571. Remember, in every question in this rule the ratio is always the same between the third term, and the number sought, as between the first term and the second. 572. The student will not have failed to observe, while performing the foregoing questions, that in the three terms given, there is always one of them of the same name as the number sought; this term should in all cases form the third term in the statement of the question; the other two terms given are always of the same name. In the placing of these, the student should consider, from the nature of the question, whether the term sought will be larger or less than this third term; if it will be larger than the third term, place the larger of the other two terms for the second term, and the lesser for the first; if the term sought will be less than the third, then place the lesser of the other two terms for the second, and the larger for the first. To illustrate this, take the following question. If 10 cwt. 3 qrs. 14 lbs. of sugar cost £39 14s. 9d, what will 7 cwt. cost? Here are three terms given to find a fourth. From the nature of the question, we perceive that the fourth term must be in money; therefore we place money for the third time; by inspecting the other two terms, we perceive that the answer must be less than the third term, therefore we place the lesser of the two for the second term, and the larger for the first. The question should therefore be stated thus. and the answer, whatever it be, will bear the same proportion to the third term which the second term bears to the first. Now the ratio between the first and second terms is not easily perceptible; in such cases as this, we pursue a method somewhat different to that hitherto pursued, but founded on the same principles. We first reduce the two first terms to the lowest denomination contained in them, and each of them to the same denomination; we then reduce the third term to the least denomination contained in it. The three terms thus reduced, we multiply the second and third terms together, and divide the product by the first, and the quotient will be the fourth term in the same denomination, to which the third term has been reduced. See paragraph 240 : 573. The first and second terms of the above question reduced to pounds, and the third term reduced to pence, with the answer in pence, will form the geometrical series 1211, 784, 9537, 6174,294. 574. The reason for the statement will clearly appear from what is stated above; it remains to show the principles on which we work such questions. 575. It has been shown at paragraph 545, that when four numbers are proportional, the product of the two means are equal to the product of the two extremes. Thus if a be the first term, b the second, c the third, and a the fourth or term sought, then abc, from which equation the following may be deduced, viz. : bc ax = c, ax = b, be =α, b c 576. This last equation forms the rule by which we proceed; we thus see clearly that if the first and fourth terms multiplied together be equal to the product of the second and third, the product of the second and third, divided by the first, must give the fourth, and this is the general rule of operation in what is termed the Rule of Three. 577. In order to impress the principles of proportion per manently on the mind of the student, let him at all times bear the following principles in recollection, viz. :— 1. That three numbers are proportional when the square of the middle term is equal to the product of the extremes; thus 964 for 9 X 4 = 6 X 6 = 36. 2. That four numbers are proportional when the product of the two middle terms is equal to the product of the two extremes; thus 4: 6:57. Then 4 X 7 = 6 × 5. 3. That four numbers are proportional when the first con tains the second, as often as the third contains the fourth; thus 6 : 4 :: 7 : 5, then = 7 - = 1; or when the second contains the first as often as the fourth contains the third; 71 thus 6: 7 :: 4 : 5, then == 1; or when the third contains the first as often as the fourth contains the second; 7 thus 4:5 6: 74, then = = 1; or when the first contains the third as often as the second contains the fourth, thus 7:5 :: 6:4, then 7 = 4. That when four numbers are proportional, the first, multiplied by any number, will contain the second as often as the third, multiplied by the same number, will contain the fourth; thus 4:6:: 5 : 74, thus X35X 4 71 3 2. So also the second multiplied by any number will contain the first as often as the fourth, multiplied by the same number, will contain the third. As 6:44:: 4 : 3 for 4X4 3 X 4 6 = 4 = 3; so also the third, multiplied by any number, will contain the first as often as the fourth, multiplied by the same number, will contain the second, for = 3 X 3 = 2. So also the first, multiplied by any number, will contain the third as often as the second, multiplied by the same number, will contain the 5. If four numbers be proportional, a proportion will be preserved if the extremes change place. For example, 8:6:: 28: 21 for 21 X 828 × 6=168. and 21:6 :: 28: 8 for 8 X 28 = 6 X 28168. 6. If four numbers be proportional, the proportion between the terms of the first and second couplets is the same if the first and second, the first and third, the third and fourth, or the second and fourth be divided by any particular number. To exemplify this principle, let us take the terms 8: 6:28: 21. उ 8 6 :: 28: 21. 2 28 Then 8: 6:28:21 = 8 6 8:6:28:21 = 8: : : 28: 2. Thus it will be seen that if three terms be given to find a fourth, and if the first and second, or the first and third terms be divisible by any number, the process may be much shorter, as for example: 578. If 12 acres of land maintain 16 horses, how much land will it require to maintain 36 horses? 7. If four numbers be proportional, the proportion between the terms of the first and second couplets is the same, if the first and second, the first and third, the third and fourth, or second and fourth be multiplied by any number, as 8:6:: 4:3=8×3:6 X 3::4 8' : : 3 8:6:: 4 : 3 = 8×3: 6:: 4X3: 3 579. Thus when the terms of a question consist of mixed fractional, or compound numbers, the operation may frequently be shortened by multiplying the first and second, or the first and third terms, by such a number as will reduce the fraction or compound number. As for example, If 3 acres of land let for £6 13s. 4d., what should be the rent of 12 acres of the same land? 580. If 8 men build a certain wall in 12 days, how many men could build a similar wall in 8 days? 581. This is a question in what is termed Inverse Proportion. It is usual with Arithmeticians to class questions in this rule of proportion under two separate heads, with distinct rules of statement and of operation. This separation, in my opinion, has a tendency very much to confuse the subject to beginners, Every question which is inversely proportional, may be treated as directly proportional. I have therefore thought it better to give one uniform method of statement and of operation, which is suited to all questions in this rule, whether they be directly or inversely proportional, rather than trouble the student with niceties of distinction, which will only serve at present to perplex him. After the student has passed through |