79. If a soldier be allowed 12 lbs. of bread in 8 days, how much will serve a regiment of 850 men for the year 1856? 80. If 2000 men have provisions for 95 days, and if after 15 days 400 men go away; find how long the remaining provisions will serve the number left. 81. A gentleman has 10000 acres ; what is his yearly rental, if his weekly rental for 20 square poles be 1d.? (1 year = 52 weeks). 82. If an ounce of gold be worth £4.189583, what is the value of •36822916 lbs.? 83. If 1000 men have provisions for 85 days, and if after 17 days 150 of the men go away; find how long the remaining provisions will serve the number left. 84. What is the quarter's rent of 182-3 acres of land, at £4·65 per acre for a year? 85. A grocer bought 2 tons, 3 cwt., 3 qrs. of goods for £120, and paid 50s. for expences; what must he sell the goods at per cwt. in order to clear £61. 58. on the outlay? 86. What must be the breadth of a piece of ground whose length is 401 yards, in order that it may be twice as great as another piece of ground whose length is 14ğ yards, and whose breadth is 13 yards? 87. If 3.75 yards of cloth cost £3825, what will 38 yds., 2 qrs., 3 nails cost? 88. Four horses and 6 cows together find sufficient grass on a certain field; and 7 cows eat as much as 9 horses; what must be the size of a field relatively to the former, which will support 18 horses and 9 cows? 89. A alone can reap a field in 5 days, and B in 6 days, working 11 hours a day, find in what time A and B can reap it together, working 10 hours a day. DOUBLE RULE OF THREE. 157. There are many questions, which are of the same nature with those belonging to the Rule of Three, but which if worked out by means of that Rule as before given, would require two or more distinct applications of it. Every such question, in fact, may be considered to contain two or more distinct questions belonging to the Rule of Three, and when each of those questions has been worked out by means of the Rule, the answer obtained for the last of them will be the answer to the original quest 158. The following example may serve to illustrate the preceding observations. "If the carriage of 15 cwt. for 17 miles cost me £4. 58., what would the carriage of 21 cwt. for 16 miles cost me ?" We observe that this question, though of a like nature with those which engaged our attention under the Rule of Three, is nevertheless of a more complicated description; and the student, without further explanation, would find some difficulty in obtaining an answer to it by means of a single application of the Rule. For we observe, that instead of three given quantities, we have five, every one of which must necessarily have a bearing on the answer, so that none of them can be superfluous. If however the question be divided into two distinct questions, each of these, when superfluous terms are rejected, will be found to comprise only three given terms of a proportion, from which three terms the fourth is to be ascertained; and the student would have no difficulty in working out each of these two questions by means of a single application of the Rule, so that in this way he will obtain the correct answer by applying the Rule of Three twice over. The first question may be this; "If the carriage of 15 cwt. for 17 miles cost me £4. 5s., what would the carriage of 21 cwt. for 17 miles cost me?" In this question the 17 miles would have no effect upon the answer, because the distance is the same in both parts of the question, and the answer would clearly remain unaltered, if any other number of miles, or if the words "a certain distance," had been used instead of the 17 miles. This number may therefore be neglected as superfluous, and we have then three terms of a proportion remaining, and the fourth is to be found. Solving the question by the Rule of Three, we find that the answer will be £5. 19s. The second question may be this: "If the carriage of 21 cwt. for 17 miles cost me £5. 19s., what will the carriage of 21 cwt. for 16 miles cost me?" In this question, for reasons similar to those before given, the 21 cwt. will be a superfluous quantity. Applying the Rule of Three to the question, we find the answer to be £5. 128. From the connection of the two questions with that originally proposed, we observe that £5. 12s., thus obtained through two distinct applications of the Rule of Three, must be the answer to the original question. 159. We might give still more complicated instances, in which more than two distinct applications of the Rule of Three would be needed, in order to obtain the required answer; but the practical questions which most commonly occur, of the kind we have been treating of, would require only a double application of the Rule of Three, and, like the question which has been used by way of illustration, would comprise only five given quantities for the determination of a sixth which is not given. 160. The DOUBLE RULE OF THREE is a shorter or more compendious method of working out such questions as would require two or more applications of the Rule of Three; and it is sometimes called the Rule of Five, from the circumstance, that in the practical questions to which it is applied, there are commonly five quantities given to find a sixth. 161. The method of working out examples which belong to the class under consideration, may be expressed as follows: RULE. "Set down the quantities which express the conditions of the question in one line. Under each of these quantities set its corresponding one in another line, leaving a blank in that place in which the required term would have fallen. Where two quantities stand one above the other reduce them both to one and the same denomination, if they be not already in that state; and as to the remaining quantity under which the blank lies, reduce it so as to be wholly in one denomination, if not already so; then the resulting numbers may be treated as abstract. Multiply together the numbers which arose from the producing quantities of one line and those which arose from the produced quantities of the other, or, in other words, those numbers in one line which represent causes, and those in the other which represent effects, for a dividend; and then the remaining numbers, whether arising from causes or effects, or both, for a divisor. The quotient of the division will be the answer to the question, in that denomination to which the quantity standing over the blank was reduced." The rule above laid down, though expressed at greater length, is substantially the same as that which is stated by Dr. Olinthus Gregory, in his edition of Hutton's Course of Mathematics, to have been given in 1706, by W. Jones, Esq. F.R.S., the father of the celebrated Sir William Jones. Ex. 1. If a tradesman with a capital of £2000 gain £50 in 3 months, how long will it take him with a capital of £3000 to gain £175? The quantities which express the conditions of the question are £2000, £50, and 3 months. Our arrangement may therefore be the following: £2000. £3000. 3 m. £50. The producing quantities or the causes in the first line are £2000, and 3 months; for these together, that is, £2000 employed for 3 months, produce the £50, which is the produced quantity or the effect in the first line. The produced quantity or the effect in the second line is £175. Hence, according to our Rule, Dividend = 2000 × 3 × 175=1050000 Divisor 3000 × 50 = 150000. Effecting the division, we have 15,0000) 105,0000 (7 105. The quotient is 7, which therefore, according to our Rule, is the answer to the question, in the same denomination as the term under which the blank was left, that is, in months; so that the required answer is 7 months. The tradesman, with a capital of £2000 gains £50 in 3 months. Let us first find, by the Rule of Three, how long he would be in gaining £175 with the same capital. Thus Since then the tradesman with a capital of £2000 would gain £175 in (175x 3) months, let us next find, by the Rule of Three, how long it would take him to gain the same sum with a capital of £3000, and we shall have the answer to the original question. Thus that is, if we arrange the given quantities as follows, we obtain the answer to our original question by multiplying together the producing numbers or causes of one line and the produced number or effect of the other, for a dividend, and the two remaining numbers for a divisor, and then carrying out the division; and we obtain the answer in the denomination of that term which stands over the blank. that is, if we arrange the given quantities as follows, we obtain the required time in months by multiplying together the producing numbers or causes of one line and the produced number or effect of the other for a dividend, and the remaining numbers for a divisor, and finding the quotient. Ex. 2. If a tradesman with a capital of £2000 gain £50 in 3 months, what sum will he gain with a capital of £3000 in 7 months? |