insured. And in this way is the insurance to cover loss and premium always to be found; namely, as £100 diminished by the rate £100 :: value of the property: the sum to be insured. If there be any other per-centage, as for commission, policy, &c., it must be deducted from £100 in the same way. Exercises. 1. What is the commission on £3698 12s., at 3 per cent.? 2. If a person sell out £600 stock, when the price of it is 838 per cent., how much will he receive after paying per cent. on the stock sold for brokerage? 3. What amount must be insured to cover £1880, together with the premium of £5 5s. per cent., 5s. per cent. for the policy, and per cent. for commission ? 4. What amount must be insured on £1938 12s. 6d. at 52 per cent., so as to provide also for the premium? 5. What is the commission on £876 5s. 10d. at 3 per cent.? 6. What is the brokerage on £372 7s. 4d. at 4s. 6d. per cent. ? 7. How much must be given for £912 14s. stock, at 1273 per cent.? 8. Required the brokerage on the purchase of £11675 178. stock, at per cent. 9. What will it cost to purchase £7391 14s. 9d. stock, the price being 864, and the brokerage per cent.? 10. Find the expense of insuring a cargo worth £850, at £2 12s. 6d. per cent., policy duty 5s. per cent., and commission per cent. (111.) COMPOUND INTEREST. ALL the preceding calculations respecting interest proceed on the supposition that the interest is actually paid when due. If, however, the interest be withheld for any time, then this interest so withheld becomes a new principal, and itself produces interest. The whole interest thus accumulated in any time is called compound interest; while that which arises solely from the original principal, and which has been the subject of the preceding articles, is, for distinction, called simple interest. An example will be sufficient to show you how compound interest may be calculated; but as the work, though made up of very easy and obvious steps, becomes £. 8. d. 20)550 0 ი 0 int. 1st yr. 20)577 10 0 amount. 20)606 7 6 amount. 30 64 20)636 13 10 c. int. 3rd yr. 8 amount. tedious and lengthy when those steps are numerous, tables have been contrived to save the labour. To these I must refer you for expeditious calculation; but the example worked in the margin, namely, to find the compound interest of £550, at 5 per cent., when the simple interest, which should be paid yearly, is withheld or forborne 4 years, will fully put you in possession of the mode of proceeding without tables. The simple interest for the first year, computed in the usual way, instead of being paid, is added to the principal; the amount is the principal for the second year, the interest of which is the compound interest of the original principal for the second year, and so on till the expiration of the 4th year, when the amount becomes £668 10s. 62d., which, diminished by the original principal, leaves £118 10s. 63d. for the whole compound interest: the same sum that we should get by adding the interest for the several years together. The simple interest of the proposed sum would be 4 times £27 10s., or £110; so that the difference is £8 10s. 62d. 668 10 62 amount. 118 10 6 compd. int. (112.) PROPORTIONAL PARTS. MANY useful and interesting questions depend for their solution upon the division of quantities into parts, having specified ratios to one another. I shall here give you a short article on the mode of effecting this division. To divide a Quantity into Parts, such that any one Part shall be to another, as one given Number to another. RULE. As the sum of the given numbers, is to any one of them, so is the whole quantity to be divided, to the part corresponding to that number. For example: suppose it were required to divide £80 into three parts, that should bear to one another the same relations as the numbers 2, 3, and 5. The parts would be found as follows. 10 2 :: £80: £16; 10: 3 :: £80: 24; 10 5: £80 £40. The required parts are therefore £16, £24, and £40; which together make up the £80, and which obviously bear the proposed relation to one another; namely, £16: £24 :: 2 :3; £16 £40 :: 2 : 5; £24: £40 35. It is scarcely necessary to tell you, that when all the parts but one are thus found by proportion, that one may be got by subtracting the sum of all the others from the whole. The principle of the rule can scarcely require any explanation to a person familiar with proportion: the sum of the parts of a quantity, and the sum of the parts of a number, are given; the subdivisions of the number are also known; and, since the whole of anything is to a part, as the whole of another thing to a like part, the sufficiency of the rule is obvious. Exercises. 1. Three traders, A, B, and C, contribute the following sums to the business: A, £500; B, £650; and C, £700: the year's profits are £555. Required each person's share of them. 2. Gunpowder is composed of nitre, charcoal, and sulphur, thus: nitre, 76 parts; charcoal, 14; sulphur, 10: how much of each is used in 1 cwt. of gunpowder? 3. How much pure gold, and how much alloy, are contained in a guinea? (See p. 39.) 4. Standard silver contains 37 parts of pure silver, and 3 of copper: how much of each ingredient is there in £1 7s. 6d. ; 1lb. troy being coined into 66 shillings? 5. 100 lbs. of pure water contain 88.9 parts of the gas called oxygen, and the remaining 11.1 parts of the gas called hydrogen: what weight of each is there in a cubic foot, or 1000 oz. of water? 6. A bankrupt owes A £120; B, £80; and C, £75: he possesses £165, which he is anxious to divide equitably: how much should each creditor receive? 7. Pewter is composed of 112 parts of tin, 15 of lead, and 6 of brass how much of each enters into the composition of 1 ton of pewter? 8. A person bequeathed in his will, £140 to A; 100 guineas to B; 80 guineas to C; £70 to D; and £60 to E: but, at his death, the whole amount of his property was but £311 158.: how much of this should A, B, C, D, and E. receive? (113.) THE CHAIN RULE. THE Chain-Rule is a compendious rule for working examples which involve several Rule-of-Three statings; and by which, questions in Compound Proportion may be otherwise briefly solved. The following is an example of it. If 3 lbs. of tea be worth 8 lbs. of coffee, and 5 lbs. of coffee worth 18 lbs. of sugar, how many lbs. of sugar should be exchanged for 20 lbs. of tea? Here, by two simple statings, we have 8 lbs. coffee: 5 lbs. coffee :: 3 lbs. tea : and lbs. tea: 20 lbs. tea :: 18 lbs. sugar: 3 x 5. sugar; but, by the chain-rule, the particulars ranged thus: 3 lbs. tea = 8 lbs. coffee, 5 lbs. coffee = 18 lbs. sugar, how many lbs. sugar 20 lbs. tea? 3 x 5 would be ar where no two commodities of the same kind occur in the same column; and then, by dividing the product of the numbers in the complete column by the product of those in the column which the answer would make complete, the number 8 x 18 x 20 which expresses the answer is obtained; namely, = 192, .. 192 lbs. sugar is the answer. Exercises. 3 x 5 1. If 3 lbs. of pepper be worth 4 lbs. of mustard, and 5 lbs. of mustard be worth 12 lbs. of candles, how many pounds of candles should be exchanged for 20 lbs. of pepper? 2. If 5 lbs. of tea be worth 12 lbs. of coffee, and 9 lbs. of coffee worth 28 lbs. of sugar, and 13 lbs. of sugar worth 18 lbs. of soap, how many lbs. of soap may be had for 7 lbs. of tea? 3. £1 sterling = 420d. Flemish, 58d. Flemish = 1 crown of Venice, 10 crowns Venice 6 Venetian ducats, 1 ducat = = 360 mervadies Spanish, and 272 mervadies = 1 Spanish piastre: how many piastres = £1000 sterling? The above rule is chiefly used in questions like this last, relating to exchanges with foreign countries; numerous applications of it, to matters of this nature, are given in Kelly's Universal Cambist, vol. ii. (114.) DUODECIMALS. You already know that the common notation of Arithmetic is called the decimal notation, because the local value of every figure of a number expressed in that notation diminishes at a ten-fold rate, as it is removed from place to place towards the right. If the diminution were at a twelvefold rate, the notation would be the duodecimal notation. In the measurement of lengths, the denominations feet and inches do actually descend in value, in this way :-an inch being the twelfth part of a foot; so that, for the purposes of Mensuration, it is convenient to have a duodecimal arithmetic, at least for the operation of multiplication. 12 113 16 3 7 9 29 125 11 In the decimal notation, 16.3, means 16 and 3 tenths: if these were twelfths instead of tenths, we might write the number thus: 16 3, leaving a gap between the two denominations; and, in multiplying this by any number in the same notation, it is plain, that we may proceed just as with decimals, provided we take care to carry twelves instead of tens: thus, to multiply 16 3' by 7 9, we should say, 9 times 3 are 27; 3, and carry 2 twelves: 9 times 16 are 144, and 2 are 146; which is 12 twelves, and 2: 7 times 3 are 21; 9, and carry 1: 7 times 16 are 112, and 1 are 113: so that the product is 125, 11 twelfths of one of the units in the 125, and 3 twelfths of one of the units in the 11. If we were to 12)18135 work by common arithmetic, we should, of course, get the same thing: thus, 16 × 71% 195x9; and this operation performed, as in the margin, gives the above result, namely, 12511+ of 1. = 195 93 585 1755 12) 1511 12511 |