Ex. 5. Find the value of 76 sq. yds. 5 sq. ft. 108 sq. in. at 38. 91d. per sq. ft. 176. The principles of Practice may be applied to any Compound Multiplication sum, as in the following examples. Ex. 6. Find the gross weight of 179 cast iron pipes, each weighing 2 cwt. 3 qrs. 7 lbs. The weight of 179 pipes at 1 ton each = 179 tons. Ex. 8. A bankrupt owes £3600 and pays 128. 6d. in the £, what is the value of his assets ? This means that for every £ he owes he can pay only 128. 61d.; here then we have to find the value of 128. 6d. X 3600, which must have been the value of his whole estate. If he paid 208. in the £ his assets would be worth £3600. Ex. 9. Find the weight of 252 sacks of wheat, each sack weighing I cwt. 2 qrs. 23 lbs. Here 252 sacks of wheat, each weighing 1 cwt., weight 252 cwt. 177. We may sometimes introduce a subsidiary aliquot part to facilitate calculation, and erase it from the result when the other parts have been found from it, e.g.,— Ex. 10. per ounce. Find the cost of a silver-gilt goblet weighing 3 lbs. 4 oz. 15 grs. at £2 78. 8d. In this example, as the denomination of dwts. does not occur in the question, and as a grain is part of an ounce, it is convenient to introduce the price of 1 dwt.; not, however, to affect the answer, but only to derive from it the price of 15 grs. When this has been done, the cost of the I dwt. must be erased before the products, which form the answer, are added together. 178. When both the factors in the question contain fractional parts which are complicated, it is a good plan to turn the money factor into pounds and the decimal of a pound, and then to take the aliquot parts of the other factor; e.g., let it be required to find the price of 11 tons 17 cwt. I qr. 19 lbs. at £7 78. 44d. per ton. Here, reducing 78. 44d. to the decimal of a pound, we find it is £36875. Find the value of EXAMPLES FOR PRACTICE. 1. 785 at 10s.; 785 at 5s.; 785 at 48.; 785 at 6s. 8d.; 785 at 1s. 8d. 2. 785 at 18. 3d.; 234 at £1 38. 4d.; 3. 895 at 6d.; 895 at 4d.; 895 at 1d. 1275 at £1 78. 6d.; 2101 at £2128. 1275 at 1 108.; 1275 at £1 58.; 4. 8943 at £3 178. 9d.; 4782 at £7 138. 9d; 890 at £2 os. 14d.; 164 at 88. 51d. 5. 640 at £7 48. 73d.; 320 at £3 os. 31d.; 582 at £12 108. 24d.; 1603 at 168. 10дd. 6. 7. 150 at 91d. per doz.; 265 at £63 38. 14d. per score; 85 at £8 17s. 6d. for every 40. III at 118. 11d. for every 11; 4258 at £3 178. 10d.; 12437 at £3 148. 2d. 8. 3 tons 12 cwt. 2 qrs. X 379; 2 lbs. 7 oz. 5 dwt. X 856; 14 yds. 1 ft. 6 in. x 248; 1783 at 6s. 7 d. 9. 750 tons iron plates at £8 15s. 6d.; 215 tons angle iron at £11 148.; 30 tons rivets at £22 18. 10. II. 12. 72 cwt. 3 qrs. 17 lbs. at 6s. 14d. per qr.; 250 acres 3 roods 28 poles at £2 158. 6d. per acre; 30848 at 63d. 10000 at 81d.; 7895 at 101d.; a million at 11d.; 31426 at 28. 11d; 22828 at 148. 9d. A mile at 38. 11d. per yard; a ton at 8s. 74d. per lb. ; a lb. troy at 1d. per grain. 13. The weight of a cubic yard at 1 lb. 13 oz. to the inch; the weight in tons of 257354 cubic ft. of sea water at 64 lbs. ; the income tax at 5d. in the pound for £27,000,000. 14. What is the dividend on 2045 158. 9d. at 58. 11d. in the £. 15. The weight of a sheet of lead 35 × 10 feet at 6 lb. 11 oz, to the foot, and its cost at 3 d. per foot. 16. The weight of 3 miles of iron pipes 189 lbs. per foot run, and the cost at 8s. rod. per cwt. 17. 12 tons 13 cwt. 3 qrs. 4 lbs. copper at 958. 6d. per cwt.; 7 tons 17 cwt. 2 qrs. spelter at 378. 9d. per cwt. 18. What is the difference in weight, in tons, &c., between 100 fathoms of chain cable weighing 76 lbs. to the foot, and 300 fathoms wire rope at 18 lbs. What will be the difference of cost, the chain cable being 158. 6d. per cwt., and the wire rope £139. 6d. 19. Required the cost of breaking metal for 176 miles 3 furlongs 19 chains of road at 3}d. per yard run. 20. 21. 22. How much property tax must be paid on £7964 at 28. 94d. in the L. Find the value of 157 tons at 7 guineas per ton, at £2 per ton, and at £4 118. 8d. per ton, using one aliquot part only. In the same way calculate the cost of 158 tons at 8 guineas per ton, at £1 per ton, and at £4 78. 6d. per ton. Find (using decimals) the price of 10 lbs. 11 oz. 16 dwts. 16 grs. of gold at £3 178. 10jd· per oz. 23. Find (introducing a subsidiary aliquot part) the cost of making a road whose length is 3 miles 30 poles 5 yards at £72 178. 6d. 24 Find the cost of 3 qrs. 3 bushels 3 pecks at £6 168. 8d. per quarter (artifice). 25. Find (using only one aliquot part) the cost of 18 lbs. 7 oz. 4 dwt. at £5 58. 8d. per lb. 26. A contractor having commenced a railway finds that when he has yet 36 miles 3 fur. 22 yards to finish, his working expenses are increased by a strike of his men by £14 138. 4d. per mile, and by a rise in the price of materials, by £9 138. 4d. per mile. If he had reckoned on clearing 5000 by the contract, what will he be able to clear after these misfortunes? CHAPTER VII. RATIO, PROPORTION, AND RULE OF THREE. 179. We may compare one number or quantity with another, or ascertain the relation which one bears to the other in respect of magnitude, in two different ways; either by considering how much one is greater or less than the other; or by considering what multiple, part, or parts, one is of the other, that is, how many times or parts of a time, or both, one number is contained in the other. Thus, if we compare the number 12 with the number 3, we observe, adopting the first mode of comparison, that 12 is greater than 3 by the number 9; or, adopting the second mode of comparison, that 12 contains 3 four times, and is thus 12, or four times as great as 3. Again, if we compare the number 5 with the number 11, we observe, according to the first mode of comparison, that 5 is less than 11 by the number 6, and, according to the second, that as I is one-eleventh part of 11, so 5 is fivethirteenth parts of 11, or ths of 11. 180. The relation of one number to another in respect of magnitude is called Ratio; and when the relation is considered in the first of the above methods, that is, when it is estimated by the difference between the two numbers, it is called Arithmetical Ratio; but when it is considered according to the second method, that is, when it is estimated by considering what multiple, part, or parts, one number is of the other, or, which is seen from the above to be the same thing, by the fraction which the first number is of the second, it is called Geometrical Ratio*. Thus, for instance, the arithmetical ratio of the number 12 and 3 is 9, while their geometrical ratio is or 4. In like manner, the arithmetical ratio of 5 and 11 is 6, while their geometrical ratio is 12 181. It is more common, however, in comparing one number with another to estimate their relation to one another in respect of magnitude according to the second method, and to call that relation so estimated by the name of Ratio. According to this mode of treatment, which we shall adopt in what follows, "Ratio is the relation which one number has to another, the comparison being made by considering what multiple, part or parts, the first number is of the second, or how many times, or parts of a time, or both, the second is contained in the first." The term arithmetical ratio is now generally abandoned, so that when the word ratio is used, it signifies that which is divided by the quotient of one divided by the other. 182. The ratio of one number to another is usually expressed by placing two points, or a colon (:), between the numbers compared. Then the ratio of 8 to 4 is written 8: 4; the ratio of 12 to 3 is 12: 3. The two numbers are called the terms of the ratio, the former being called the antecedent, and the latter the consequent. But to determine the value of a ratio the antecedent is written as the numerator, and the consequent as the denominator, of a fraction; for this will determine how many times the first contains, or is contained in the second, since a fraction (Def. § 81, page 63), is a simple manner of expressing the division of the numerator by the denominator; and the magnitude of the fraction thus determines the value of the ratio. The ratio of 6 to 2 is expressed by the fraction or †, which denotes that 2 is one-third of 6, i.e., 3 times less than 6. By this method ratios can be treated arithmetically, their values determined, and so compared with one another. 183. A direct ratio is that which arises from dividing the antecedent by the consequent, as 62. An inverse, or reciprocal ratio, is the ratio of the reciprocals of two numbers. Thus, the direct ratio of 9 to 3 is 9: 3, or ; the reciprocal ratio is, or, that is, the consequent 3 is divided by the antecedent 9. An inverse, or reciprocal ratio, is expressed by inverting the fraction which expresses the direct ratio; or, when the notation is by points, by inverting the order of the terms. Thus, 8 is to 4 directly as to, or inversely, as 4 to 8. 184. If the terms of a Ratio be multiplied or divided by the same quantity, the magnitude of the ratio will not be altered. Let the ratio be 3: 8, then its magnitude is , which is equivalent to fe, or, or i, &c., that is, the ratio 3: 8 is equal to each of the ratios 6: 16, 9: 24, 12 : 32, &c., which arise from the equal multiplication of its terms; and, conversely, each of the latter ratios is reducible to the original one by the equal division of its terms. 185. If the antecedents of two or more ratios be multiplied together for a new antecedent, and their consequents be multiplied together for a new consequent, the resulting ratio is said to be compounded of the others, and it is called their Compound Ratio. Thus, if the ratios be 2: 3, 4: 7, and 8: 13, the ratio which arises from their composition will be 2 X 4 X 8 : 3 X 7 X 13, or 64: 2.73. 186. If the two numbers compared are equal, the ratio is a unit, or 1, and is called a ratio of equality. If the antecedent of a ratio is greater than the consequent, the ratio is greater than a unit, and is called a ratio of greater inequality. Thus, the ratio of 12 4 is 3; for the value of Y = 3. T |