IO 9x8 In the same way it could be shown that 25 days, when the rate is 3 per cent., are equivalent to 25 x6 days when the rate is 5 per cent.; that 9 days, when the rate is 4 per cent., are equivalent to days when the rate is 5 per cent.; and that 13X7 13 days, when the rate is 33 per cent., are equivalent to days when the rate is 5 per cent. So that 2 per cent. for II days, 11 per cent. for 17 days, 2} per cent. for 8 days, 3 per cent. for 25 days, 4 per cent. for 9 days, and 33 per cent. for 13 days—all taken together—are equivalent to 5 per cent. for ib of (44+51+40+150+72+91=) 448 days; and we take 1 of 448 by cutting off the last figure. IO IO PXDX2R=1; ; R=73,000 XI Principal, Interest, Rate, Time: Any one of these can be found, when the remaining three are given. For, putting P for the Principal, I for the Interest, R for the Rate, and D for the Time-expressed in days, we have ($179) 73,000 [PXDX2R=73,000 x I;] P=73,000 x I. =x ] = and DX2R PXD X2 D_73,000 XI PX2R 180. To find the Principal, when the Interest, Rate, and Time are given: Set down, for dividend, the product of the Interest by 73,000; and, for divisor, the product of the number of Days by twice the Rate. The resulting quotient will be the answer. EXAMPLE IV. Find the principal which, at 3 per cent., would produce 19s. 70. in 124 days. Writing 19. 7d. as £.979, and multiplying by 73,000, we obtain, for dividend, (73,000 X '979=) 71467. For divisor, we take the product of 124 by the double of 3—that is, (124*6=) 744. Then, dividing 71467 by 744, we find the required principal to be (71467-744=) £96 1s. 2d., nearly. 181. To find the Rate, when the Principal, Interest, and Time are given: Set down, for dividend, the product of the Interest by 73,000; and, for divisor, twice the product of the Principal hy the number of Days. The resulting quotient will be the answer. EXAMPLE V.-At what rate would £125 produce £ 1 10s. in 219 days. Writing £1 ios. as £1.5, we set down, for dividend, 73,000 X15=109500. For divisor, we take twice the product of 125 by 219—that is, (125X219X2=) 54750. Then, dividing 109500 by 547 50, we find the required rate to be (109500; 547 503) 2 per cent. 182. To find the Time, when the Principal, Interest, and Rate are given: Set down, for dividend, the product of the Interest by 73,000 ; and, for divisor, the product of the Principal by twice the Rate. The resulting quotient will be the answer in days. EXAMPLE VI.-In what time would £750 produce £40 11S. rid., at 41 per cent. ? Converting £40 IIS. 11d. into £40.596, we have, for dividend, 73,000 X 40.596=2963508. For divisor, we write twice the product of 750 by 43 —that is, (750 x 41 x 23) 67 50. Then, dividing 2963508 by 6750, we find the required time to be (2963508 - 67 50=) 439 days, or 1 year and 74 days. Note.—Principal, Rate, Time: It will be observed that, in finding any one of these three, we simply divide twice the product of the remaining two into the product of the Interest by 73,000. 183. Todetermine the Principal, when the Amount, Rate, and Time are known: Say—as the amount of £100 for the stated time, and at the stated Rate, is to 2100, so is the given Amount to the required Principal. ExamPLE VII.—What principal would amount, in 7 years, at 5 per cent., to £1320 68. ? At the given rate, £ 100 would produce £5 in one year, and (£5X7=) £35 in seven years; so that, for the given time, and at the given rate, the "amount” of £100 would be (£100+ £35=) £135. The question now assumes this form: If £100, principal, would become £135, amount, under certain circumstances, what principal would become £1320 68., amount, under the same £100- £135 circumstances? The proportion is -£1320 6s. £135: £1320 6s. : : £100 : a; or, by Alternation, £135 : £100 :: £1320 6s. : a P. A. ? а If, instead of £100, we take any other sum--say £30, we shall obtain the same result, but at the expense of a little more trouble. At 5 per cent., £30 would produce 30s., or £i ios. in one year, and (£ i jos. X7=) £10 10s. in seven years; so that the amount of £30 for the given time, and at the given rate, would be (£30+£10 1os.=) £40 10s. We therefore say—as £,40 1os., amount, is to £30, the principal which would produce it, so is £1320 6s., another amount, to the principal which would produce it under the same circumstances: £40-5: £30 : : £1320'3: a DISCOUNT. a 184. When a debt is paid before the time originally agreed upon, an abatement is usually made in the amount. This abatement is termed DISCOUNT; and the sum accepted in discharge of the debt is known as the present-worth of the debt. So that the full amount exceeds its present-worth by the discount. EXAMPLE I.-A person holds a claim for £420, payable a year hence; what sum paid now would satisfy the claim, the rate of interest being 5 per cent. ? It is evident that the true present-worth of this claim is the sum which, if invested—as principal—at 5 per cent., would amount to £420 in a year. Now, at the given rate, £ 100 would amount to £105 in a year: consequently, the principal which, at the same rate, and in the same time, would amount to £420, is the fourth term of the proportion £105: £100 :: £420 : a a=(420 X 100:-105= £400. The true discount, therefore is £20—the difference between £420 and £400. In the commercial world, however, discount is never calculated in this way; the amount of a debt being invariably treated (not as an "amount,” which it really is, but) as a principal, and the discounter charging interest, not merely : upon what he advances as present-worth, but also upon a sum which he does not advance—that is, upon the discount itself. In the case under consideration, the sum charged as discount would be £21—the interest of £420 for a year, at 5 per cent. ; so that the present-worth would be (£420-£21=) £399.* 185. Rule for the calculation of Discount: Find what interest the amount of the debt, if invested as principal, would produce in the given time and at the given rate; this interest will be the Discount required. Bills of Exchange.—A bill of exchange may be described as a written engagement, by a debtor, in obedience to the order of a creditor, to pay a sum of money at a future time. The following is the usual form of a bill: [ Stamp. ] £500 0 0 Sackville-street, Dublin, 21st April, 1870. Three months after date, pays to me or my order the sum of five hundred pounds sterling-for value received. James Browne. To Mr. John Jones, Patrick-street, Cork. Jones would accept this bill by simply writing his name across the “ face” (not the back) of it; and the money would be presumed to be payable at his house in Cork unless he-on the face of the bill—made a notification to the contrary, such as“ Payable at the National Bank.” In relation to this bill, Browne (the creditor) and Jones (the debtor) would be spoken of as the DRAWER and the ACCEPTOR, respectively. When a bill is accepted, and returned to the drawer, what usually occurs is this : (a) The drawer "holds” the bill until it comes to “maturity”—that is, until the day on which payment a * It will be seen that £399 is the true present-worth—not of £,420, but—of (£399+£19 198.=) £418 198. ; the interest of £399 for a year, at 5 per cent., being £19 198. + The acceptor is sometimes called the drawee. runs is to be demanded; (b) or he at once converts bis claim into ready-money, by getting the bill discounted; (c) or he“ passes the bill, as an equivalent for cash, to a creditor of his own this creditor, in his turn, being then at liberty to adopt, with respect to the bill, any one of the three courses just mentioned. Before, however, parting with a bill which has not arrived at maturity (in other words, before negotiating a bill), the drawer -as well as everybody else into whose hands it comes after leaving the drawer's-has to indorse it; that is, write his name on the back of it. So many as ten, fifteen, and even twenty names-representing an equal number of indorsers-are frequently met with on the back of a bill : a fact which enables us to understand the very important part which bills play in mercantile transactions. In the United Kingdom, a bill" " three days--called DAYS OF GRACE—in addition to the time mentioned on the face of it. So that if a three months' bill were drawn on the ist of July, the amount would not be legally due until the third day after the ist of October—that is, until the 4th of October. If, however, the last of the days of grace happened to be a Sunday, the amount would be due on the preceding (Satur-) day. Moreover, if a four months' bill, say, were drawn on the 31st of May, the days of grace would be reckoned (not from the 31st of September, there being no such day, but) from the 30th of September; so that the amount would be due on the 3rd of October. Again, if a two months' bill were drawn on the 31st, 30th, or 29th of December, the two months would expire on the last day (28th or 29th, as the case may be) of February ; so that the amount would be due on the 3rd of March. EXAMPLE II.-A three months' bill for £240 was drawn on the 24th of June, and discounted on the 13th of July, at 6 per cent. per annum ; find the discount. The three months having expired on the 24th of September, the bill arrived at dys. inaturity three days July, 18 £240 x 76 x 12=73000= later—that is, on the Aug., 31 2218880-73000= £2.998 = 27th of September; Sept., 27 £2 19 111: so thatwhen dis 218880 counted (on the 13th 76 72960 of July), the bill bad 7296 76 days to 729 namely, the last 18 days of July, the 2'99865 whole 31 days of August, and the first 27 days of September. The required . run |