terms are distinguished, as in Simple Proportion, into terms of supposition, and terms of demand; the number of the former always exceeding that of the latter, by one. Each term of supposition, except one, has a corresponding term of the same kind given in the demand; and the odd term of supposition, which has no corresponding term of demand. given, is of the same kind with the answer or term required. All questions in Compound Proportion may be solved by the following RULE. 1. State the given question as follows: Write that term of supposition which is of the same kind or quality with the answer sought, for the third term of the proportion. Take one of the other terms of supposition and one of the demanding terms of the same kind, and place one of them for a first term, and the other for a second, according to the directions given in the Single Rule of Three. Do the same with another term of supposition, and its corresponding term of demand; and so on with all the remaining terms, (if any,) of each kind, writing the terms in the first and second places in columns, under each other. 2. When any of the terms are compound numbers, reduce them to simple numbers; and always reduce the first and second terms of each line to the same denomination when they are of different denominations. Then, multiply together all the terms in the second and third places for a dividend, and those in the first place, or column, for a divisor. Lastly, divide the dividend by the divisor, and the quotient will be the answer, in the same denomination as the third term, or as that which the third was reduced to. Note. According to the foregoing Rule, when the third term is a compound number, it is to be reduced to a simple number. It may be proper, however, to inform the learner, that it is not absolutely necessary, and sometimes not so convenient, to perform this reduction: for the third *The principles on which the operations in Compound and in Simple Proportion depend, are the same. In solving any question by the Double Rule of Three, all the dividends that would be used in solving the same question by two or more statements by the Single Rule of Three, are collected into one dividend; and all the divisors, into one divisor; by which means the answer is found by one statement, or set of operations. term may be multiplied by the product of all the terms in the second place, and the result then divided by the continued product of the terms in the first place, as in Compound Multiplication and Division, and the quotient will be the answer. When this method is used, if there are decimal figures in the multiplier, (i. e. the product of the terms in the first place,) and divisor, (i. e. the product of the terms in the second place,) or in either of them, it will be proper to equalize the numbers of decimal places in them, and then reject the decimal points, as directed in the Note immediately after exam. 40th, in Simple Proportion. PROOF. By the Single Rule of Three; or by varying the order of the question. EXAMPLES. 1. If 6 men can build a wall 96 rods long in 18 days, when the day is 11 hours long; how many men must be employed to build 64 rods of the wall in 2 days, when the day is 12 hours long? 64×18x11x 6=76032, the dividend. 96× 2×12= 2304, the divisor. Then, 76032÷2304-33 men, Ans. Explanation. In the foregoing question, the supposition is, that 6 men can build 96 rods of wall in 18 days, when the day is 11 hours long; and the demand is, to find how many men must be employed to build 64 rods of the wall in 2 days, when the day is 12 hours long; there being four numbers, or terms, in the supposition, and three in the demand.* The 96 rods of wall, mentioned in the sup *The number of terms in the question may be reduced to five, thus: 18 days of 11 hours each 198 hours, and 2 days of 12 hours each = 24 hours. Then the question is as follows: If 6 men can build 96 rods of wall in 198 hours, how many men must be employed to build 64 rods of the wall in 24 hours? Stated thus; As 96 rods: 64 rods :: 6 men: 33 men, Ans. position, is a term which is of the same kind with the 64 rods in the demand; the 18 days in the supposition and the 2 days in the demand, are terms of the same kind; and, the 11 hours in the supposition and the 12 hours in the demand, are corresponding or like terms. The 6 men is the term of supposition which has no corresponding term of demand given, and which is of the same kind with the answer required.-I state the question thus: I first set down 6, the given number of men, for the third term, because this is the term of supposition which is of the same kind with the answer. I then dispose of the other given terms in the first and second places, as follows: I begin with the 96 rods in the supposition, and the corresponding term of demand, 64 rods; and because it would not require as many men to build 64 rods of wall as it would to build 96 rods, in the same time, I write the less number of rods for the second term, and the greater for the first. Then, because it would require more men to do the required work in 2 days than it would to do it in 18 days, I write the greater number of days in the second place and the less in the first. Lastly, because it would not require as many men to perform the work when the day is 12 hours long as it would when the day is only 11 hours long, I write the less number of hours in the second place and the greater in the first.— Having thus stated the question, I proceed to resolve the statement; and, as all the terms are simple numbers, and the first and second terms of each line are of the same denomination, there are no reductions to be performed; so I multiply together all the terms in the second and third places, for a dividend, and those in the first column, or place, for a divisor: I then perform the division, and the quotient is the answer. Any question in Compound Proportion may be solved, by two or more statements, by the Single Rule of Three. The foregoing question may be solved by three statements* by the Single Rule of Three, viz. as follows: 1st, As 96 rods : 64 rods :: 6 men: 4 men, the number of men that must be employed to build 64 rods of wall in 18 days, when the day is 11 hours long. *If the number of terms in the question be reduced to five, as in the note at the bottom of page 165, the answer may be found by two state ments by the Single Rule of Three. 2dly, As 2 days: 18 days :: 4 men : 36 men, the number that must be employed to build 64 rods of wall in 2 days of 11 hours each. Lastly, As 12 hours: 11 hours :: 36 men : 33 men, the answer, or number of men that must be employed to build 64 rods of wall in 2 days of 12 hours each. It will be well for the learner to solve some of the following questions by the Single Rule of Three. 2. If 5 men in 8 days cat 27 lb. 12 oz. of bread, how much bread will 4 men eat in 15 weeks! Stated thus As 5 men: 4 men lb. oz. :: 27. 12: the ans. After performing the proper reductions, according to the Rule, the statement is as follows: 3. If 9 bushels of oats will serve 7 horses 10 days, how many bushels, at the same rate, will serve 20 horses 3 weeks? Ans. 54 bush. 4. If a family of 19 persons expend $235 in 8 months, how much, at the same rate, will a family of 12 persons expend in 5 months? Ans. $92.763+ 5. If 6 oxen, in 5 days, plough 11 acres, how many oxen would plough 44 acres in 12 days? Days, 12: 5) Acres, 11: 44 :: 6 oxen 10 oxen, Ans. 6. If 3 masons, working 7 hours a day, build a wall in 6 days; how many hours a day must 4 masons work, to build it in 5 days? Masons, 4:3) 7. If 36 yards of cloth, 7 quarters wide, cost $504, what cost 120 yards, of the same quality, but only 5 quarters wide? Ans. $1200. 8. If the carriage of 13 cwt. 65 miles cost $9, how many cwt. may be carried 40 miles, at the same rate, for $15? Ans. 35 cwt. 2 9. If $100, in one year, gain $6 interest, how much will $75.28 gain in 9 months? Int. Int. ::$6 $3.3876, Ans. 10. If $100 will gain $6 interest in 12 months, in what time will $75.28 gain $3.3876 ? Ans. 9 months. 11. If 3000 copies of a history of the United-States, each containing 11 sheets, require 66 reams of paper, how much paper will 5000 copies require, if the work be extended to 12 sheets? Ans. 125 reams. 12. If 6 men can build a wall 20 feet long, 6 feet high, and 4 feet thick, in 16 days; in what time will 24 men build a wall 200 feet long, 8 feet high and 6 feet thick? 24 6 da. days. Men, ft. 20: 200 Height, ft. 6: 8 Thickness, ft. 4: 6 J :: 16 80, Ans. 13. Suppose 30 men perform a piece of work in 20 days; how many men will accomplish another piece of work 4 times as large, in a fourth part of the time? of 20 days 5 days. Days, 5:20 1: 4 :: 30 men : 480 men, Ans. METHOD OF CONTRACTION. The work may often be very much abbreviated as follows: 1. State the question as usual, and perform such reductions as may be necessary, in order to prepare the terms for multiplying and dividing. Then draw a horizontal line, and place all the numbers which are to be multiplied together for a dividend, (viz. all the terms in the second and third places,) in a row, above this line, with the sign of multiplication between every two of them; and place all the terms which are to be multiplied together for a divisor, below the line, in the same manner. 2. If any factor, or number, above the line, be the same as a factor below the line, cancel or strike out both the num |