As this result is below £5 there is no need of correction, and we have, as before, 15s. 8d. for the answer. In calculations requiring exactness to a farthing, in which the regular process must be followed, it will be found of advantage to double one of the factors-the period or the rate per cent. -and to divide by 730 instead of by 365. The system of decimalising fractions of a pound, as explained on page 93, confers great facility in the working of questions in Interest, but if it is essential to obtain the result correct to a single farthing it will be necessary in most instances to use at least five places of decimals. Interest at 5 per cent. is so readily computed that it is often advisable to solve the question as if that were the given rate, and then to make such abatements or additions as each case may require. 5 per cent. that is, 5 for every 100, or 1 for every 20, of the same kind, whether pounds, shillings, pence, or farthings, is equal to a shilling in the pound, sixpence in 10s., and so on. Accordingly, to find any amount of interest or duty at this rate per cent. it is only requisite to divide the whole pounds by 20, as in the reduction of shillings to pounds. The calculation of 5 per cent. on sums less than one pound may be effected with great ease, even mentally, by the following rule-Disregard fractions of a penny; multiply pence by 05, and shillings by 6, forming a continuous product. The result expresses the amount at 5 per cent. in pence and decimal parts of a penny. Example (6.) What is the interest at 5 per cent. on £1,377 12s. 101d. ? It is evident that as 5d. or 20 farthings produces exactly one farthing at 5 per cent. no fractions of a penny in the given sum need be taken into account, unless it should be necessary to bring out the result to a fraction of a farthing. Then instead of reducing the shillings and pence to total pence before multiplying by 05, it is sufficient to multiply the pence by .05 and the shillings by 6, which is the product of 12, the number of pence in a shilling, multiplied by .05. If this abbreviation were not adopted, the work would stand as under, 8. d. 12 10 12 -- 154 .05 7.70d. Example (7.) Suppose now it were required to apply the preceding rule to the computing of the interest at 31 per cent. for 3 months, on £936 17s. 5d., we should first obtain the amount at 5 per cent. and then correct for the difference between the rates, by the method of aliquot parts, thus, The greater number of practical questions in interest may be treated with advantage in the same manner. Thus, for 4 per cent. deduct 1th of the amount at 5 per cent. For 6 per cent., add 1th. For 41 per cent. deduct 1th and so on. As that which is technically called the amount in questions of interest, is merely the sum of the principal and the interest that has accrued on it within any given time, there can be no necessity to furnish examples of the mode of calculating such amounts. It will generally be found less laborious to compute the interest separately and to add it to the principal, than to use as the multiplier or second term of the proportion, as might otherwise be done, the sum of the rate per cent. and £100, that is, the amount of £100 for one year. In actual business, the question that most commonly occurs, is,-Given the principal, the rate, and the time, to find the interest or the amount; but it must be evident that if any three of the four quantities, viz., the Principal, the Rate, the Time, and the Interest, be taken as data, the remaining quantity may be found by a simple calculation in proportion. The amount, being the sum of the principal and the interest, it follows that if two of these be known the third can at once be determined merely by addition or subtraction. Thus, A few examples of the mode of statement will sufficiently indicate how exercises of this kind are to be wrought. Example (8.) What sum of money lent at 33 per cent., will in 4 years produce £75 interest ? £100 at 33 per cent. will in one year yield £3 15s., and in 4 years £15 interest. We say accordingly, Example (9.) In what time will £500 amount to £575, if put out to interest at 33 per cent. ? The principal and amount being assigned, the interest, £75, is found by subtrac tion; and in order to arrive at the period necessary to produce that interest, the operation plainly is to divide £75 by one year's interest. Example (10.) At what rate per cent. will £500 produce £75 interest in a period of 4 years? As £500 yields £75 interest in 4 years, it will yield £18.75 in one year; and £500 : £100 :: £18.75 £3.75 Answer. 33 per cent. DISCOUNT, is properly the abatement made in consideration of the payment, of money before it is due. If A has to receive from B, £1000 at the end of twelvemonths, and B wishes to discharge the debt at once, the latter would plainly be entitled to some allowance for giving up a year's use of the money. The sum payable under these circumstances by B, is such a sum as would at the current rate of interest amount to £1000 in a year's time; and the difference between such sum or present worth of the debt, and £1,000 represents the true discount on the transaction. Supposing the rate of interest to be 5 per cent. then the question is, what sum at 5 per cent. interest will in one year amount to £1000? (See Example 8 above.) Amt. £105 : Amt. £1000 :: Prin. : Prin. £952 7s. 7&d. It appears, therefore, that the immediate payment of £952 7s. 73d. is equivalent to the payment a year hence of £1000, the rate of interest being 5 per cent. From £1000 take the present value, viz., £952 7s. 7 d., and the difference, £47 12s. 4 d., gives us the discount or allowance that should be made for the premature settlement of the debt. In practice, however, discount is never estimated on this principle, but is used merely as another name for simple interest. The calculation of the discount on a sum of money paid before it is due is understood in all affairs of commerce or the public finances, to be the same as the calculation of the interest on that sum for the given time and at the given rate per cent. So, whenever the word "discount occurs in a General Order, or in instructions issued by the Board of Inland Revenue, the meaning to be attached to it, unless the contrary be specified, is that of simple interest, in accordance with general usage. Thus, if a maltster were called upon to pay £1000 duty, six weeks before it would otherwise be due, and "discount were allowed at the rate of 4 per cent. per annum, the deduction to be made would be the interest on £1000 for six weeks at 4 per cent., and not the true discount or difference between £1000 and the sum which would amount to £1000 at 4 per cent. in the given time. In this case the interest is £4 12s. 01d.; the discount, £4 11s. 73. The maltster, therefore, benefits to the extent of 43d. by the mode of calculation adopted, which gives him in fact not only the true discount, but also the interest upon that discount for the period in question. EXERCISES IN INTEREST. 2 (1.) What is the interest on £2,755 15s. for 3 years 110 days, at 3 per cent. ? Answer. £284 6s. 1d. (2.) What is the amount of £1158 17s. 6d. for 1 year 115 days, at £2 108. per cent.? Answer. £1196 19s. 6d. (3.) If £450 amount to £523 10s. in 1 year 8 months, calculate the rate per cent. Answer. 94d. Answer. 10 years. (4.) In what time will £750 amount to £1125 at 5 per cent. ? Answer. £720. (8.) PER-CENTAGES.-To find how much per cent. one quantity is of another of like kind, is simply to determine a number which shall have to 100 the same ratio which the first of the given quantities bears to the second. Thus, if it were asked, What per cent. £13 is of £52? the operation would be 52: 13 :: 100 : 25. The answer, 25 per cent., means that 25 parts out of 100 are equal to 13 parts out of 52, or that 25 is the per centage, as it is said, of 13 in 52. In every case of per centage, 100 being the fixed term of comparison, the process to be performed is to supply to the denominator 100, such a numerator as shall form with it a fraction equal in value to a given fraction. In the example just adduced, 13 is the given fraction or ratio, and 25 is found to be its equivalent in the denomination required, namely, hundredths. 100 As per centage in all its varieties is merely an application of the Rule of Three, or of the principle of division into proportional parts, it is unnecessary to prescribe any special method for the treatment of questions occurring under this head. It may, however, be useful to observe, that 100 is always either the second or the third term of the proportion, and that the first term or divisor is that quantity of which the per centage is required. Bearing this in mind, a formal statement of the question may generally be dispensed with. In some instances of gain or loss, increase or decrease, per cent., it is requisite before making the calculation, or as a final step, to deduct a rate or price from a total given value; but the proper manner of using the data on each occasion will readily suggest itself without the assistance of rules adapted to the different cases. The advantage of expressing results generally in terms of per centage, instead of stating particular instances and quantities, is so obvious as to need little illustration. In almost every branch of science, commerce, and finance, the great utility of the system, and the ease with which the necessary calculations can be made, have caused it to be extensively adopted. A man who has doubled his capital by trading, is said to have realised a profit of "100 per cent.," that is, 100-100ths of the whole. Again, 1, 1, 1, 1, 2, of anything, are commonly spoken of as 25, 331, 50, 20, 75, 121 per cent. respectively, and so on.* Persons in business find it much more explicit and convenient to reduce the result of their dealings to per centage equivalents, than to specify the amount actually gained or lost on each separate transaction. For example, the declaration that "an investment yielded 35 per cent. profit," conveys a far clearer idea of the proportion of the gain to the outlay, than would be derived from hearing it announced that «an investment of £1365 yielded a profit of £447 15s. Od.,” a fact which might have supplied the data of the per centage in question. Similarly, a better judgment is formed as to the relative loss sustained on a quantity of spirits, from the statement that «5 per cent. was wasted by leakage," than from the more particular account that "in a cask originally containing 84 gallons, 4-2 gallons had been found deficient." Indeed, on every occasion of loss or gain, increase or decrease, we instinctively seek to assist our estimate of the proportional magnitude of the event, by reducing the given numbers, either roughly or exactly, to an equivalent per centage. By this method we generalize the records of results, throwing aside the details as no longer of 100 is, for most practical purposes, the best number that could be taken as a standard in these calculations, since the adoption of a smaller number, such as 1 or 10, would involve the frequent use of fractions, if any approach to accuracy were required; while a higher number, as 1000, would make it necessary to quote hundreds of value, when tens might ordinarily suffice. importance, and, at the same time fixing a convenient basis for future calculations. In the various revenue departments, the facility conferred by the use of per centage amounts, makes itself highly conspicuous. So much is this the case as regards the Excise Service, that the arithmetical attainments of an officer seeking promotion are always tested with special reference to his ability to compute per centages. A few examples will show the proper mode of treating every question that can be proposed. Example (1.) How much per cent. is £9 18s. 9d. of £16 10s. 4d. ? £16 10s. 4d. 3964d. £9 188. 9d. =2385d. 3964 : 2385 :: 100 : 60.17. Answer. 60.17 per cent. Or, by decimalizing the fractions of a pound. (See page 93.) 16.516 : 9.937 :: 100 : 60.17. Example (2.) If 811 bushels of dry barley increase during steeping to 100 bushels, what is the increase per cent.? Or, first deducting the given quantity of dry corn from what it measures after steeping, to obtain the amount of swell, we may say 81.5 : 100 :: 18.5 : 22.7. Example (3.) If barley increase 22.7 per cent. by steeping, and the steeped corn increases 63 per cent. by subsequent vegetation, what is the total per centage swell of the dry barley? By the terms of the question, 100 bushels of dry barley measure 122.7 after steeping, and 100 bushels of steeped barley grow to 163 by germination. Therefore, to find the proportional increase of 122.7 steeped bushels; we say Increase. Steeped Bush. Steeped Bush. Increase. 100. Answer. 100 per cent. That is, the dry barley, on this supposition, doubles its bulk in the process of malting. Example (4.) How much per cent. is deducted in reducing "floor charges to net, each bushel of a floor charge being equal to half a bushel net ? = 50 Answer. 50 per cent. To divide by 2 is to deduct Example (5.) How much per cent. is deducted in reducing «couch bushels" to net, each couch bushel being charged 815 bushel net? If from 1 couch bushel, 185 bushel (= 1 — •815) be deducted, then from 100 couch bushels, the deduction will be 100 x 185 18.5 bushels. Answer. 18.5 per cent. Example (6.) The first gauge of a steeping of corn amounted to 95.8 bushels; the last gauge to 117.5 bushels. Required the increase per cent. ? Given increase, 117.5 95.8 - 21-7 bushels. Then, 95.8: 100 :: 21.7 22.65. : Answer. 22.65 per cent. Example (7.) The depth of grain in a couch frame is at first found to be 11.8 inches. After throwing out and returning the grain, an increase of one inch in the depth is obtained. By how much per cent. does this result exceed or fall short of the legal allowance of 7 per cent. ? And 8.477 1.47. Answer 1.47 per cent. illegal excess. |