Such questions are the same as those treated of in Chapter VII. The method there pursued of solving such questions is more intelligible to young students: for advanced students, however, the principles enunciated in this chapter are intrinsically valuable, and form an introduction to their more complete treatment in Algebra. In most treatises on arithmetic they are made the foundation of Rule of Three, both simple and compound. EXERCISE LXXVI. CHAPTER XII. PROPORTIONAL PARTS. 92. By the expression 2:34: 5, we denote the ratio of each one of the numbers 2, 3, 4, 5 to each of the others. By the expression 2 3 4 = 14: 21 : 28, we denote that 23 14:21, 2 : 4 = 14: 28, and 3: 4 = 21: 28. = It is evident that as in simple ratios, a successive ratio as 23:45 is not altered in value by multiplying or dividing each term of the ratio by the same quantity. We can thus simplify a successive ratio, as; for multiplying each term by 30, 2 20 24: 9. 93. It frequently becomes necessary to divide quantities into two or more parts, which shall have to one another the same ratio that certain given numbers have respectively. The quantity is then said to be divided proportionally, and the parts are called proportional parts. Thus if £108 is divided into parts, £24, £36, £48, it is divided into parts which have the same ratio as 2, 3, 4 respectively. To divide a given quantity (£144 for instance) into parts proportional to 3, 4, 5. Now 3+ 4+ 5 = 12, and 3 4 5 12' 12' 12' 3 are proportional to 3, 4, 5, and their sum is 1. Hence if we take ths, 4ths, ths, of the 12 12 12 given quantity, these parts will together be equal to the whole given quantity, and be proportional to 3, 4, 5. Thus of £144 = £36, of £144 = £48, of £144 = £60, and £36, £48, £60 are together equal to £144, and are proportional to 3, 4, 5, and are therefore the parts required. 94. If the terms of the given ratios are expressed as fractions, it is better to reduce them to integers, and then proceed as above. Example. Divide £1580 into parts proportional to 11⁄2, 1, 1}. Multiplying by 20; 13: 1: 1; 30: 25: 24, and the parts required are 3 of £1580 = £600, 25 of £1580 = £500, of £1580 = £480. = 95. Sometimes the question requires modification before the rule is applicable. Example 1. A merchant employs £700 in trade, and at the end of 3 years takes a partner who advances £1900. At the end of 4 years more they have gained £500; how ought it to be divided? Here the merchant employs £700 for 7 years, which is equivalent to employing £4900 for 1 year. The partner employs £1900 for 4 years, which is equivalent to employing £7600 for 1 year. Hence their profits must be divided in the ratio of £4900: £7600, that is of 49 76. Example 2. A person buys wheat at 39s. a quarter, and some of better quality at 6s. a bushel; in what proportion must he mix them, so as to gain 25 per cent. by selling the mixture at 57s. 6d. a quarter? The cost price of the mixture is of 57s. 6d., or 46s. a quarter. 100 The best wheat cost 48s., the inferior 39s. a quarter. They must therefore be mixed in the ratio 46 - 39: 48 — 46, or 7: 2, for 7 × 48 + 2 × 39 = 9 × 46. EXERCISE LXXVII. 96. In the English mint gold is used of which 22 parts are pure gold and 2 parts alloy; such gold is called standard gold, and is said to be 22 carats fine. (The carat was an old weight, the 24th part of a mark or half-pound, but not the same as the carat used for weighing diamonds; 22 carats were mixed with 2 of alloy.) From 40 lbs. troy of standard gold 1869 sovereigns are coined, and therefore a sovereign contains 123'2744 grains of this gold. Similarly standard silver contains 37 parts out of 40 pure metal and 3 parts alloy; a pound troy of this metal is coined into 66 shillings, each of which contains 87 grains. The mint or tariff price for standard gold is £3 178. 10d. per ounce. The price of silver varies, but is never so great as the value of the same weight of silver coinage. Hence silver is not allowed to be a legal tender for more than 40 shillings. 97. France, Belgium, Switzerland, and Italy, by the treaty of 1866, have the same system of coinage. Their standard gold is 9 parts pure out of 10, and from 1 kilogram 155 Napoleons or 20 franc pieces are coined, each of which therefore weighs 6:45161 grams. Their silver 5-franc pieces are 9 parts pure out of 10, and weigh 25 grams each. Their smaller silver coins are 835-1000ths-fine, and a franc contains 5 grams. These smaller coins are not a legal tender for more than 50 francs, except in payment to the government which has issued them. A charge of 6 francs a kilogram of gold is made by the French mint for defraying the expense of coinage, and of 2 francs a kilogram for silver. Gold bullion, or uncoined gold, is generally worth more in France than the mint price; it is then said to be at a premium, and this is estimated at the rate per mille. Thus if 1007 francs have to be given for the weight of standard gold in 1000 francs (3225805 grams) gold is at a premium of 7 per mille. EXERCISE LXXVIII. CHAPTER XIII. EXCHANGES. 98. Suppose the following question were asked 66 'If 3 lbs. of tea be worth 4 lbs. of coffee, and 6lbs. of coffee be worth 20 lbs. of sugar, how many lbs. of sugar are worth 9 lbs. of tea ?" Here 3 lbs. of tea are worth 4 lbs. of coffee. therefore I lb. of tea is Ι and I lb. of coffee is therefore I lb. of tea is and therefore 9 lbs of tea are وو lbs. of coffee. 9 × × 20 or 40 lbs. of sugar. Now arrange the terms of this question thus— required number of lbs. of sugar = 9 lbs. of tea. 3 lbs. of tea = 4 lbs of coffee. Comparing this statement with the preceding, we find that the required number of lbs. of sugar will be found by multiplying all the terms in the right-hand side together, and dividing by the product of the known terms on the left-hand side. This mode of solving the question is called the Chain Rule. 99. The principal use of this rule is in calculating foreign exchanges, that is, in finding how much foreign money can be obtained for a given sum of English money, and vice versâ. The value of a sum of English money in foreign money estimated from the quantity of gold and silver they respectively contain, is called the par of exchange. From several causes, the consideration of which belongs to Political Economy, the actual sum of foreign money which can be obtained for a given sum of English money varies from day to day. This is called the course of exchange. If for a given sum of English money a greater sum of any foreign money can be obtained than the par of exchange, the exchange is said to be in favour of England, and vice versa. Thus if the par of exchange between London and Paris is 25:22 francs for £1, then if the course of exchange is 25 30 francs for £1, the exchange is in favour of England; if the course is 25 15 francs for £1, the exchange is against England. 100. Transactions in exchange are conducted by means of Bills of Exchange. A bill on London entitles the holder to receive a certain sum of English money in London, at a certain date, from a certain person, named in the bill. Short exchange is when the bills are payable immediately. At London the exchange is on Paris, when a bill on Paris is bought in London. The short exchange on Paris will be lower than the usual one of 3 months, because £1 will purchase fewer francs to be paid immediately, than to be paid at the end of 3 months. So also in Paris the short exchange on London is greater than the long exchange, because it will require more francs to purchase £1 to be paid immediately than to be paid at the end of 3 months. The difference between the two exchanges will depend upon the prevailing rate of interest and the prospects of trade. IOI. From the variation in the course of exchange between different countries, it frequently happens, that a fixed sum of money of one country A will exchange for more money of another B, if it is first exchanged for money of a third country C and the proceeds are exchanged for money of B, than if the exchange were made directly between A and B. The rate of exchange calculated on this supposition is called the simple arbitrated rate: when two or more intermediate countries are used it is called a compound arbitrated rate. The following coins are used in the Examples to this chapter. In Hamburg 16 schillings = 1 mark, and 13 marks 8 schillings = £1. In Holland 20 stivers = 100 cents = 1 florin, and 11.95 florins = £1. An American dollar equals about 48. 2d. Example. I buy bills upon Hamburg at 13 marks 10 Schillings for £1, and sell them at Amsterdam at 40 marks for 35 florins, and buy bills on Genoa at 46 florins for 100 francs, which are discounted at I per cent. The proceeds are laid out in bills on Madrid at 5:25 francs for 1 dollar, and I sell these bills in London at 4s. 2d. the dollar. What is my profit per cent.? |