If more is required, the lesser} is the divisor. If less, the greater EXAMPLES FOR EXPLANATION. Example 1. If 4 men plane 250 deal boards in 6 days, how many men will plane them in two days? If 6 days require 4 men, what 2 days? 6 2) 24 12 men. Answer. If a board be 9 inches broad, how much in length will make a square foot? in. b. If 12 in. b. 12, what 9 inches broad? 9) 144 Answer 16 inches in length. In these examples, the first and second numbers are multiplied together and the product is divided by the third; for, in the first example, it is most certain that 2 days will require more hands to perform the work than 6 days; therefore the lesser of the extreme numbers is the divisor; and declares the question is in the indirect proportion. Likewise, in the second example, 9 inches in breadth, must needs require more in length to make a foot than 12 inches in breadth: therefore it is in the same proportion with the first example, because the divisor is the third number. Example 3. How many yards of sarcenet, of 3 qrs. wide, will line 9 yards of cloth of 8 qrs. wide? qrs. wide yds. long qrs. wide Example 4. If a quartern loaf weigh 44lb. when wheat is 5s. 6d. the bushel, what must it weigh when wheat is 4s. the bushel? If in 12 months 1007. principal gains 57. interest, what principal will gain the same interest in 5 months? M. 12 1. P. -100 M. -5 12 5) 1200 Answer, 2407. principal. THE DOUBLE RULE OF THREE DIRECT. In this rule there are five numbers given to find out a sixth, which is to be in proportion to the product of the fourth and fifth numbers, as the third number is to the product of the first and second numbers. Questions in this kind of proportion are wrought either by two operations in the Single Rule of Three Direct, or by the rule composed of the five given numbers; and the one may be a proof to the other as may be seen in the example following. Example 1. If 1007. principal in 12 months gain 57. interest, what will 2461. gain in 7 months? If 1007. gain 47. what will 2467.? Then 5 100) 1230 20 1100) 600-Answer, 121. 6s. say again, if 12 months gain 127. 6s. what 7 months? 20 246 7. 12) 1722 (6 214) 14/3-Answer, 71. 3s. 6d. In the first stating, the answer is this, that if 1007. gain 57. then 2461. will gain 127. 6s. Then I say, in the next stating, if 12 months gain 127. 6s. what will 7 months? And the answer is 71. 3s. 6d. And so much will 2467. gain in 7 months, if 1007. gain 57. in 12 months. You must particularly note, that in all operations where the answer to the question is found by two statings of the Rule of Three, the answer of the first stating is over the middle number of the second stating: as in the preceding example. The foregoing question may be answered by a stating composed of the 5 given numbers thus: In this work, in the stating the question the first and fourth numbers are made of one name, as are the second and fifth; then the two first numbers are multiplied together for a divisor, and 1200) 8610 (71. 210 20 the last three numbers are multiplied to- 1200) 4200 (3s. gether for a dividend, and the quotient or answer is of the same name with the middle numbers, viz. pounds interest. In the work I find the first quotient, 71. interest; and so I proceed from one denomination to another, till I find the same answer as in the work at two statings, 71. 3s. 6d. 3600 600 12 1200) 7200 (6d. 7200 This method of operation serves to answer all questions in the Double Rule of Three Direct. THE DOUBLE RULE OF THREE REVERSE. In this rule you must place your numbers in such order, that your second and fourth numbers may be of one name or denomination, and your third and fifth. Example. If 100l. principal in 12 months gain 61. interest, what principal will gain 201, interest in 8 months? Stated thus: In this work, the third and fourth numbers are multiplied together for a divisor; and then the first is multiplied by the second, and that product by the fifth number, and the product, 24000, is divided by 48, and the quotient is 5001. principal, which is the answer to the question. RULES OF PRACTICE. These rules are so called, from their frequent use and brevity in casting up most sorts of goods or merchandise. Note, That any question in the Rule of Three, when the first number in the stating is 1, it is more briefly done by these rules, called Practice. But previous to these rules, it is necessary to have the follow ing tables by heart: 101. 13s. Answer. R Here 6d being the price of each pound, and the half of a shilling; therefore the half of 426 is taken, and gives 213s. or 10. 13s. Example 2. 4d. is of 1s. 512lb. of cheese, at 4d. per lb. } 20) 170 8d. Answ. 81. 10s. 8d. Here 4d. is of a shilling; therefore the third part of 512 is of a shilling, or 8d. remains. 170s. and Here the yards are divided by 4, because 3d. is the fourth of a shilling, and it quotes 61 shillings, and 2 remains, or two 3-pences so the answer is, 31. 1s. 6d. And thus may any proposed question be answered, belonging to the first table, or parts of a shilling; that is, by dividing the given number by the denominator of the Fraction, and the quotient will be always shillings, which (the remainder being known as above) bring into pounds, by dividing by 20, &c. When the price of the integer is at a farthing, a halfpenny, or three farthings, more than the value of the pence mentioned, then for those farthings take a proper part of the foregoing quotient found for the pence, and add them together. In this example I divide by 3 for the groats, as being the third of one shilling, and it quotes 83s. then I consider that a halfpenny is the eighth of 4d. therefore I take the eighth part of the groat line, or 838. and that produces 10s. and of a shilling, or 4 d. then the two lines being added together, make 93s. 44d. or £4 13s. 44d. as in the work. |