3. Multiply the second and third terms together, divide the product by the first, and the quotient will be the answer. plication and division. It is shown in multiplication of money, that the price of one, multiplied by the quantity, is the price o£ 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 teaches, 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 rule of three inverse. 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 NOTE 1. It is sometimes most convenient to multiply and divide as in compound multiplication and division; and 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. 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. What quantity of shalloon, that is three quarters of a yard wide, will line 7 yards of cloth, that is 1| yard wide? lyd. 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 examplc; 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 : 6×10 -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 twelf part of that time, or 5 days; and this reasoning is applicable to any other instance whatever. sometimes it is expedient to multiply and divide according to the rules of vulgar or decimal fractions. But 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. If the first term and either the second or third Gan be divided by any number without a remainder, let them be divided, and the quotient 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. 1. Let it be proposed to find the value of 14oz. 8dwt. of gold, at 31. 19s. lid. an ounce. EXPLANATION. The three terms being stated by the general rule, as above, the second term is reduced to pence, and the third to penny-weights, these being their lowest denominations, as directed in the first note. The first term is also reduced to dwts. that it may agree with the third, by the same note. The second term is then multiplied by the third, and the product divided by the first, according to the general rule, when the answer comes out 13809 pence, and 12 remaining; which remainder being reduced to farthings, and these divided by the same divisor 20, by the second note, the quotient is 2 farthings, 8 remaining. Lastly, the pence are divided by 12, to reduce them to shillings, and these again by 20 for pounds; when the final sum comes out 571. 10s. 9d. 2q. for the answer. 2. How much of that in length which is 44 inches broad, will make a square foot ? |