making it 960, and as 960 pence immediately suggests £4, which multiplied by 81 gives £34, it is only necessary to deduct the price of 4 at 81d., or 28. 10d., and the answer is reached. In the latter solution, advantage is taken of the nearness of the number of articles to a number of pence exactly equal to a whole number of pounds, which circumstance permits the use of a very easy method of calculation; and in a great many cases it will be possible, by a similar increase or abatement of the giren rates or quantities, to evade the difficulty of dealing with fractional values, such as parts of a £, a shilling, a cwt., &c. But, since an artifice of this kind cannot be profitably employed on all occasions, it is necessary to be possessed of some good general rule applicable to every variety of data. Perhaps the best process to follow, when no special device suggests itself, is to find the amount separately at a single unit of each denomination of money in the given rate, to multiply by the number of such units, and to add the results together. It is essential, however, to the successful practice of this, or indeed, any other system of compound mental arithmetic, that the computer should have his memory stored with extensive tables of equivalents in the various denominations of money, weight, measure, &c. He should, for instance, need little more time or thought to arrive at the perception that 1000 pence - £4 3s. 4d., or that 180 lbs. =1 cwt. 2 qrs. 12 lbs., than that 9 times 7 = 63. Example (3.) What is the drawback on 128 barrels, 2 firkins of beer, at 78. 9d. per barrel ? In adding together two numbers, either simple or compound, it is sometimes very convenient first to increase one of the numbers and to diminish the other by the same amount. Example (4.) The sum of 175 and 156 is more easily found in the mind by changing the numbers to 180 and 151, previous to making the addition. To add £2 8s. 9d, and £1 ls. 10d., raise the first to £2 10s., and take 1s. 3d. from the second ; then, £2 108. + £1 Os. 7d. £3 10s. 7d. Subtraction is often facilitated by increasing both the numbers or antities equally, a process which evidently does not affect their difference. Examplo (5.) 69 24 70 25 £7 13s, 2d. 45 £4 58. 2d. There are special methods, also, of abridging the work of direct multiplication, such as, doubling a factor which ends with 5 and halving the other, or if that should be an odd number, halving the product; increasing one of the factors to the nearest number ending with 0, then multiplying and deducting the product of the augment by the unaltered factor, &c. Thus, = 2205 2 Example (6.) 58 x 35 29 x 70 2030 49 x 90 49 x 45 96 x 27 = 100 x 27 — 4 x 27 2592 In division, if the divisor be any number ending with 5, it will be of advantage to double the dividend and divisor, halving the remainder, if any. 63 Example (7.) 276 • 35 552 ; 70 = 7 with 2 or 31 over. To square a number, that is, to multiply a number by itself, mentally, reduce one factor to the nearest number ending with a cipher, and increase the other equally, multiply, and add the square of the number subtracted. 2 Example (9.) To find mentally the number of farthings corresponding to any number of shillings and pence. If the pence be exactly half the number of shillings, then 100 times the pence will give tho number of farthings in the entire value, thus, 158. 7 d. reduced to farthings. Here, as 71 is the half of 15, the number of farthings is 72 hundreds or 750. If the pence exceed or fall short of half the number of shillings, calculate what the value would be if the pence were half that number, and for tho farthings in the difference, subtract or add as the caso may require, e. 9. 16s. 33d :-at 16s. 8d. the farthings would be 800, from which deduct 17 for the farthings in the difference between the given and the assumed sums, and the result is 783. * The principle of this method is readily explained. If the number of pence be half the number of shillings, and the shillings and pence be reduced to total penco, it is evident that the latter will always be 12 times the number of shillings; and this multiplied by 4, to convert it into farthings, will be tim the number of sbil ogs, or 100 times the number of pence. An account of all the possible methods of shortening and simplifying ordinary arithmetical operations so as to adapt them the better for being carried on partly or wholly by a mental process, would occupy much more space than can be properly or usefully devoted to the purpose in the present work. It may be observed, that every computer who is bent on saving himself needless labour and attaining to an intelligent quickness in the use of figures, will do well not to trust altogether to a set of ready-made rules, but to exercise his own ingenuity in reasoning out and discovering personally those expedients which may enable him to arrive with the least trouble at the results most frequently required in his particular business. He will thus acquire a degree of mechanical expertness such as the teaching of books, however aptly framed or carefully followed, must fail to bestow, and will also advance to an intimate knowledge of principles, which far exceeds in importance any amount of mere technical efficiency. A few additional examples of short methods of calculation adapted for mental exercise, will be found in the ensuing articles on Interest, Per-centage, &c. 7. INTEREST. Interest is the general name given to any premium or consideration charged for the use of a sum of money, whether this be advanced as a loan or allowed to remain unpaid as a debt. The rate of interest is always expressed at so much per cent. per annum according to agreement between the parties—that is so much for the use of £100 for one year, and so proportionally for greater or less sums and longer or shorter periods. It is customary to call the sum lent or kept back, the principal, and the principal together with the interest, the amount. Money is said to be lent at simple interest, when the interest is paid as soon as it becomes due, at the end of each year, quarter, or other stipulated term ; or when, if such payment be deferred, interest is not demanded upon interest. When a charge of this kind is made, the interest designated compound interest. It is unnecessary in the present work to treat of the subject of compound interest, as a knowledge of it is of no professional utility to officers. Every question of simple interest may be solved by an application of the Rule of Three, as all the data concerned in it are evidently proportional to one another. Thus, if the principal, rate, or period be doubled, the amount of interest will be doubled. If any of these be made three times as great as before, the result will also be trebled, and so on. The interest on £100 for six months is half the interest at the same rate on $100 for twelve months. £500 invested for one year produces as much interest as £100 invested at an equal rate for five years, or £50 for ten years. It must be obvious, therefore, that it is immaterial in what order the steps of the operation are executed, so long as the principle is rightly apprehended. When the interest for one year only is required at any given rate, a single statement in the Rule of Three is sufficient. Example (1.) What is the interest on £764 18s. 5d. for one year at 4 per cent. per annum ?* * The words per annum are usually omitted in statements of the rate of interest, as it is always understood that the computation is to be made by the year. Principal. : Principal. Given rate. Int. required, $764 18 5 £4 : . 4 20 12 11,24 4 96 $ 8. : :: £3 : 6 Interest for any period greater or less than a year is found, by first computing what the amount would be for one year, and then multiplying this result by the given time. Example (2.) Required the interest on 673 68. 6d. for six years at 3} (=£3 108.) per cent. d. £? 36 12 9 £2 118. 3.d. for one year. £15 7 9 Answer. 4 2,04 Interest for fractions of a year expressed in months or twelfths may be readily found by the use of aliquot parts, thus, Example (3.) Suppose in the last example, the given time were 6 years and 10 months, what would be the interest ? £ 8. d. 6 15 7 9 6 months = } year 1 5 74 4 months year 0 17 1 10 £17 10 5 Answer. When it is required to compute interest for a number of days which cannot be resolved into aliquot parts of a year, the only available process, if extreme exactness be sought, is to multiply the amount for one year by the given number of days, and then to divide by 365. This is equivalent to taking so many 365ths of the annual amount. Example (4.) Find the interest on £115 178. 8d. for 76 days (common year) at 34 per cent. ? As £100 is always the divisor or first term in proportions of this kind, it is needless to go through the form of stating the questions by the Rule of Three. £ d. 3 15 3.89 31 11 347 13 0 41 86.79 28 19 5 7 100)3,76 12 6 289 19 11:53 20 3 15 3.89 15,32 365)286 4 7.64 12 20 )5724( 15 12 75 Answer. 158. 8d. 302 (for •64d. add id.) Several artifices may be resorted to for abridging the labour of the calculation of interest on days. Accountants who have frequent occasion to arrive at such results usually avail themselves of interest tables, many of which are published. When a table is not at hand, and an error of a farthing or two in the final amount is immaterial, one of the best methods of shortening the work is as follows First compute the interest for a year. Then suppose the year to consist of 360 instead of 365 days ; take aliquot parts of 360 corresponding to the given number of days, and from the result subtract its 72nd part, which is easily found by dividing in succession by 8 and by 9. The correct process would be to deduct its 73rd part, but for practical ends it will be sufficient to operate as above directed, and to add a penny for every $20, or a farthing for every £5 in the amount thus obtained. Example (5). (See the last Example.) d. 3 15 3.89 Interest for a year, days. 2 6.13 10-04 12 = 76 15 10.82 8) 9) 1 11.85 2.65 15 10.82 15 8.17 |