and the other on the left for the first number or term; but if less, write the less of the two remaining numbers in the third place, and the other in the first. the quotient, of the remainder after division, and of the product of the second and third terms, when it cannot be divided by the first, is obvious. 6. If the second and third numbers be multiplied together, and the product be divided by the first; it is evident, that the answer remains the same, whether the number compared with the first be in the second or third place. Thus is the proposed demonstration completed. There are four other methods of operation beside the general one given above, any of which, when applicable, performs the work much more concisely. They are these : 1. Divide the second term by the first, multiply the quotient by the third, and the product will be the answer. 2. Divide the third term by the first, multiply the quotient by the second, and the product will be the answer. 3. Divide the first term by the second, divide the third by the quotient, and the last quotient will be the answer. 4. Divide the first term by the third, divide the second by the quotient, and the last quotient will be the answer. The general rule above given is equivalent to those, which are usually given in the direct and inverse rules of three, and which are here subjoined. The RULE OF THREE DIRECT teacheth, by having three numbers given, to find a fourth, that shall have the same proportion to the third, as the second has to the first. RULE. 1. State the question; that is, place the numbers so, that the first and third may be the terms of supposition and demand, and the second of the same kind with the answer required. 2. Bring 3. Multiply the second and third terms together, divide the product by the first, and the quotient will be the answer. t NOTE I. 2. Bring the first and third numbers into the same denomination, and the second into the lowest name mentioned. 3. Multiply the second and third numbers together, and divide. the product by the first, and the quotient will be the answer to the question, in the same denomination you left the second number in; which may be brought into any other denomination required. EXAMPLE. If 24lb. of raisins cost 6s. 6d. what will 18 frails cost, each weighing net 3qrs. 18lb. ? 24lb.: 6s. 6d. :: 18 frails, each 3qrs. 181b. : The rule is founded on this obvious principle, that the magnitude or quantity of any effect varies constantly in proportion to the varying part of the cause: thus, the quantity of goods bought is in proportion to the money laid out; the space gone over by an uniform motion is in proportion to the time, &c. The truth of the NOTE 1. It is sometimes most convenient to multiply and divide as in compound multiplication and division; and sometimes it is expedient to multiply and divide according to the rules of vulgar or decimal fractions. But when the rule, as applied to ordinary inquiries, may be made very evident by attending only to the principles of compound multiplication and division. It is shewn in multiplication of money, that the price of one, multiplied by the quantity, is the price of the whole; and in division, that the price of the whole, divided by the quantity, is the price of one. Now, in all cases of valuing goods, &c. where one is the first term of the proportion, it is plain, that the answer, found by this rule, will be the same as that found by multiplication of money; and, where one is the last term of the proportion, it will be the same as that found by division of money. In like manner, if the first term be any number whatever, it is plain, that the product of the second and third terms will be greater than the true answer required by as much as the price in the second term exceeds the price of one, or as the first term exceeds an unit. Consequently this product divided by the first term will give the true answer required, and is the rule. There will sometimes be difficulty in separating the parts of complicated questions, where two or more statings are required, and in preparing the question for stating, or after a proportion is wrought; but as there can be no general directions given for the management of these cases, it must be left to the judgment and experience of the learner. The RULE OF THREE INVERSE teacheth, by having three numbers given, to find a fourth, that shall have the same proportion to the second, as the first has to the third. If more require more, or less require less, the question belongs to the rule of three direct. But if more require less, or less require more, it belongs to the fule of three inverse. NOTE. when neither of these modes is adopted, reduce the compound terms, each to the lowest denomination mentioned in it, and the first and third to the same denomination; then will the answer be of the same denomination with the second term. And the answer may afterward be brought to any denomination required. NOTE 2. When there is a remainder after division, reduce it to the denomination next below the last quotient, and divide by the same divisor, so shall the quotient be so many of the said next denomination; proceed thus, as long as there is any remainder, till it is reduced to the lowest denomination, and all the quotients together will be the answer. And when the product of the second and third terms cannot be divided by the first, consider that product as a remainder after division, and proceed to reduce and divide it in the same manner. NOTE 3. NOTE. The meaning of these phrases, " if more require more, less require less," &c. is to be understood thus: more requires more, when the third term is greater than the first, and requires the fourth to be greater than the second; more requires less, when the third term is greater than the first, and requires the fourth to be less than the second; less requires more, when the third term is less than the first, and requires the fourth to be greater than the second; and less requires less, when the third term is less than the first, and requires the fourth to be less than the second. RULE. 1. State and reduce the terms as in the rule of three direct. 1 2. Multiply the first and second terms together, and divide their product by the third, and the quotient is the answer to the question, in the same denomination you left the second number in. The method of proof, whether the proportion be direct or inverse, is by inverting the question. : EXAMPLE. NOTE 3. If the first term and either the second or third can be divided by any number, without a remainder, let them be divided, and the quotients used instead of them. Direct and inverse proportion are properly only parts of the same general rule, and are both included in the preceding. Two or more statings are sometimes necessary, which' may always be known from the nature of the question. The method of proof is by inverting the question. EXAMPLES. EXAMPLE. What quantity of shalloon, that is 3 quarters of a yard wide, will line 7 yards of cloth, that is 1 yard wide? Iyd. 2qrs.: 7yds. 2qrs. :: 3qrs. : The reason of this rule may be explained from the principles of compound multiplication and division, in the same manner as the direct rule. For example: If 6 men can do a piece of work in 10 days, in how many days will 12 men do it? As 6 men: 10 days :: 12 men : 6X10 12 = 5 days, the answer. And here the product of the first and second terms, that is, 6 times 10, or 60, is evidently the time in which one man would perform the work; therefore 12 men will do it in one twelfth part of that time, or 5 days; and this reasoning is applicable to any other instance whatever. |