If the antecedent is less than the consequent, the ratio is less than a unit, and is called a ratio of less inequality. Thus, the ratio of 3 6 is, or, for. 187. We have hitherto given instances only of the ratios of abstract quantities or numbers to one another; but we may similarly obtain the ratios of concrete quantities. Thus, the ratio of £54: £72, of 9 cwt.: 12 cwt., of 15 gals.: 20 gals., of 33 ft. 44 ft., are by (§ 113, Rule XL) represents,,,, each of which reduces to, and we say, therefore, that the ratio of £54: £72 is the same as that of 3: 4, or, meaning that £54 is of £72, and so on with the other ratios; but Ratio can exist only between quantities of the kind, for otherwise, one of them could not be a fraction of the other. The idea of relative or comparative magnitude which is essential to the correct notion of proportion, forbids our considering the term ratio in the same unrestricted sense as the term quotient; the two terms are to be regarded as meaning the same thing only when dividend and divisor are quantities of the same kind; ratio is always an abstract number; but the name quotient, as we have seen, is applied not only to abstract numbers, but to the concrete quantities that arise from taking the third, fourth, fifth, &c., part of concrete quantities. Be careful to observe this distinction, and to remember that the ratio of an antecedent to its consequent always has reference to the number of times, and parts of a time, which the former contains the latter; so that it would be absurd to speak of the ratio of one thing to another of a different kind, as for instance, of the ratio of £6 to the number 3, of 8 cwt. to £4, or 9 cwt. to 12 gals., &c. 188. The ratio between two quantities of the same kind cannot be expressed by an abstract number unless both are expressed in one denomination. If the concrete numbers, though of the same kind, be not in the same denomination of that kind, they must be reduced to one and the same denomination, in order to find their ratio. Thus, if one number be 7 days and the other 13 hours, the ratio of the former to the latter will not be that of 7 to 13 but that of 7 days to 13 hours, that is, 168 hours to 13 hours, which will clearly be that of the abstract number 168 to the abstract number 13, and so will be expressed, not by, but by 15. We see that 7 dys. 13 hrs. is the same as 168: 13, and that each is 16. Thus it is plain that when the numbers are concrete, we may reduce them to one and the same denomination, and then, in considering their ratio, treat them as abstract numbers. 13 16 EXAMPLES FOR PRACTICE. I. 2. 3. What is the simplest expression of the magnitude of the ratio 35, 4: 12, 9:21, Which of the ratios is greatest, 5: 9 or 7: 11, 10: 17 or 17: 23, and 34: 27 or 37: 31. 4. What is the ratio of £2 6s. 8d. to £3 10s., of 3 tons 5 cwt. 2 qrs. to 5 tons 7 cwt., of an oz. troy to an oz. avoirdupoise. 5. Find which is the greatest ratio, or 3, or 14, 1 or 4. 189. Proportion in an equality of ratios. If two ratios are equal (i.e, "If the fraction expressed by the antecedent and consequent of the first ratio be equal to the fraction expressed by the antecedent and consequent of the second ratio ") they are said to form a Proportion, and the four terms of which it is composed are called Proportionals, or said to be proportional to one another.* Thus, if we have two numbers, as 6 and 2, of which we know the first is 3 times greater than the second, and two other numbers, 15 and 5, of which the first is again 3 times greater than the second, the ratio of 6 to 2 is equal to the ratio of 15 to 5, and the four numbers 6, 2, 15, 5, constitute a proportion which is usually written and is read 6:2 :: 15 : 5, or 6 : 2 = 15 : 5 as 6 is to 2, so is 15 to 5. And since a ratio may be written as a fraction the proportion may be written = , and is read-6 divided by 2 is equal to 15 divided by 5. 190. The first and last terms of a proportion are called the Extremes, the two middle terms the Means. While the two terms of a ratio, if they be not abstract numbers, must be quantities of the same kind, it is not necessary that all the four terms of a proportion should be of the same kind; it will be sufficient that the quantities in the first ratio be of one kind, and the quantities in the second ratio of one kind; thus 3 cwt.: 12 cwt. = 7 in. 28 in.; and generally we may say that any four quantities are proportionals when the first is the same number of times greater or less than the second that the third is greater or less than the fourth. 191. Since a proportion expresses the equality of ratios, and the value of these ratios may be denoted by fractions, the properties of ratios are made to depend directly upon the properties of fractions. Hence, if we take any two ratios which are equal to one another, for instance, 4: 12 and 7: 21, where it is true that 4 12: 7:21, we may say, A = 1 next, reducing these fractions to equivalent fractions having a denominator, we have and, as the numbers taken were not particular but general, we deduce the following general rule, viz., that in every proportion the product of the extremes is equal to the product of the means. The following definition should be remembered- -Four quantities are said to be proportionals when the first is the same number of times greater or less than the second that the third is greater or less than the fourth." The usual form of writing the process is as follows: N.B. In taking the aliquot parts a little judgment should be exercised to prevent an awkward quantity from remaining at last-e.g., we might have taken the aliquot parts in the above example thus: We should now have had d. to calculate for. But d. is 1 of 6s. 8d., so that we should have been compelled to divide "the cost at 6s. 8d. by 160 to find the cost at d." This would have been a comparatively awkward operation, and it will be found better to avoid such remainders when it is possible to do so. When a remainder of this kind does occur, however, it is sometimes easier to make a separate calculation for it, thus: 324 articles at d. each 162 pence 138. 6d. Ex. 2. Find the cost of 8794 at 28. old. each. In this instance, instead of taking 28. od. = 1% of a £, and a farthing of 28., the last being an inconvenient divisor, we take 18. 8d. = 11⁄2 of £, 4d. = } of that, and a } = ✩ of that. 169. There is another method of finding the value of a number of articles when the price of one is less than £1, thus, Since the 750 articles at 12 we obtain the cost at at 10s. Next, 68. 8d. is see by the table that 28. is cost at 28. the cost at the cost at the cost at the cost at of each. the cost at £12 18 10 each. 1 each would obviously cost £750, if we multiply that sum by 12. As 10s. is of 1, by dividing 750 by 2, we find the cost of £1, by dividing 750 by 3 we get the cost at 6s. 8d. Next, we of 10s.; hence, by dividing the cost at 10s. by 5 we find the 170. If the price be less than a shilling, the best way is to take parts of a shilling so as to find the price in shillings, and afterwards reduce to pounds, thus Ex. 5. Find the cost of 10374 articles at 9fd. The cost of 10374 articles at 18.103748. 171. The following artifice may sometimes be employed to simplify the solution of questions in practice, viz. Taking the price as if somewhat higher than that which is given, and subtracting a convenient aliquot part; thus, Ex. 6. Find the price of 218 cwt. at £5 18s. 4d. per cwt. Here the given price, viz., £5 18s. 4d. = £6, less is. 8d. Therefore, find the cost at £6, and subtract the cost at is. 8d. the cost of 218 cwt. at 10 £218 o per cwt. 172. When the price is an even number of shillings, multiply the number of articles by half the number of shillings, cut off the unit's figure of the result, and double it: reckon this doubled figure as the shillings, and the rest of the result as the pounds of the answer. 313 9 Answer £281 14 Here the of 18s. is 98.; then 9 times 3 are 27, and doubling this we have 54; 54 shillings are 2 pounds 14 shillings; write 148. down and carry the 2 pounds: 9 times I are 9, and 2 carried are 11; set down 1 and carry 1: 9 times 3 are 27, and I carried are 28, set down 28; hence the answer is £281 148. 173. When it is required to find the value of a certain number of things, and a fraction, we proceed as follows: Answer £724 16 8 the cost at £2 13 8 each. The above is the most expeditious method of finding the answer, which, however, might have been obtained as follows, Since 270 cwt. will cost 270 times £2 138. 8d., cwt. will cost times of 2 138. 8d., In the last line of division the student must observe that the id. is the that £16 18. 1d. is divided by 15, instead of the line immediately above. of Is. 3d., and |