(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 £587 168. 4d. for 7 years, at 4, 5, and 6 per cent. ? d. £. d. 587 16 4 587 16 4 587 16 4 7 7 7 £. 1,12 3.68 Hence, the interest at 4 per cent. is £164 118.91d.+f; at 5 per cent., £205 148. 8.d.+*f.; and at 6 per cent., £246 178. 73.+Hf. 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 far-thing, has here been rejected: if it had been retained, the quotient by 5 would have been increased by ho f.; so that the resulting interests for 4 and 6 per cent. would have been increased, the former by sf., and the latter by Hf.; 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 is 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. accuracy S. £. 8. 2. Find the interest of £619 98. 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. d. d. £. d. 619 9 6 2,0)61,99 6 309 14 9 5 11 30 19 11, int. 5 per cent. 3097 7 6 iul 3 1 114, at i ditto. 34 07 2 3 | 309 14 9 20 34 15, at 5f ditto. 34,07 2 3 1,42 the tenth part of the interest 5,07 :. £34 18. 5d.=interest. 5,07 3. What is the interest of £500 for 4 years, at £5 78. 6d. per cent. ? The work is given below in four different ways. Here, £5 78. 6d. = £5% = £5•375 or = 48£. £500 £500 £500 £500 4 4 4 4 12 2000 53 2000 5•375 2000 43 2000 5 10000 107 50.000 8)860,00 10000 750 20 58.=# 500 £107 108. 2s.6d.=) 250 107,50 10,00 20 107,50 20 10,00 :.£107 108.=interest. 10,00 In the first of these methods we have to take of 2000, *+, 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 or, since 8. 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 2 years, at 24 per cent. ? Here 2 = 1, and 2} = 5 :: 2 x 2 = = 7-3. d. £. d. d. 212 10 4 8) 212 10 4 212 10 4 7 28 26 11 31 1487 12 4 55=11x 5 11 425 0 8 26 11 31 (106 5 2 292 4 2 53 2 7 14,61 1 0 5 20 584 8 5 14,61 10 twice 2}= 5 12,21 20 12 2)2922 2 1 12 21 2,52 12 14,61 1 20 4 12,21 12 2,10 2.52 4 2,10 Hence the interest is £14 128. 2 d., and, therefore, the amount is £227 28. 67d. (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 45 per cent. ? By Rule 1. By Rule 2. £. s. 325 10 365 : 89 :: 14 12 11) 2929 10=1464 15 20 89 8. x 2 11 40 126558 4 112496 834,510(11 73 104 73 31510 12 £. 8. d. 1051 14 12 114 + ff. int. 1 yr. 730 12)857 1f. 378 120(5 365 3218 2,0) 7,1 5d. 43 18 104 int. 3 yrs. 2920 13120 3 11 54 int. 89 days. £ 3 11 54 4 298 47 10 31 whole int. 52,480(1 nearly (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 x quotient=dividend, it is plain, that if, by means of any 3)73000 division-operations upon 73000, we can reduce it 24333 to unity, the same operations upon the dividend 2433 will reduce it to the quotient, that is, to the in- 243 terest for the proposed number of days. The number 73000 is reduced to unity, or, as nearly 1:00009 to unity as is necessary for the degree of accuracy |