INTEREST AND DISCOUNT. Interest is money paid for the use of money for a given time; the factors are: p, the sum loaned, or the principal: t, the time in years; r, the rate of interest; i, the amount of interest for the given rate and time; Formulæ : If the rate is expressed decimally as a per cent,-thus, 6 per cent = .06,the formulæ becoine Rules for finding Interest.-Multiply the principal by the rate per annum divided by 100, and by the time in years and fractions of a year. principal rate × no. of days If the time is given in days, interest = 365 X 100 In banks interest is sometimes calculated on the basis of 360 days to a year, or 12 months of 30 days each. Short rules for interest at 6 per cent, when 360 days are taken as 1 year: Interest of 100 Dollars for Different Times and Rates. 1 day 386 year .0055 .0083 .01113 .01388 .01663 .02222 .0277 = 3 year .005479 .008219 1 day .010959 .013699 .016438 .0219178 .0273973 Discount is interest deducted for payment of money before it is due. True discount is the difference between the amount of a debt payable at a future date without interest and its present worth. The present worth is that sum which put at interest at the legal rate will amount to the debt when it is due. To find the present worth of an amount due at future date, divide the amount by the amount of $1 placed at interest for the given time. The discount equals the amount minus the present worth. What discount should be allowed on $103 paid six months before it is due, interest being 6 per cent per annum? 103 1+1 x .06 X 1 2 = $100 present worth, discount = 3.00. Bank discount is the amount deducted by a bank as interest on money loaned on promissory notes. It is interest calculated not on the actual sum loaned, but on the gross amount of the note, from which the discount is deducted in advance. It is also calculated on the basis of 360 days in the year, and for 3 (in some banks 4) days more than the time specified in the note. These are called days of grace, and the note is not payable till the last of these days. In some States days of grace have been abolished. What discount will be deducted by a bank in discounting a note for $103 payable 6 months hence? Six months = 182 days, add 3 days grace = 185 103 X 185 days' 6000 $3.176. Compound Interest.-In compound interest the interest is added to the principal at the end of each year, (or shorter period if agreed upon). Let p = the principal, r = the rate expressed decimally, n = no of years, and a the amount: (Value of one dollar at compound interest, compounded yearly, at 12345 1.03 1.04 1.05 1.06 1.1236 17 1.6528 1.9479 2.2920 2.6928 1.7024 2.0258 2.4066 2.8543 1.3382 20 1.8061 1.4185 21 1.8603 2.2787 1.9161 2.3699 2.9252 3.6035 23 1.9736 2 4647 24 40 3.2620 4.8009 7.0398 10.2858 5.8410 8.9847 13.7648 50 4.3838 7.1064 11.4670 18.4204 At compound interest at 3 per cent money will double itself in 231⁄2 years, at 4 per cent in 17% years, at 5 per cent in 14.2 years, and at 6 per cent in 11.9 years. EQUATION OF PAYMENTS. By equation of payments we find the equivalent or average time in which one payment should be made to cancel a number of obligations due at different dates; also the number of days upon which to calculate interest or discount upon a gross sum which is composed of several smaller sums payable at different dates. Rule.-Multiply each item by the time of its maturity in days from a fixed date, taken as a standard, and divide the sum of the products by the sum of the items: the result is the average time in days from the standard date. A owes B $100 due in 30 days, $200 due in 60 days, and $300 due in 90 days. In how many days may the whole be paid in one sum of $600 ? 100 × 30+ 200 × 60 + 300 × 90 = 42,000; 42,000 ÷ 600 = 70 days, ans. A owes B $100, $200, and $300, which amounts are overdue respectively 30, 60, and 90 days. If he now pays the whole amount, $600, how many days' interest should he pay on that sum? Ans. 70 days. PARTIAL PAYMENTS. To compute interest on notes and bonds when partial payments have been made: United States Rule.-Find the amount of the principal to the time of the first payment, and, subtracting the payment from it, find the amount of the remainder as a new principal to the time of the next payment. If the payment is less than the interest, find the amount of the principal to the time when the sum of the payments equals or exceeds the interest due, and subtract the sum of the payments from this amount. Proceed in this manner till the time of settlement. Note. The principles upon which the preceding rule is founded are: 1st. That payments must be applied first to discharge accrued interest, and then the remainder, if any, toward the discharge of the principal. 2d. That only unpaid principal can draw interest. Mercantile Method.-When partial payments are made on short notes or interest accounts, business inen commonly employ the following method: Find the amount of the whole debt to the time of settlement; also find the amount of each payment from the time it was made to the time of settlement. Subtract the amount of payments from the amount of the debt; the remainder will be the balance due. ANNUITIES. An Annuity is a fixed sum of money paid yearly, or at other equal times agreed upon. The values of annuities are calculated by the principles of compound interest. 1. Let i denote interest on $1 for a year, then at the end of a year the amount will be 1+i. At the end of n years it will be (1 + i)n. 2. The sum which in n years will amount to 1 is present value of 1 due in n years. 1 or (1+i)n, or the (1 + i)n (1 + i)n − 1 i 3. The amount of an annuity of 1 in any number of years n is 4. The present value of an annuity of 1 for any number of years n is 1-(1+í)-n i 5. The annuity which 1 will purchase for any number of years n is i 1 − (1 + i) − n' 6. The annuity which would amount to 1 in n years is i (1 + i)n - 1' Amounts, Present Values, etc., at 5% Interest. TABLES FOR CALCULATING SINKING-FUNDS AND PRESENT VALUES. Engineers and others connected with municipal work and industrial enterprises often find it necessary to calculate payments to sinking-funds which will provide a sum of money sufficient to pay off a bond issue or other debt at the end of a given period, or to determine the present value of certain annual charges. The accompanying tables were computed by Mr. John W. Hill, of Cincinnati, Eng'g News, Jan. 25, 1894. Table I (opposite page) shows the annual sum at various rates of interest required to net $1000 in from 2 to 50 years, and Table II shows the present value at various rates of interest of an annual charge of $1000 for from 5 to 50 years, at five-year intervals and for 100 years. Years. Table II.-Capitalization of Annuity of $1000 for 5 4,645 88 4,579.60 4,514.92 4,451.68 4,389.91 4,329.45 4,268.09 4,212.40 10 8,752.17 8,530.13 8,316.45 8,110.74 7,912.67 7,721.73 7,537.54 7,360.19 15 12,381.41 11,937.80 11,517.23 11,118.06 10,739.42 10,379.53 10,037.48 9,712.30 2015,589.215 14,877.27 14,212.12 13,590.21 13,007.88 12,462.13 11,950.26 11,469.96 25 18,424.67 17,413.01 16,481.28 15,621.93 14,828.12 14,093.86 13,413.82 12,783.38 35 40 30 20,930.59 19,600.21 18,391.85 17,291.86 16,288.77 15,372.36 14,533.63 13,764.85 23,145.31 21,487.04 20,000.43 18,664.37 17,460.89 16,374.36 15,390.48 14,488.65 25,103.53 23,114.36 21,354.83 19,792.65 18,401.49 17,159.01 16,044.92 15,046.31 45 26,833.15 24,518.49 22,495.23 20,719.89 19,156.24 17,773.99 16,547.65 15,455.85 50 28,362.48 25,729.58 23,455.21 21,482.08 19,761.93 18,255.86 16,931.97 15,761.87 100 36,614.21 31,598.81 27,655.36 24,504.96 21,949.21 19,847.90 18,095.83 16,612.64 Additional measures of length in occasional use: 1000 mils = 1 inch; 4 inches = 1 hand; 9 inches = 1 span; 2 feet = 1 military pace; 2 yards = 1 fathom; 5% yards, or 161⁄2 feet = 1 rod (formerly also called pole or perch). Old Land Measure.-7.92 inches = 1 link; 100 links, or 66 feet, or 4 rods = 1 chain; 10 chains, or 220 yards = 1 furlong; 8 furlongs = 1 mile; 10 square chains = 1 acre. *The British Admiralty takes the round figure of 6080 ft. which is the length of the "measured mile" used in trials of vessels. The value varies from 6080.26 to 6088.44 ft. according to different measures of the earth's diameter. There is a difference of opinion among writers as to the use of the word "knot" to mean length or a distance--some holding that it should be used only to denote a rate of speed. The length between knots on the log line is 1/120 of a nautical mile, or 50.7 ft., when a half-minute glass is used so that a speed of 10 knots is equal to 10 nautical miles per hour. |