The two following questions are Sexcesimals. 7. If 2 places differ in longitude 2° 12′; what is their difference of time? Mult. 2° 12′ 00′′ 00′′ by 3′ 59′′ 20" the time in which the sun passes through 1° 8' 46" 32" Answer. 8. Two places differ in longitude 31° 27′ 30′′; What is the dif ference, in time, of the sun's coming to the meridian of those places, the sun passing through 15° in an hour? 31° 37' 30" 4' 00" In 4' of a solar day, or day of 24 hours, the sun passes 1o 2o 6' 30" 00′′ Answer. 9. Bought a load of wood, which was 3 feet wide, 2 feet 8 inches high, and 8 feet long; what part of a cord of wood did it conAns. Half a cord. tain ? 10. A load of wood was 4 feet 6 inches wide, 3 feet 10 inches high, and 7 feet 8 inches long; how many feet more than a cord did it contain? Ans. 41 feet. 11. A stick of timber is 1 foot 8 inches in depth, and 2 feet 3 inches in width, and 42 feet 8 inches long; how many solid feet Ans. 160. of timber does it contain? 12. Multiply £3 6 8 by £2 5 7. £ S. d. 2 5 7 £3× £2= £6 = 6 0 0 6s.X £2=12s. = 0 12 8d.x £2=16d. £3X5s. =15s. 0 15 0 = 0 4 0 6 0 2 8d.x7d. d. 0 0 0 13. A, B and C bought a drove of sheep in company; A paid £14 5s. B, £13 10s. and C, £11 5s. They agreed to dispose of them at the market; that each man should take 18s. as pay for his time, &c. and that the remainder should be divided in proportion to their several stocks: At the close of the sale, they found themselves possessed of £46 5s. what was each man's gain, exclu&c. sive of the pay for his time, £14 5+ £13 10+ £11 5= £39, and £46 5—£39= £7 5, and £75-18s.X3= £4 11s. whole gain, and £4 11-÷39=23. 4d. gain THE SINGLE RULE OF THREE, IS so called, because three numbers are given to find a fourth, which shall have the same ratio to one of the given numbers, as there is between the other two. It is usually distinguished into Direct and Inverse. The reason of this distinction, and the particular rules, will be given hereafter. It will be more easy however, for the student to proceed according to the following General Rule for stating and working questions in the Rule of Three. GENERAL RULE.* 1. Place that number, which is of the same name or quality as the answer sought, for the second term. 2. Consider whether the answer should be greater or less than the second term. If it must be greater, place the greater of the two remaining numbers in the question on the right for the third *This Rule, on account of its great and extensive usefulness, is sometimes called the Golden Rule of Proportion: For, on a proper application of it and the preceding rules, the whole business of Arithmetick, as well as every mathematical enquiry depends. The rule itself 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. As the idea, annexed to the term, proportion, is easily conceived, the truth of the rule, as applied to ordinary inquiries, may be made evident by attending to principles, already explained. It has been 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 a unit; consequently, this product, divided by the first term, will give the true answer required. Note 1. When it can be done, multiply and divide as in Compound Multipli. cation, and Compound Division. 2. 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 quotient used instead of them. The following methods of operation, when they can be used, perform the work in a much shorter manner than the general rule. 1. Divide the second term by the first: Multiply the quotient into the third, and the product will be the answer. 2. Divide the third term by the first; multiply the quotient into the second, and the product will be the answer. 3. Divide the first term by the second, and the third by that quotient, and the last quotient will be the answer. 4. Divide the first term by the third, and the second by that quotient, and the last quotient will be the answer. R term; but if the answer must be less, place the less of the two numbers on the right for the third term, and, in each case, place the remaining number on the left for the first term. 3. Divide the product of the second and third terms by the first term, and the quotient will be the fourth term or answer sought. Note. As all questions in the Rule of Three, are readily solved by this process, all the statements, unless specially mentioned, will be made according to this rule. The method of proof is by inverting the question. But, that I may make the method of working this excellent Rule as intelligible as possible to the learner, I shall divide it into the several cases following: 1. The fourth number is always found in the same name in which the second is given, or reduced to; which, if it be not the highest denomination of its kind, reduce to the highest when it can be done. 2. When the second number is of divers denominations, bring it to the lowest mentioned, and the fourth will be found in the same name to which the second is reduced, which reduce back to the highest possible. 3. If the first and third be of different names, or one or both of divers denominations, reduce them both to the lowest denomination mentioned in either. 4. When the product of the second and third is divided by the first; if there be a remainder after the division, and the quotient be not the least denomination of its kind; then multiply the remainder by that number, which one of the same denomination with the quotient contains of the next less, and divide this product again by the first number; and thus proceed till the least denomination be found, or till nothing remain. 5. If the first number be greater than the product of the second and third; then bring the second to a lower denomination. 6. When any number of barrels, bales, or other packages or pieces are given, each containing an equal quantity, let the content of one be reduced to the lowest name, and then multiplied by the given number of packages or pieces. 7. If the given barrels, bales, pieces, &c. be of unequal contents, (as it most generally happens) put the separate content of each properly under one another, then add them together, and you will have the whole quantity. EXAMPLES. 1. If 6 of sugar cost 9s. what will 30 cost at the same rate? th S. # 9 Here the answer must be money, As 6:9: 30: the Answer. therefore 9s. is the second term ; as 30 must cost more than 6, 30 must be placed on the right of 9s. for the third term, and 6 on the left for the first term. 6)270 458.£2 58. Aus. Again, By inverting the order of the question, it will be, 2. If 9s. buy 6 of sugar, how much will £2 5s. buy at that rate? 8. S. As 96: 45: the Ans. 6 9)270(30 Ans. Again, 3. If 30% of sugar be worth £2 5s. how much may I buy for 9s. S. 站 S. As 45: 30: 9: the Ans. 9 45)270(6 the Ans. 270 Again, 4. Suppose £2 5s. will buy 301 of sugar: What will 6 of the same sugar cost? S. As 30 30)2710 9s. Ans. N. B. The last three questions are only the first varied, being put merely to show how any question, in this Rule, may be in verted. 5. If 5 yds. of cloth cost $10 what will 20 yds. come to ? yds. $ yds. As 5: 10 :: 20 10-5-2 $40 Ans. Here I divide the 2d term by the 1st, and multiply the quotient into the 3d, for the answer. Here I divide the 3d term by the 1st, and multiply the quotient into the 2d, for the answer. 7. If 20yds. cost $120, how many yards may I have for $30? $ yds. $ As 120 20 :: 30 120÷20 6 quot. and 30-6=5 yards, Answer. Here I divide the 1st term by the 2d, and then, the 3d term by the quotient for the answer. $ yds. $ Again, 8. As 120 : 20 :: 30 120 30 4 quot, and, 20-45 yards, Ans. Here I divide the 1st term by the 3d, and then, the 20 term by that quotient for the answer. 9. If 1cwt. of tobacco cost £5 12 9; what will 8cwt. ditto cost? cwt. £ s. d. cwt. As 1: 5 12 9:: 8 Ans. £45 24 Here there is no need of reducing the middle term, because it can be performed by compound multiplication, the first term being an unit. 10. If 8cwt. of tobacco cost £45 2 4d; what is that per cwt.? £ s. d. 8)45 2 4 Ans. 5 12 9 Here there is no need of reducing the middle term, because it may be performed by compound division only, the 3d term being an unit. 11. If 9cwt. 3qrs. sugar cost £27 17s. 6d. what will 2cwt. 1qr. 1 cost? 2cwt. 1qr. 1 9cwt. 3qrs. £ s. d. 27 17 6 4 |