amount for £: Ex. 3 might 48 at 1s. 1 d. have been worked in this way:. 1 d. 13 | 6 but I shall illustrate what I mean by taking Ex. 5. Here we see, 54s. that by increasing the price by £312-£2 148.= 18. 11d., it becomes £6 10s. : the correct result will therefore be £309 6s. found by computing the value of 48 articles at £6 108., as at the bottom of last page, and subtracting from it the value of 48 at 18. 12d. I shall work out for you only one more example, serving to show how you may sometimes combine Practice with Compound Multiplication to advantage: you will see that one part of the operation is performed by both methods. 7. Find the value of 7cwt. 3 qrs. 11 lbs., at £2 138. 1d. per qr. As the price is per qr., we reduce the 7 cwt. 3 qrs. to 31 qrs., so that we shall have to multiply the price by 31, and then to take parts for the odd 11 lbs. : the first of these operations is here given in two distinct forms. Of the two ways of finding the £82 58. 7d., the first is to be preferred as the easier. NOTE 1. It is worth while to observe, that whenever the price of a number of articles at an even number of shillings each is to be computed, the shortest way is to take only half the number of shillings, to multiply the given number by this half, and put down the double of the first figure of the product for shillings, the remaining part of the product will be pounds : thus, if you have to compute the value of 123 articles at 188., the work is 123 at 18s. comprised in only the few figures in the 9 margin. By doubling the first figure 7 of the product, for shillings, and not £110 148. doubling the other figures, you get the same result as you would do if you were to multiply by 9 and by 2, and were then to divide the resulting number of shillings by 20, as is sufficiently obvious. Note 2. It may also be noticed here, that in the purchase of some kinds of goods, certain commercial allowances are made for the packages, chests, &c. containing them, as also for waste; and these allowances are deducted from the gross weight. The deduction for the package is called tare, and is generally at so much per cwt. This deduction is made first. The deduction for waste, which is made next, is called tret, and is an allowance of go, or } of is of the weight, when diminished by the tare. Besides these, a trifling allowance is sometimes made to retailers, for what is called “the turn of the scale;" it goes by the name of cloff. The amount of these deductions may always be computed by Practice ; and no special directions are necessary for executing the work : the deductions being made, the result is the net weight. Exercises. * 1. 37 at 2 16 6 7. 243 at 2 4 74 2. 41 at 3 17 8 8. 317 at 4 3 9 3. 79 at 5 11 7 9. 353 at 7 18 41 4. 83 at 6 3 62 10. 417 at 3 9 2 5. 133 at 1 13 8? 11. 358 at 6 11 5? 6. 211 at 3 9 53 12. 519 at 7 19 10 13. 4 cwt. 2 qr. 5 lb. at 168. 4d. per cwt. 14. 2 cwt. 1 qr. 13 lb. at £1 13s. 9d. per cwt. 15. 38 yds. 1 ft. 7 in. at £2 38. 7d. per yard. 16. 17 yds. 2 ft. 8 in. at 138. 3 d. per foot. 17. 7 ož. 11 dwt. 18 gr. at 38. 84d. per oz. 18. 14 weeks 3 days at £1 12s. 6d. per week of 7 days. 19. 3 months 3 weeks 3 days at £1 3s. 10d. per week of 6 days. 20. 127 ac. 3 roo. 37 per. at £3 68. 8d. per acre. 21. 17 cwt. 3 qr. at £5 5s. per ton. 22. 86 lb. 3 oz. 15 dwt. 18 gr. (troy) at £4 16s. 4d. per oz. 23. The Sydney Morning Herald (Aug. 1851) reports, that gold to the amount of from £5000 to £10000 reaches Sydney, in Australia, daily, from the gold fields : * In working these exercises, the results are to be brought out to the Dearest farthing, and are not to be encumbered with fractions of a farthing. £3 88. 4d. per ounce has been offered for all the gold Government might receive during two months : among the recent arrivals from Ophir was a lump weighing 51 oz. 15 dwt. ; what would be the price of it at the above rate? 24. The number of acres growing hops in England, in 1850, was 43127: the duty on these was at the rate of £9 178. per acre: required the whole amount of duty ? 25. In the year ending Jan. 5, 1851, 2623656 gallons of spirits were imported into England from Scotland, and 828138 gallons from Ireland; the duty paid on the transfer was 78. 10d. per gallon : what was the whole amount of duty ? 26. The prices paid at St. Thomas's Hospital, London, for beef and mutton during the year 1850, were for the former 28. 8d. per stone, and for the latter 38. 4d. per stone: how much was paid for 13 cwt. of each ? 27. The average pay of the crew of a foreign-going trading vessel is as follows: captain, £10 per month ; mate, £5; second mate, £2 158. ; carpenter, £4; able sea man, £2 58. The number of able seamen is 2 to every hundred tons registered : what is the pay of the company of a ship of 800 tons for 107 days, allowing 30 days to the month ? 28. Every person in Great Britain who receives an annual income of £150, or more, must pay an income-tax of 7d. in the £: what amount of tax must a person pay whose incor is £2625 ? 29. The dearest year for provisions ever known in England was the year 1813; the contract price of butchers' meat paid by Government for the supply of Greenwich Hospital was then £4 58. per cwt.: as the average charge for beef and mutton in 1850 was 3s. per stone, calculate the reduction in price per cwt. and per stone. 30. The quantity of wheat sold in the United Kingdom in 1850 was 4688246 quarters; of barley, 2235271 quarters; and of oats 866082 quarters: the average prices for the year were, wheat, 40s. 3d. per quarter ; barley, 238. 5d. per quarter; and oats, 168. 5d. per quarter : what sum was received for the whole ? PROPORTION. (67.) Four quantities are said to be in proportion when the first contains the second as many times and parts of a time, as the third contains the fourth, or when the complete quotient of the first, divided by the second, is the same abstract number as the complete quotient of the third divided by the fourth.* (68.) The complete quotient arising from dividing one quantity by another of the same kind is called the ratio of the former to the latter : thus the ratio of 6 to 3 is 2; the ratio of 6 to 2 is 3; the ratio of 7 to 4 is 1; and so on. Ratio is thus only another name for quotient : the first term of the ratio, that is the dividend, is called the antecedent; and the other term, that is the divisor, is called the consequent. When, therefore, four quantities are said to be in proportion, or to form a proportion, you are merely to understand, that the ratio of the first to the second is the same as the ratio of the third to the fourth ; in other words, that if you were to divide the first by the second, you would get the same complete quotient, as if you were to divide the third by the fourth. For example, the four numbers, 12, 2, 18, 3, form a proportion, because the ratio of 12 to 2, namely 6, is the same as the ratio of 18 to 3. Instead of saying in words that these four numbers are in proportion, it would be sufficient to write them in a row, with dots between them, as follows: 12 : 2 :: 18 : 3, which would be read, 12 is to 2, as 18 to 3; or, as 12 is to 2, so is 18 to 3. You see that the two dots which separate the terms of each ratio, differ from the sign for division (=) only by the little mark between them; and, in fact, the notation just employed is only the same as 12+2 = 18-3, and you may always regard a proportion in * The learner will observe, that I say here the same abstract number, because a ratio, implying the relation of one quantity to another with respect to the magnitude of them, can exist only between quantities of the same kind. It has already been seen (page 62), that division and quotient, unlike multiplication and product, are terms that are used in different senses : we are said to divide a concrete by 4, when we take the fourth part of that quantity; which fourth part we are accustomed to call the quotient. If division were restricted to mean the finding of how many times one quantity is contained in another, the operation just noticed would be excluded; the term quotient is to be understood in this restricted sense when used for ratio. this light; indeed, you will often find the term proportion to be briefly defined as “an equality of ratios," or quotients. (69.) I think, from this explanation, you clearly see the meaning of ratio, as applied to two numbers, and of proportion, as applied to four; that if two numbers were proposed, you could tell the ratio of the first to the second; and that if four were proposed, you could find out whether they were in proportion or not: thus, if the two numbers 24 and 6 were proposed, you would know that the ratio of the first to the second is 4 ; that the ratio of 8 to 2 is also 4; and you would thus infer, that the four numbers, 24, 6, 8, 2, are in proportion; or, that 24 : 6 :: 8 : 2. Again ; the ratio of 17 to 4 is 3, so that the four numbers, 24, 6, 17, 4, are not in proportion; the ratio of the first to the second being greater than the ratio of the third to the fourth. In like manner, the ratio of 6 to 8 is g or ; and the ratio of 15 to 20 is 16 ore; therefore, 6:8 :: 15 : 20; also, 6 : 8 :: 3 : 4. When you have once got an antecedent and consequent, that is, the two terms of a ratio, you can easily get another antecedent and consequent, that is, two other terms, in the same ratio ; and can thus form a proportion, having the given antecedent and consequent for the first two terms of it: suppose, for instance, you had 6 and 8 for antecedent and consequent, and you wanted a proportion in which 6 and 8 should stand first; you have only to remember, that the ratio of 6 to 8 is expressed by the fractions, and that numerator and denominator of a fraction may be multiplied or divided by any number you please, in order to get as many other suitable pairs of numbers as you choose : thus, since = g = 1 = 48 = 40 &c., you infer at once that all the following are proportions, namely, 6 : 8 :: 3:4; 6:8 :: 12 : 16; 6:8 :: 30 : 40; 12 : 16 :: 36 : 48; and so on. And whenever you have two fractions equal to one another, you may always convert the equality into a proportion, and say, the numerator of the first is to its denominator, as the numerator of the second to its denominator; or, instead of saying numerator is to denominator, you may, if you please, say denominator is to numerator, because if two fractions are equal, the equality remains, though we make the numerator and denominator of each change places; and it is because of this, that in any proportion we may make the antecedent and consequent of each ratio change places : thus, since it is true, that 3:19 :: 9 : 21, it is also true, that 7 : 3 :: 21 : 9; and |