exhausted. Each of its factors has become 1 by division, and consequently will not affect the Quotient, or value of the Fraction. Taking, therefore, the last obtain. ed factors in the Numerator, (which are of course, those which stand highest,) and mnltiplying them (rejecting 1s) with the terms unaltered by the process, (viz. in this case, the 13 and 8) we shall obtain the answer. Thus, 4×2×13×2×8=1,664. Ans. as before. We have been particular in explaining this mode, because it is usually much the shortest, and is never longer than any other. It also saves the trouble of a regular statement, for each ratio may be taken, (after the third term is written in the Numerator,) and written by itself, one term in the Numerator and the other in the Denominator, according to the direction given in Simple Propor tion. 27 ANALYTIC SOLUTION. If it take 64 men to dig 27 cellars, it will take as many to dig 1 cellar; that is, it will take 4 men. If it take 4 to dig one 24 ft. long, it will take as many to dig one 1 ft. long; that is, it will take men, If it take to dig one 18 ft. wide it will take as many to dig one 1 ft. wide that is, it will take rass=ys men. If it take men to dig one 16 ft. deep, it will take as many to dig one 1 ft. deep; that is, it will take T7-2016 men. If 2016 T 36 6 will do it in 36 days, it will take 36 times as many to do it in one day; that is, it will take, 3 men. If will do it in a day of 8 hours long, it will take 8 times as many to do it in a day of 1 hour long; that is, it will take men. 81 81 81 27 Now if dig 1 cellar, it will take 48 times as many to dig 48 cellars; that is it will take 384=128 men. If it take 12 when the length is 1 ft. it will take 54 times as many, when the length is 54 ft.; that is, it will take 6912=256. If it take 256 men, when the width is 1 ft. it will take 30 times as many, when the width is 30 ft; that is, it will take 7,680 men. If it take 7,680, when the depth is 1 ft., it will take 13 times as many, when the depth is 13 ft.; that is, it will take 99,840 men. But if 99,840 do it in 1 day, it will require only the tenth as many for 10 days; that is, it will require 9,984 men. If it require 9,984 men, when the days are 1 hour long, it will require but the part as many, when the days are 6 hours long; that is it will require 1,664 men. Ans. as before. By the analytic method, the pupil will perceive that we take each term in the given or supposed case, and find what the answer would be, if that term were made 1. We then, consider what the answer would be, if we increase each term separately, from 1 to the num. ber in the case required to be solved. The teacher should make this clear by illustration and he should require his pupil to find many answers by analysis, for the purpose of mental discipline. For common practice the mode recommended before is best. The pupil will observe, that, by the analytic method we sometimes obtain, in the course of the process, expressions for things which can, actually, have no existence; as, for instance, of a man, in the last example. But, he should recollect that by this expression is only meant of the labour of a man, and not of a man's person. Similar explanations may be given in all cases, where these quantities occur. As we were speaking just now, of supposed and required cases, contained in a question, it may be well to notice the distinction. It will be observed, that, in the last example, is contained a case complete in every term; viz. that "64 men can dig 27 cellars, 24 ft. long, 18 ft. wide and 16 ft. deep in 36 days, working 8 hours a day." This is complete in every term, because every number concern. ed is given. This is the supposed case, or supposition of the question. In the same question is contained another case, incomplete in one term, which term is required to be supplied; and hence, the case is called the required case, or the requisition, or the demand of the question. In the last example, the requisition or demand is, "how many men can dig 48 cellars, that are 54 feet long, and 30 feet wide, and 13 feet deep, in 10 days, working 6 hours a day?" Every term in this case is complete except the number of men, which is required, or demanded. The terms of the supposed case are called TERMS OF SUPPOSITION. NOTE. The expression "required case," is not strictly proper, since but one term is required. Its conciseness, however, recommends it to use, and if it is understood, nothing more is necessary. From the preceding we derive the following rules: 1. BY PROPORTION. ARRANGE EACH COMPLETE RATIO, BY THE DIRECTIONS IN SIMPLE PROPORTION. COMPOUND THESE RATIOS, BY MULTIPLYING ANTECEDENTS TOGETHER, AND CONSEQUENTS TOGETHER. THE RESULT WILL BE THE RATIO OF THE REMAINING TERM TO THE ANSWER; WHICH OBTAIN AS IN SIMPLE PROPORTION. Of this rule we have the following abbreviations. I. EXPRESS EACH COMPLETE RATIO FRACTIONALLY, AND REDUCE IT FIND THE CONTINUED PRODUCT OF ALL THE REOr, TO ITS LOWEST TERMS. DUCED RATIOS AND THE REMAINING TERM. II. BY ANALYSIS. CONSIDER WHAT THE ANSWER WOULD BE, EXAMPLES FOR PRACTICE. 12. If 20 bushels of wheat are sufficient for 8 persons, 5 months, how many will be sufficient for 4 persons, 12 months? A. 24. RULE I. PROPORTION, 8 persons: 4 persons 5 months: 12 months } :: 20 bu. : *** bu. 1. Ratio of persons Ratio of mo.. Then 20×12X124 RULE II. ANALYTIC SOLUTION. If 20 bushels are enough for 8 persons, of 20 will answer for 1 person, that is, 20 bushels. If answer for 5 months, } as much will answer for 1 month, that is bushel. If answer 1 person, 4 persons will need 4 times as much, that is =2 bushels. If they need 2 bu. in 1 month, they will need 24 bu. in 12 mo. Ans. |