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the process, and thus dispenses with the trouble, otherwise necessary, of reducing to the lowest denomination, and finally bringing back to the highest denomination concerned. Thus, if it were required to find the cost of 11 cwt. 3 qrs. 23 lbs. at £6 158. 91d. per cwt., those who are unacquainted with Practice could arrive at the reeult only by reducing to total lbs. and to halfpence, respectively, multiplying these quantities together, dividing by 112, and reducing the quotient to £ s. d.; whereas by Practice the answer would be obtained in a few figures without changing either of the given denominations, as follows :

£ 8. d. 6 15 91

11

74 13 81 2 qrs. = 1 cwt..... 3 7 103 1 qr. = 1 2 qrs.... 1 13 111 16 lbs. = I cwt..... 0 19 43 7 lbs. = 1 1 qr. ... 0 8 55

£81 3 5* No formal directions can be laid down for the application of the method of Practice, as the discovery of the shortest process in each instance must depend on the judgment and memory of the operator; and almost every question admits of being treated in numerous distinct ways, some one of which is preferable in point of simplicity and despatch to the others. It is necessary to get off by rote short tables of aliquot parts of the various denominations of money, merchandise, &c. Such tables will be found in all ordinary manuals of arithmetic, or may easily be formed by each person for his own purposes.

In excise calculations the rule of Practice is chiefly useful as a means of finding the amount of duty, drawback, or allowance payable in respect of quantities of goods or numbers of licences. A few examples will sufficiently indicate to the young officer the course which should be followed on these occasions.

Example (1.) What is the duty on 375 bushels of malt at 25. 7d. per bushel, and 5 per cent. additional ?

8 do
375 bushels at £1 per bushel -
28. Od. is £77; 375 at 28. each = 6970 = 37 10 0
6d. is 4th of 2s.; amount at 6d. is

4th of the amount at 28. ...... = 9 7 6
id. is * of 6d.; amount at id. is a

of the amount of 6d. i = 1113

28. 7d.

Amount at 28. 78. = 48 8 9
5 per cent. or at of this sum = 2 8 51

Total duty £50 17 23 The principle of the foregoing process needs no explanation beyond that which accompanies each of the steps. A little practice will enable any person to compute amounts at 5 per cent.--that is, to divide a sum of money by 20—with great

* The reason for the successive steps of this process will appear from the explanations appended to example (1) below,

ease. It is obvious, also, that at £5 for every £100, or £1 for every £20, there is 18. for each whole pound, 6d. for 108., 1d, for 1s. 8d., 4d. for 5d., &c.*

Other aliquot parts may be employed in calculating amounts of malt duty, but none perhaps so advantageously as those above chosen. Thus, instead of finding

Loth of the amount at £1, and then 4th of this quotient, we might by a single division by 8 arrive at the same result, since 28. 6d, is 4th of a pound. In this case, however, it would be necessary to work for the additional penny, either by taking

both of the amount at 28. 6d., or by reducing separately as many pence as there are bushels to pounds, shillings, and pence-operations which to most computers would present less facility than the plan of division here adopted, although it is very desirable to test the correctness of the work at various points whenever a ready method of verification suggests itself.

In order to judge properly how far a saving of time or labour is effected in any instance by having recourse to the system of aliquot parts, it is requisite to compute the same amount by different processes, and to compare the number of figures in each, as well as the character and number of the several steps taken to arrive at the result. It will thus be seen also whether it is advisable to work by practice alone, or to combine that rule with other modes of computing. In commercial arithmetic, the superior brevity of Practice is exhibited on almost every occasion of calculation, but in excise accounts, especially those which contain decimal fractions, it is not always clear that despatch is promoted by keeping the given rates or quantities in their highest denomination and operating by means of aliquot parts. To illustrate this, the last example may be wrought as follows, and the two processes contrasted in point of facility.

375 x 31 (2s, 7d. = 31d.)
1125
12)11625
20)968–9

£48 8 9 Answer. It would thus appear that in the present instance, at least, there is little if any gain in solving the question by Practice.

If it were required to calculate the duty on bushels and fractions of a bushel, or on fractions of a bushel only, the advantage to be derived from Practice is still more doubtful. Example (2.) What is the duty on 4-37 bushels of malt at 2s. 7d. per bushel P By Practice.

Otherwise.

4:37 (x 31)

131 1
10 4

12) 135.47
For ease of work-
62 ing the decimals of

11,3.47 (47d, is evidently 3.1

& penny are taken
instead of dividing

greater than a 1.55 out to farthings and

farthing or -25d.,

and less than a -02 = 1 of 10 0:62

thing.
Fractions of a far.

halfpenny or

500.) 11 3.47) Answer. 11s. 31d.

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* Another short method of finding 5 per cent. on any number of shillings and penco, will be given under the head of INTEREST.

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Example (3.) A useful mode of checking the separate calculation of the duty and additional 5 per cent., is to multiply the number of bushels by 32.55 (= 31d. + .05 x 31d.) Thus, as in Example (1.)

32 55 x 375 = 12206-25d. = £50 178. 21d. By Practice the operation is best conducted as under,

375 28. = 1£

37 10 0 6d. = 1 of 28. 9 7 6

= of 6d. 3 2 6 •55 = .50 + .05

0 15 = it to x l •05d. = „ of ld. 1 0 1 68

£50 17 21* Example (4.) What is the amount of drawback on 1199.8 proof gallons of spirits at 2d. and 3d. a gallon respectively ? 1199s.

11998. 2d. = 43. = 1993. 10 (65. = 10d.) 3d. = 18. = 2998. 9 (s. = 9d.) .8 gall. at 2d. = 1.6

-8 gall. at 3d. = 2.4 20)1998. 11.6

20)299.11.4 Answer. £9 19 114 at 2d.

Answer. £14 19 114 at 3d. Example (5.) Required the amount of duty on 79 licences at £3 6s. 13d. each ?

79 at £1 = £79

237
58. = 1€ 19 158.

= 1 of 58. 8 19
ld.= is. „ 6 7d.
1d.= ld. 3 31

id.= 1 of 4d. , 177

Answer. £2615 61 As a general rule it will be found preferable in questions like this, to adopt the method of compound multiplication throughout. The number of licences, in the present instance, may be resolved at a glance into 11 x 7 + 2. It is only necessary, therefore, to multiply the given rate by 11, the product by 7, and to add to the last result the given rate multiplied by 2.

£ 8. d.

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• Tables showing the separate and total amounts of malt duty, at every five bushols, and also the net couch or cistern bushels at every tenth of a bushel, are published by W. R. Loftus, price 6d.

Surveying officers of all grades have occasion to make themselves particularly expert at calculations of this kind.*

The amount may sometimes be very conveniently found, by computing what it would be at the nearest integral rate higher than the given one, and deducting the amount at the difference between the actual and the assumed rates.

Example (6.) What is the duty on 43 maltsters' licences at 78. 101d. ?

43 at 88. = 344.
11d. = 18. 545

20)338 71
Answer. £16 18 7

Example (7.) Reduce to proof, 136 gallons of spirits at 24.4 and 25-6 per cent. O.P., respectivelyAt 244 O.P.

At 25.6 0.P.
8 ) 136

.8) 136
Deduct .006
170

Add .006

170 of 136

.8

of 136 Answer. 169·1 galls. at proof. Answer. 170.8 galls. at proof.

We, first, suppose the given strength to be 25 O.P., and reduce from that strength to an equivalent at proof by using the divisor •8 instead of the multiplier 1.25 (See Reduction of Spirits, page 98,) since 1= .8. The reason of the rest of the process is obvious.

The example given under the head of Proportion, page 112, Example (2.) may be solved by Practice as follows

Example (8.) What should be the net weight of a consignment of molasses weighing, gross, 55 cwt. 3 qrs. 17 lbs., if a deduction of 11 lbs. per cwt. be made for the tare of the casks ?

cwt. qrs. lbs.

65 3 17
28 lbs. = 1 cwt. .. 13 3 25.25
7 lbs. = iqr. .. 3 1 27.31
4 lbs. = gr. .. 1 3 27.60

Tare .. 5 i 26.91
Gross .. 56 3 17
Net ., 60 1 18.09

* Tables showing the amounts of Licence Duty at the various rates, whole and fractional, and sufficiently extensive for all purposes of business, may be had of the publisher of this work. Price 1s. Sd.

In business the fractions of a pound would, of course, be neglected, and the net weight shown, 50 cwt. 1 qr. 19 lbs. It will probably be found more expeditious when the rate of tare cannot be resolved into convenient aliquot parts of a cwt., as in the present instance, to make the calculation thus,

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Other examples of short modes of calculation applied to the various branches of an officer's duty, and referable to the general head of Practice, will be found in the succeeding parts of this work.

Mental Calculations. Most of the simple arithmetical questions which it is possible for persons of ordinary training and quickness to solve without the aid of paper, are best wrought by the method of Practice, as being that which admits of the greatest ease of operation and imposes the least burden on the memory. Officers will find it extremely serviceable to be prompt and exact at « head-reckonings” of every description connected with their business, especially at the estimation of amounts of gauges, duties, allowances, averages, per-centages, &c. It is to be borne in mind, however, that constant exercise, and the adoption of a good method are indispensable to the success of efforts in this direction where natural aptitude is wanting. No one who does not possess that aptitude to some extent, and who makes merely a few occasional, immethodical attempts at working a sum in his mind, can hope to bring out his results with any degree of certainty or expedition.

One of the most common and useful expedients resorted to in the mental reckoning of prices and other values, is to calculate the amount by the nearest round number or integral rate, and then to apply a correction for the difference, as in the manner exemplified on page 124, examples 6 and 7. There is no doubt that the secret of the great rapidity displayed by some accountants lies mainly in the fact of their having familiarised themselves with numerous convenient standards or points of reference extending over the area of ordinary commercial calculations, and especially with “money equivalents," far beyond the limits of the tables learnt at school. By the additional rote-knowledge thus acquired, and always instantly available, such persons are enabled to furnish prompt and correct solutions of all questions not involving very large quantities, without finding it necessary to write down any of the steps of the operations. If, for instance, it were asked, What is tbe cost of 956 articles at 81d. each ? it would at once occur to them, that 956 at 9d., or 8ths of a shilling, must amount to 956s., less one-fourth of itself, that is, to £35 178. Od., from which should be subtracted the half of 956 pence, or £1 19s. 10d., leaving as the answer, £33 178. 2d.

A different mode, leading even more readily to the same result, might be adopted in this particular instance. Let 4 be added to the number of articles,

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