or since 3:7 :: 1? : 3}, so also 7:3 :: 31 : 1ų; you, of 3 18 course, see that since the latter fraction arises from 7 33 dividing numerator and denominator of the former by 2. From all this, it appears, that a proportion may always be converted into a pair of equal fractions, and a pair of equal fractions into a proportion : the word fraction being here used in its widest sense, and applying to every quantity expressed in a fractional form. By thus replacing a proportion by two equal fractions, many properties of proportional quantities may be easily deduced; I shall here notice only two. Since first term third term fourth term second term second term fourth term? third term first term second term it follows that fourth term = x third term, first term a property which indicates to us how we are to find the fourth term of a proportion, when only the first, second, and third terms of it are given. Of these three terms, the first two, that is, the first antecedent and consequent, must always be quantities of the same kind; I mean, that both of them must be either abstract numbers, or both of them concrete quantities, belonging to one class of quantities: if the first term, for instance, be money, the second must be money ; if the first be time, the second must be time; and so on. You must see the necessity of this from the very nature of ratio, as defined at the beginning of this section ; it is the number of times the antecedent, or first term of the ratio, contains the consequent, or second term; and therefore ratio is always an abstract number. (70.) This you must be careful to bear in mind, else you will run the risk, in finding the fourth proportional, as it is called, to three given concrete quantities, of committing an absurdity at every step of your work, though you may bring out the correct result. I have shown to you above, how the fourth term, or fourth proportional, may always be found in an unobjectionable manner; you there see that you are to divide the second term by the first, and to multiply the third term by the resulting abstract number : you will thus get the sought fourth term, and always in the same denomination as the third term. If the terms of both the equal ratios constituting the proportion, or if the terms of only one of these equal ratios, were abstract numbers, then we might deduce from the general property above, this other particular property, namely, first term x fourth term = second term x third term ; or, that the product of the extremes is equal to that of the means ; a statement which, you see, would be absurd, if the first and fourth, or the second and third, were both concrete quantities. (71.) If the first and second of the three given terms of a proportion were always abstract numbers, then, from what has just been shown, we could find the fourth term by multiplying the second and third together, and dividing by the first, or cutting the product up into as many equal parts as there are units in the first term: and this would, in general, be an easier way of getting at the fourth term, than by dividing the second term by the first, and then multiplying the third term by the quotient, as directed above. Now I wish you particularly to observe, that we may always proceed in the latter easier way, even when the first and second terms are concrete. You know you cannot divide one concrete quantity by another, till both are brought to the same denomination : having done this, your quotient is an abstract number;—the very same abstract number that you would get if the common denomination of dividend and divisor were wholly disregarded, and the division performed on the abstract numbers simply. In the case, therefore, of the first and second terms of a proportion being concrete quantities, all you have to do is to reduce these quantities to the same denomination ; then, leaving denomination altogether out of consideration, to employ only the resulting abstract numbers, which you see may always be put instead of the quantities themselves : for the ratio, or quotient, remains unaltered, whether the denomination (or the concrete unit) be preserved or suppressed. (72.) What is called the Rule of Three, or the Rule of Simple Proportion, is merely the method of finding the fourth terın of a proportion when the first three are given. In most of the questions coming under this rule, the three given quantities do not occur in the order in which you would write them as three of the four terms of the proportion : in the question you will usually find that the quantities which stand first and third must stand first and second, or second and first, when arranged as terms of a proportion; the following, for instance, is such a question : If 3 lbs. of sugar cost 164..., what will 10 lbs. cost? Here, it is plain that the 3 lbs. and the 10 lbs. must bear the same relation to one another, that is, must have the same ratio, as the price of the 3 lbs. to the price of the 10 lbs. Hence the statiny of the question, as it is called, would be: 3 lbs. : 10 lbs. :: 16.d.; or, suppressing the common denomination, lbs., of the terms of the first ratio, 3 : 10 :: 16d. As the third term here is pence, the wanting fourth term, which is to complete the proportion, must also be pence. It is found agreeably to what is said above, by multiplying the 16d. by the 10, and dividing the pro 16 d. x 10 duct by the 3; that is, it is = 55d. The general 3 rule for all questions of this kind is as follows : (73.) RULE OF THREE, OR SIMPLE PROPORTION. Rule 1. Write down the three given quantities as the first three terms of a proportion, taking care that the third term is a quantity of the same kind as the required fourth term; and according as this fourth term is to be greater or less than the third, so let the second term be greater or less than the first. 2. Having thus stated the question, reduce the first and second terms to the same denomination, if they are not already the same as well in denomination as in kind; and then, disregarding the denomination, consider the first and second terms to be abstract numbers, divide them by any number that will obviously divide both, and use the quotients instead; which, as they will be smaller numbers, may in general be more easily worked with. 3. Multiply now the third term by the second, and divide the product by the first term, and the quotient will be the answer, or fourth term of the proportion; and it will be in that denomination, whatever it be, in which the third term was used. (74.) Note 1. It may be proper here to remind you, that the first and second terms of a proportion may be regarded as the two terms of a fraction ; and the third and fourth terms as the two terms of another fraction equal to the former. It is by viewing an antecedent and consequent in this light, that we perceive our right to divide both by any number we please; and further, that according as the first term of a proportion is greater or less than the second, so inust the third be greater or less than the fourth. This fact will be a sure guide to the correct stating of a Rule-of-Three question. A little attention to the question will always enable you to see whether the fourth term you seek ought to be greater or less than the third, which is given; so that you need never fall into the mistake of putting that term first which ought to be second. NOTE 2. As the fourth term or answer to the question is always got by multiplying the third term by the second, and dividing the product by the first, using the first and second as abstract numbers, you may regard the work as indicated by a fraction, of which the pumerator is the product of the third term by the second, and the denominator the first term; and as you may divide the numerator and denominator of a fraction by any number you please, you may obviously divide not only the first and second, but also the first and third of the terms, by any number that will really simplify them; and may work with the simplified results instead of with the terms themselves; but you must be careful not to take this liberty with the second and third terms. An example or two will make you familiar with the process. Ex. 1. If 8 articles cost £21 4s., how much will 26 cost ? Here the fourth term of the proportion, that is, the answer £. £. to the question, must be money; 8 : 26 :: 21 4 : 68 18 we therefore make money the 4 : 13 20 third term; and as the required fourth term must evidently be 424 greater than the third, we take 13 care that the second term is greater than the first; and 1272 therefore state the question as 424 in the margin. We now look at the first and second terms, 4) 5512 and readily see that both will divide by 2; we therefore re- 2,0) 137,88. place 8 : 26 by 4 : 13; and as £21 4s. is not easily multiplied Ans. £68 18s. by 13, we reduce the compound quantity to the denomination shillings before using it; we then multiply the shillings thus obtained by 13, and divide by the 4, as the rule directs. We thus find that the fourth term of the proportion, or the answer to the question, is £68 188.; and we make the original stating complete by inserting the fourth term, now found, in its proper place. The work would have been a little easier if, instead of first multiplying by 13 and then dividing by 4, we had first divided by 4 and then multiplied by 13; but if, after having replaced 8 : 26 by 4 : 13, we had divided the first and third terms by 4, the work would have been easier still; for the stating would then have been 1 : 13 :: £5 68., and we should have got the answer at once, by multiplying £5 68. by 13, which of course is easily done; the first term being 1, no division is performed. 2. If 19 cwt. of sugar cost £57, how much may be bought for £111 ? Here the answer must be weight; £. £. cwt. cwt. we therefore put the given weight 57 : 111 :: 19 : 37 for the third term; and as the re- 19 : 37 quired weight is greater than the 19 given weight, of the other two given quantities we take care to put the 333 greater second; the stating is there- 37 fore as in the margin : and as we easily see, that instead of 57 : 111 19) 703 (37cwt. Ans. we may put 19 : 37, we accordingly 57 use this simplification. As 19 times 37 is the same as 37 times 19, we 133 find the product in the former way 133 for convenience. But you see there was not, in reality, any occasion for multiplying at all; for dividing the first and third by 19, the stating becomes simply 1:37 :: 1 cwt. : 37 cwt.; so that the answer is got at once. From the remarks appended to the two examples here solved, you cannot but perceive the advantage of a little preliminary examination of your stating before you begin to apply the rule. The work given at length in the margin is intended more for your avoidance than for your guidance in similar cases. A little thought and ingenuity on your part will often do more for you than all the rules of arithmetic. 3. What is the yearly rent of 47 ac. 3 roo. 21 per. at £1 4s. 6d. per acre ? |