157 be equal to the product of the two extremes, or that of two means equally distant from the middle terms: as 2, 4, 8, 16, 32; here, 32×2=64; 16×4=64; and 8×8=64. Also, 1, 3, 9, 27, 81; now 81x1=8i; 27x3=81; and 9×9=81. This brief explanation of the nature of proportion, both arithmetical and geometrical, with the obvious distinctions by which they are respectively characterized, will essentially aid the learner in his progress in the subsequent rules. This is clearly evident from the consideration, that the Rule of Three; or as it is often denominated, the Rule of Proportion, has its radical foundation on the nature and principles of geometrical progression. Questions relative to Proportion. 1. To discover what, are numbers compared together? 2. How many numbers are required to form a comparison? 3. By what name, is the number compared and first written, called? 4. By what name is that called, with which it is compared? 5. In how many ways are numbers compared with each other? 6. What does the one comparison consider, and what is it called? 7. What is the difference often called? 8. What does the other comparison consider, and what is it called? 9. What is the difference, or arithmetical ratio of 4 and 12? 11. When two or more couplets or numbers have equal ratios or differences, what is the equality denominated, and how are their terms placed? 12. Must all the greater or all the less terms be taken as antecedents, and the rest as consequents? 13. When thus taken, what are they called? 14. What proportionals are 2, 4, 6, 8? or 8, 6, 4, 2? or 4, 2, 8,6? 15. What proportionals are 2, 4, 8, 16? or 16, 8, 4, 2? or 4, 2,16, 8? 16. In what manner does an arithmetical ratio increase, or decrease! 17. How does a geometrical ratio increase, or decrease? 18. Of what are ratios ever considered as the result, without regarding either term as the antecedent? 19. What kind of progression may any rank of numbers more than two, increasing or decreasing by a common difference, be called? 20. Of what series, ascending or descending, are 1, 2, 3, 4, 5, 6? or, 6, 5, 4, 3, 2, 1? also, 2, 4, 6, 8, 10? or 10, 8, 6, 4, 2? 21. In the first rank of numbers, whether of ascending or descending series, what is the common difference, whether by addition or subtraction? 22. In the second rank, what is the difference, whether by addition or subtraction? 23. What are the numbers which form the series called? 24. What are the first and last term of the series called? 25. In any series of numbers in arithmetical progression, what is the sum of the two extremes equal to ? 26. When the number of terms is odd, what is the double of the middle term equal to ? PROPORTION; OR RULE OF THREE DIRECT. IT is called the Rule of Three, in consequence of three numbers being given to find a fourth, which will be proportional ; viz. which will bear the same proportion to the third, as the second does to the first; or, the fourth term will bear the same proportion to the second, which the third does to the first. Of the three terms given, two are called terms of supposition, and the other that of demand. There are always two of the numbers given in a question, which are of the same name or 1 kind, one of which is the first, and the other the third term, in stating the question: the remaining number, which is of the same name or kind with the answer sought, will possess the second place. In stating a question in the Rule of Three Direct, 1. Put that number in the third place, the value of which is sought, and which would follow some such inquiry, viz. what will? what cost? how many? how far? how long? how much? &c. 2. Place the term of the same name or kind with the third, in the first place; and if these terms be of different denominations, reduce them to the lowest denomination, in either of the given terms. 3. Put the remaining term in the second place, which will be of the same name with the answer; and if it consist of several denominations, reduce it to the lowest mentioned. Having thus stated the question, proceed by the following RULE. Multiply the second and third terms together, and divide that product by the first, and the quotient will be the fourth term, or answer; and of the same denomination with that of the second, or to which the second was reduced. When there is a remainder, multiply it by the next lower denomination, and divide by the first term, &c. The proof is easily found by inverting the question. Note.-There are various methods by which the work may be abridged, and which are oftentimes preferable to the general rule. 1. Divide the second term by the first, multiply the quotient into the third, and the product will be the answer. Or, 2. Divide the third term by the first, multiply the quotient into the second, and the product will be the answer. Or, 3. Divide the first term by the second, and the third by that quotient, and the last quotient will be the answer. Or, 4. Divide the first term by the third, and the second by that quotient, and the last quotient will be the answen So in the first example, under this rule, 8×12=96, is a product as much too great, as the first term 4, contains more units than 1. Therefore, 96-4-24, the fourth term proportional. Then 4×24-96 ; and 8×12=96 ; so that the result is, 4 : 8 :: 12: 24. The same will be true of every operation in the Rule of Three, whether direct or inverse; the product of the means and extremes will be equal. For, whether the divisor be the first, or third term, as the statement shall fall under direct or inverse proportion, the effect will be the same, and the true answer obtained. These several examples may also be wrought by the rules given in the last note. Take the first question, and obtain the answer agreeably to the first rule, viz. ; divide 8 by 4, and multiply the quotient 2 by 12, the answer is 24; by the second rule, 12÷÷4×8=24, answer: or take the second question; and by rule third, divide the first term 24, by 12, the second term, and divide the third term 8, by the last quotient: thus, 24÷12=2÷ 8=4, answer. Or the same example by the fourth rule. Divide the first by the third term, and the second term by this quotient, and the last quotient is the answer. Thus, 24÷8-3 Example. -12-4, answer. 5. If 5cwt. of sugar cost £18 15s. what will 567b. cost? Reduced 50wt. : £18 15s. : : 56lb. 560)210010(37 Here the first and third terms are lbs. and the second term reduced to its lowest denomination, shillings. 168 420 392 28 12 56)336(6 336 Ans. £1 17s. 6d. 6. If 3 pairs of stockings cost 138. what will 6 dozen pairs cost? prs. S. 6 doz. is pairs 72. 3:13: 72: 312s. 2,0)31,2 shillings. £15,12 Ans. 7. At 1s. 5d. a pound, what will 138lb. of butter cost? lb. d. lb. £ s. d. 1 : 17 :: 138: 9 15 6 138×17-2346d. 8. If a person spend 4s. 6d. a day, how much is that in a year? ds. £ s. d. d. d. 54x365-19710 9. If 7 days' board cost 22s. what will it come to by the year? £ s. d. d. S. 365x22÷7-1147s. 11⁄2d. 10. If a man's salary be £250 a year, what is that by the calendar month? M. £ M. £ s. d. 12: 250 :: 1: 20 16 8 11. How much will a grindstone, 30 inches in diameter, and 5 inches thick, come to, at 5s. per cubic foot? Diameter, 30+15×15×5=3375 Cubic inches. Then, 1728 5s. : : 3375×5÷1728 : =9s. 9d.+Ans. 12. What is the expense of a grindstone, 34 inches in diameter, and 4 inches thick, at 6s. the cubic foot? Diameter, 34+17×17×4=3468 Then, 1728 6s. : : 3468 : =12s. Od. 2gr. Ans. |