£. S. d. 20)550 0 0 20)577 10 O amount. c. int. 2nd yr. 20)606 7 6 amount. c. int. 3rd yr. 20)636 13 10 amount. 668 10 6 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. 63d., which, diminished by the original principal, leaves £118 10s. 6d. 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. 63d. 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 3 5. 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 sesses £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: 3 x 5. and lbs. tea: 20 lbs. tea :: 18 lbs. sugar: 8 3x5 sugar; but, by the chain-rule, the particulars would be arranged thus: 8 lbs. coffee, 3 lbs. tea = how many 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 which expresses the answer is obtained; namely, 192, 192 lbs. sugar is the answer. Exercises. 8 × 18 × 20 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? 1 3. £1 sterling 420d. Flemish, 58d. Flemish = 1 crown of Venice, 1 crown Venice 60 Venetian ducats, 1 ducat = 360 mervadies Spanish, and 272 mervadies 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. 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 work by common arithmetic, we should, of course, get the same thing: thus, 1632 × 71% = 195 × 93; and this operation performed, as in the margin, gives the above result, namely, 12511+ of 12. 12 12 113 16 7 29 ප ය 3 9 3 125 11 3 195 93 585 1755 12)18135 12) 1511 3 12511 ft. in. 22 5 In our multiplicand, 16 3, above, the 16 might have been the number of feet in the length of a floor or wall, and the 3, the number of inches besides; and the multiplier, 7 9, might have been, in like manner, the number of feet and inches in the width or height: the first principles of Mensuration show us, that the product, namely, 125 11 3, would be the number of square feet, twelfths of a square foot, and twelfths of a twelfth, that is, ths, or square inches, in the surface of that floor or wall. The twelfths of a square foot are called parts; so that the measure of the surface in question would be 125 sq. ft., 11 pts., 3 sq. in. Suppose, for instance, it were required to find the measure of a ceiling, which is 22 ft. 5 in. long, and 16 ft. 11 in. wide: a workman would compute the surface, as in the margin, using the number in the multiplier, connected with the higher denomination, first, instead of second, as above; and he would thus find the surface to contain 379 sq. ft., 2 pts., 7 sq. in. (See the remarks at p. 62.) Exercises. 16 11 138 8 22 20 6 7 379 2 7 Find the surface-measures from the following linear measures. 1. Length, 32 ft. 9 in.; 3. Length, 65 ft. 10 in.; breadth, 8 ft. 3 in. 8 in.; breadth, 3 ft. 11 in. 4. Length, 97 ft. 9 in.; 5. Length, 75 ft. 9 in.; 6. Length, 97 ft. 7. Length, 59 ft. 6 in.; 8. Length, 87 ft. 5 in.; Examples such as these belong more properly to Mensuration. I give them here, as a conclusion to this Rudimentary Treatise on Arithmetic, in order that you may have a little experience, before commencing the subject just named, in what is commonly, but improperly, called the multiplication of length by length, or by width; and, at the same time, be furnished with a practical illustration of the remarks at p. 62. |