(104.) INTEREST, DISCOUNT, INSURANCE, &c. INTEREST is the sum paid for the use of money by the borrower, or him who holds it, to the lender, or him who deposits it. The consideration agreed upon is usually at so much for the use of every £100 for a year, or, as men of business call it, at so much per cent. per annum; or, simply, at so much per cent., per annum, or per year, being understood. The money lent or deposited is called the Principal; the consideration for £100 for a year, the Rate of Interest; and the principal, together with the interest, for any length of time, is called the Amount in that time: thus, if £100 be lent under an agreement that £5 is to be paid for the use of it for one year, then, £100 is the principal lent; £5 is the rate per cent.; £10 is the interest for 2 years; and £110 is the amount in that time. The finding of the interest of a given sum of money at a given rate per cent. for 1 year, is obviously nothing more than a simple Rule-of-Three operation: for, as £100 is to the rate, so is the principal lent to the interest upon it; or, as £100 to the sum lent, so is the interest of £100 to the interest of the sum. And if the interest, thus determined for 1 year, be multiplied by any number of years, the product will be the interest accumulated in that time; or, instead of multiplying the interest for 1 year by the number of years, we may commence by multiplying the principal by that number, and make the stating afterwards. A formal Ruleof-Three stating is, however, generally dispensed with, and the operation conducted as follows. (105.) To find the Interest of a given Sum at a given Rate per Cent., for a given Number of Years. RULE. Multiply the principal by the number of years, the product by the rate, and divide the result by 100. NOTE. When the rate is 5 per cent., —a rate very commonly charged, the operation, simple as it is, becomes still simpler for the multiplication by 5, and the subsequent division by 100, may be replaced by a single division by 20; so that having multiplied the principal by the number of years, the 20th part of the product will give the interest. Ex. 1. What is the interest of £587 16s. 4d. for 7 years, Hence, the interest at 4 per cent. is £164 11s. 91d. +25ƒ. ; at 5 per cent., £205 14s. 84d.+f.; and at 6 per cent., £246 178. 73d.+f. The additional work in the middle column, shows how the interest at 4 and 6 per cent. may be obtained from that at 5 per cent., namely, by subtracting a fifth part of the latter interest, for the 4 per cent., and adding that fifth part for the 6 per cent. The fraction, of a farthing, has here been rejected: if it had been retained, the quotient by 5 would have been increased by f.; so that the resulting interests for 4 and 6 per cent. would have been increased, the former by f., and the latter by 17f.; and there would have been an exact agreement with the determinations of the other method. I have been thus particular, more for the purpose of showing you the strict conformity between the results of different processes than because such minute accuracy is necessary in actual practice. Fractions of a farthing are, of course, always disregarded in business; and in calculations respecting interest, pence is in general the lowest denomination noticed. See art. 107. 2. Find the interest of £619 9s. 6d. for 1 year, at 5 per cent. Either of the following three ways may be employed: in the last of these, the principal is halved, and the rate doubled. 3. What is the interest of £500 for 4 years, at £5 7s. 6d. per cent.? The work is given below in four different ways. Here, £5 7s. 6d. = £5% = £5·375 or = 43 £. 12 5,07 :. £34 1s. 5d.=interest. = .£107 10s.=interest. In the first of these methods we have to take of 2000, or, since +, we may take of 2000, which is the 500 above, and then add the half of that fourth, namely, 250, which gives 750. Instead of multiplying by 4 and by 5 in the first and third methods, we may, of course, multiply by 20 at once. The product of the numbers expressing the years and the rate may always be used instead of those numbers separately, whenever it is thought convenient to do so, as in the next example. 4. What is the interest and amount of £212 10s. 4d. for 23 years, at 2 Here 24 = £. S. d. 11 per cent.? Hence the interest is £14 12s. 2 d., and, therefore, the amount is £227 2s. 6d. (106.) When the Interest for any Number of Days is required. RULE 1. As 365 is to the number of days, so is the interest for one year to the interest required; or, since twice 365 is 730, and since, moreover, in finding the interest for 1 year, we always have to divide by 100, after multiplying by the rate, we may proceed as follows: RULE 2. Multiply twice the product of the principal and rate by the number of days, and divide by 73000. Ex. 1. What is the interest of £325 10s. for 3 years and 89 days at 4 per cent.? By Rule 2. £. 8. £. 8. 36589 14 12 11 2929 10-1464 15 52,480(1 nearly. 47 10 3 whole int. (107.) In this example, as well as in the examples which have preceded, I have been unnecessarily exact in computing the interest to the nearest farthing: but the odd farthings are disregarded in business-transactions, and interest calculations are considered sufficiently accurate when the true results are reached to the nearest penny. With this understanding, the operation by Rule 2 may be shortened. In dividing by so large a number as 73000, the odd shillings in the dividend may be disregarded; or, if above 10s., the pounds may be increased by a unit. Now, since 73000 × quotient=dividend, it is plain, that if, by means of any division-operations upon 73000, we can reduce it to unity, the same operations upon the dividend will reduce it to the quotient, that is, to the interest for the proposed number of days. The number 73000 is reduced to unity, or, as nearly to unity as is necessary for the degree of accuracy 3)73000 24333 2433 243 1.00009 |