(4.) 31 cub. met. 725 cub. decim. contained in 45684 cub. met. ? (5.) 345 millig. contained in 165 kilog. 6 hectog.? (8.) 4 kilog. 5 grams, contained in 38 myriag. 4 kilog. 480 grams? (9.) 8 centimes contained in 10 francs ? 9. A merchant bought 95 litres of wine for 118 fr. 75 c., and sold it at a loss of 10 c. per litre. What was the selling price per kilolitre ? 10. To make 12 suits of clothes, it required 40 metres of stuff 90 centim. wide. How much stuff will it take if the width is 80 centim.? 11. How many cubic decimetres of iron are there in a bar weighing 280 kilog. 368 grams, when one cubic centim. weighs 7 grams 788 millig. ? 12. An iron wire, 126 metres long, is cut into pieces 3 centim. 2.5 millim. long. How many pieces are there? a Relation between the Metric Units and the English System of Weights and Measures. 34. We shall work a few examples to show how quantities expressed in the metric system may be expressed in the English system, and vice versa. Ex. 1.-Reduce 10 kilom. 321 metres to English measure. 10 kilom. 321 metres 10321 metres. = (10321 x 1.094) yards. x 1.094 miles. = 6 miles 731•174 yards. Ex. 2.-Express 2 miles, 309 yards in the metric system. 2 miles, 309 yards = (2 x 1760 + 309) yards. 3829 yards 3 8 2 9 metres. 1032 1 1760 1.094 7000 20 X 12 Х 15.4 3 2 3 Ex. 3.- Reduce 1 ton to kilograms, having given 1 gram 15.4323 grains. 1 ton = 20 x 112 x 7000 grains. grams. 1016050•7507 grams nearly. = 1016 kilog. 50 grams, 750-7 millig. nearly. Ex. 4.-Express £13. 178. 41d. in the pound and mil system. (£1 = = 10 florins, 1 florin = 10 cents, 1 cent = 10 mils.) Reducing the given sum to the decimal of a pound, we have£13. 178. 4!d. £13.86875. £13. 8 fl. 6 cent. 8 mil. Ex. XIV. 1. Express a mile in the metric system, having given that a metre = 39.3708 inches. 2. An are contains 1076.43 square feet. Reduce 53 ares 25 centiares to English measure. 3. The area of a room is 22 sq. met. 26 square decim. Express this in English measure (1 metre 39.3708 inches). 4. A block of marble measures 3 feet, 3 inches in length, 2 feet, 6 inches in depth, and 3 feet, 9 inches in width. What is the solid content expressed in cub. centim. ? 5. In 1235 litres how many gallons, when 50 litres 11 gallons nearly? 6. Supposing a franc to be equivalent to 93d., reduce £44. 13s. to francs. 7. Taking £1 sterling as equal to 25•22 francs, reduce £2. 13s. 7 d. to francs. 8. In 1852 France reaped about 47850000 hectol. of wheat. Express this in gallons, assuming 1 gallon = 4:543 litres. 9. The ceiling of a room contains 83 sq. met. 53.96 sq. decim. What will be the expense of painting it at 10d. a square yard (1 metre 1.094 yard)? 10. Find the cost of 2000 kilog. of sugar at 0 fr. 50 c. per lb. 11. In England the unit of work is the foot-pound, and in the metric system it is the kilogram-metre. Reduce 62 metric units of work to English units, taking 1 gram 15.4323 grains, and 1 metre 39.3708 inches. 12. The pressure of the atmosphere is 143 lbs. upon the square inch. Find the pressure in kilograms upon the square centimetre. CHAPTER V. 15 PROPORTION. 38. Proportion is the equality of ratios. Thus, since the ratio 6 : 8 we have ratio 6 : 8 ratio 15 : 20; and we say that the numbers 6, 8, 15, 20 form a proportion, We generally express the fact thus 6 8 : : 15 : 20. It is easy to find by trial that the product of the extreme terms is equal to the product of the means. Thus, we have 6 x 20 8 x 15. We may prove this property of the terms of a proportion to hold generally as follows : Suppose we have given the proportion 12 : 21 :: 20 : 35. It follows, from our definition above, that ļi 20, and multiplying each of these fractions by the product of their denominators, viz., by 21 x 35, we have Hi x 21 x 35 33 x 35 x 21. Now (Art. 8), it x 21 12, and 38 x 35 ° =: 20, and we hence have 12 x 35 20 x 21. Now, we have not in our reasoning taken into account the actual value of the terms of the given proportion; and it is therefore evident that a similar result will follow from every proportion, and we may hence conclude generally : In every proportion the product of the extremes is equal ta the product of the means, = а 28 X 30 24 : 1 = 39. Having given any three terms of a proportion, to find the remaining one. Since the product of the extremes is equal to the product of the means, the following rule is evident : RULE.—If the required term be a mean, divide the product of the extremes by the other mean; but if the required term be an extreme, divide the product of the means by the other extreme. Ex. 1.—28, 24, 30 are respectively the 1st, 3rd, and 4th terms of a proportion, required the 2nd term. We have— 28 : required term :: 24 : 30 .. required term 35. Ex. 2.—10, 45, 16 are respectively the 1st, 2nd, and 3rd terms, required the 4th term. We have 10 : 45 : : 16 : required term ... required term = Ex. 3.—2 hours, 45 minutes, 8 men are respectively the 1st, 2nd, and 3rd terms, required the 4th term. We must express (Art. 6) the 1st and 2nd terms in the same denomination, and the proportion will stand thus Min. Min. Men. 120 : 45 ::8 : required term. Now, the ratio of the first two terms is the same as the ratio of the abstract numbers 120 and 45; and the 4th term must be of the same denomination as the 3rd term, otherwise the 3rd and 4th terms could not form a ratio. We have therefore Required term = 9 X 8 45 X 16 10 72. 1 a 45 X 8 3X1 men = men 1 3 men, 120 Simple Proportion. 40. In Arithmetic we divide Proportion into Simple and Compound. Simple Proportion is the equality of two simple ratios, and therefore contains four simple terms; and the usual problem is to find the fourth term, having given the first three terms. When we know the exact order of the given terms, the fourth term is, of course (Art. 38), found thus RULE.—Multiply the 2nd and 3rd terms together, and divide by the 1st. The formal arrangement of the three given terms in their proper order is called the statement; and the only difficulty, therefore, in working a sum in Simple Proportion, or Single Rule of Three, as it is called, consists in stating it. We shall work a few examples to illustrate the mode of doing this. Ex. 1.-If 12 men earn £18, what will 15 men earn under the same circumstances ? We have here two kinds of terms, men and earnings, and whatever ratio any given number of men bears to any second given number of men, it is evident that it must be equal to the ratio of the earnings of the first lot of men to the earnings of the second lot, and we may therefore write Men. Men. 12 : 15 = £18 : 2nd earnings. Men. Men. 12 : 15 :: £18 : 2nd earnings. As the first two terms are of the same denomination, their ratio is not altered by treating them as abstract quantities, and the denomination of the 4th term must be the same as that of the 3rd. Hence we have Ans. : £ = £22. 10s. Ex. 2.—If 18 men do a piece of work in 25 days, in what time will 20 men do it? The two kinds of terms we have here to consider are men and time. In doing work we know that the time will diminish exactly as the number of men increases, and hence the ratio of the second lot of men to the first lot will be equal to the ratio of the given time to the time required. We therefore have Men. Men. Days. 20 : 18 :: 25 : required time. .. Ans. : 18 * 25. days * days = 22.5 days. or 18 X 13 12 £3* 15 х 9 X 5 2 20 |