State the questions as directed in the Rule of Three Direct; then multiply the second term by the first or third term accordingly as the answer ought to be greater or less; divide the product by the other term, and the quotient will be the answer. COMPOUND PROPORTION, OR DOUBLE RULE OF THREE, TEACHES to resolve such questions as require two or more statements by Single Proportion, and hence its name. There is always an odd number of terms given, as five, seven, &c. All questions in Compound Proportion may be stated and wrought by the following GENERAL RULE.* 1. Place that term, which is of the same kind or quality with the answer sought, for the second term. 2. Then, of the two terms in the question of the same kind, place the greater or less on the right for the third term, and the other on the left for the first term, according to the directions under the General Rule for Simple Proportion. Arrange the two remaining terms under the first and third, on the same principle. 3. Find the fourth term from the first statement, and place it for the second term in the second part of the statement, and find the fourth term from this statement, and it will be the answer required. Note. If there be more than five terms in the question, the same mode of statement must be continued, and a third proportion formed, and so on, and the fourth term found from the last statement, will be the answer as before. *This rule is evident from the General Rule of Three, for each statement is a particular statement under that Rule. If, then, all the separate dividends be collected into one dividend, and all the divisors into one divisor, their quotient must be the answer. Thus, in Ex. 1. M. and as 12: D. Int. D. Int. 400 X 6 As 1006: 400: 100 Int. 400 X 6 M. 400×6×9 :: 9: 13 Int. Aus. 100 100 x 12 EXAMPLES. 1. If a principal of $100, gain $6 interest in one year, what will $400 gain in 9 months. Statement and Operation. In this question, the answer sought is interest, and therefore $6 must be the second term. As $400 will gain more interest in the same time than $100, $400 must be placed on the right for the third term, and $100 on the left for the first term. And as the same sum will gain more interest in 12 months than in 9 months, the 9 must be placed under the third term, and the 12 under the first term. The operation is obvious on inspecting it. Note. Instead of working two proportions, the whole may be reduced to one, by multiplying the first terms together, and also the third terms, and using their products for the first and third terms. This is merely changing the order of the operation, as will be seen in the preceding example. D. Int. 100 12: D. 6: 400 :: 490 This becomes, evidently, } 100x1218, as before. The work may also frequently be contracted by dividing the first and third terms by a common divisor, or the first and second terms, and using their quotients, for the divisor will diminish the terms in the same ratio, and the proportion be still preserved. Thus, in the preceding example, 100 6 400 becomes 1: 6:4, by dividing by 100. :: 9 1: 6:4 12: And 4: :: 3 becomes 1: 6:3: 18, Ans. as before. Ex. 2. If 950 soldiers consume 350 quarters of wheat in 7 months, how many soldiers will consume 1464 quarters in 1 month? Ans. 27816 soldiers. Ex. 3. If 1464 quarters of wheat be used by 27816 soldiers in a month, in what time will 950 soldiers consume 350 quarters? Ans. 7 months. Ex. 4. If 144 men, in 6 days of 12 hours each, dig a trench 200 feet long, 3 wide and 2 deep, how many hours long is the day, when 30 men dig a trench 350 feet long, 6 wide and 3 deep, in 259.2 days? Ans. 7 hours. The following Rule for the Double Rule of Three, involves the consideration of Direct and Inverse Proportion. Though the General Rule will enable the student to solve all questions with ease, this Rule is retained for the satisfaction of those who might desire to use it. RULE. Always place the three conditional terms in this order: That number, which is the principal cause of gain, loss or action, possesses the first place; that, which denotes the space of time, distance of place, rate, medium or mean of action, the second; and that, which is the gain, loss or action, the third: This being done, place the other two terms which move the question, under those of the same name, and if the blank place, or term sought, fall under the third place, then the question is in direct proportion: therefore, RULE I.* Multiply the three last terms together, for a dividend, and the two first for a divisor:-But, if the blank fall under the first or second place; then, the proportion is inverse; therefore, RULE II. Multiply the first, second and last terms together for a dividend, and the other two for a divisor, and the quotient will be the an swer. EXAMPLES. 1. If $100 gain $6 in a year; what will $100 gain in 9 months? D. P. Mo. D. Int. 100 12 6 Ferms in the supposition, or conditional terms. 400 : 9 Terms which move the question. Here, the blank falling under the third place, the question is in direct proportion, and the answer must be found by the first Rule; therefore, 400X 9×6=21600 For the dividend, and, 1200 For the divisor. 1. When the blank falls under the third term by this mode of statement, it is obvious on inspecting the statement that the proportion is direct, and the same terms are taken to form the dividend and divisor as in the preceding rule, or, by two statements in the Single Rule of Three direct. 2. But in Example 2nd, and when the blank falls under the first or second term, the proportion is inverse. In this Example, more principal and interest require less time, and, every statement according to the rule will make more require less. The operation by the rule is the same as from two statements by the Single Rule of Three Inverse. These statements on this Example would be thus 2. If $100 will gain $6 in a year; in what time will $400 gain $18? D. Mo. D. 100 12 6 Terms in the supposition. :: 400 : :: 18 Terms which move the question. Here, the blank falling under the 2d place, the question is in reciprocal or inverse Proportion, and the answer must be sought by the second rule; therefore, 100×12X18=21600 For the dividend. Here, the blank falling under the first place, the proportion is inverse, and the answer found by the second rule, as in the last example. 5. If 8 men spend £32 in 13 weeks; what will 24 men spend in 52 weeks? Ans. £384. 6. If the freight of 9hhds. of sugar, each weighing 12cwt. 20 leagues, cost $50; what must be paid for the freight of 50 tierces ditte, each weighing 24cwt. 100 leagues? Ans. $289 35c. 12m. 7. There was a certain edifice completed in a year by 20 workmen; but the same being demolished, it is necessary that just such an one should be built in 5 months. I demand the number of men to be employed about it? Ans. 48 men. 8. If 6 men build a wall 20 feet long, 6 feet high and 4 feet thick, in 16 days, in what time will 24 men build one 200 feet long, 8 feet high, and 6 feet thick? Ans. 80 days. COMPARISON OF WEIGHTS AND MEASURES. EXAMPLES. 1. If 78 pence Massachusetts be worth 1 French crown, how many Massachusetts pence are worth 320 French crowns? F. cr. d. F. cr. As 1 78: 320 78 2560 2240 24960 Ans. 2. If 24 yards at Boston make 16 ells at Paris, how many ells at Paris will make 128 yards at Boston ? Bost. Par. Bost. 3. If 60 at Boston make 56 at Boston will be equal to 350 at Par. 16ells: 128yds.: 85 ells, Ans. Ans. 375 Boston. 4. If 95 Flemish make 100 pounds are equal to 550 Flemish ? CONJOINED PROPORTION, IS when the coins, weights or measures of several countries are compared in the same question; or, in other words, it is joining many proportions together, and by the relation, which several antecedents have to their consequents, the proportion between the first antecedent and the last consequent is discovered, as well as the proportion between the others in their several respects. |