1. How much must I abate of £5394 10s. due 3 years hence, at 23 per cent. per annum? £5394 10s. 23 27+3-9)10789 0 8, therefore, X2, and÷27 3)1198 15 6 £399 11 10 Ans. 2. What is the discount of $546 62c. 5m. for 8 years, at 1 per cent. per annum, (or for one year, at 8 per cent. per annum ?) $ c. m. 13)546-62 5 Ans. $42.04 8 3. What is the discount of $125 at 1 per cent. per annum, for four years, (or, at 4 per cent. per annum, for 14 year?) Ans. $5 95c. 2m. * These contractions are obvious from any example, wrought according to the General Rule. Thus let the sum to be discounted be 300 dollars. 3. At 21 per cent. Then, 1025 25 300: discount, or 108 8:300: discount, or, 272: 300: Answer required. In the same way, may all the contractions be made. The contractions are made on the two terms of the proportion which are invariable, when the rate is DISCOUNT BY DECIMALS. The sum to be discounted, the time and the ratio given, to find the present worth. RULE. Multiply the ratio by the time, add unity to the product for a divisor by which sum divide the sum to be discounted, and the quotient will be the present worth.* Subtract the present worth from the principal, or sum to be discounted, and the remainder will be the discount. Or, as the amount of £1 for the given time, is to £1, so is the interest of the debt for the said time, to the discount required. Subtract the discount from the principal, and the remainder will be the present worth. EXAMPLES. 1. What is the present worth of 6007. due 3 years hence, at 67. per cent. per annum? ·06×3+1 Product='18 Divisor 1.18)600(508 4745 present worth. -=£508 9s. 5‡d. Ans. Present worth=508-4745=£508 9s. 53d. which, subtracted from the principal, will give the discount=£91 10s. 64d. 600 *As the sum to be discounted is, in fact, the amount of some principal at the given rate and time; to find the principal, which is now the present worth, you have only to employ the rule for Case 2, in Simple Interest by decimals. This is the rule in the text. Thus in Ex. 1, by said Case 2,-=the principal, in this case the present worth. The remainder of the rule is evident from what has been said under Discount, Rule 1. 1+06X3 Note 1. In the method used in Simple Interest by Decimals, you may easily find rules for obtaining either of the four terms, present worth, ratio, time, or sum to be discounted, when the other three are given. Note 2. When the ratio is 06, or six per cent. per annum, and the given time is expressed in months, if the debt be divided by unity added to half as many hundredths of an unit as there are months in the given time, the quotient will be the present worth. Thus for 3 years or 36 months, the divisor, we have just seen to be 1+06x3=1+18, or I added to half as many hundredths as there are months; if the time be 3 years or 42 months, the divisor is 1+06×3.5,= 1+21; if 10 months, then 1+06×1+05, as before; if 2 months, then 1+06×2,=1+01; if 1 month, then 1+06x=1+005, and so on. Second Method. Ratio='06 Multiply by 3 As 1·18: 1 :: 108 .18 Add 1. Amount of 17. for the given time } =1.18 1 1.18)108-00(91.5254 And 600×06×3=108=interest of the debt for the given time.Discount=91-5254=£91 10s. 6d. which taken from the principal will leave the present worth-£508 9s. 6d. 2. What is the present worth of $558 62c. 5m. due 2 years hence, at 4 per cent. per annum? And $558-625-$512.5=846 12c. 5m.-discount. Or, As $1·09 (=amount to $1 for the given time): $1 :: $50-27625 (= interest of the debt for the given time): $46-125-discount as above. And, $558 625-$46 125-$512 5-present worth, as above. 3. Required the present worth and discount of $4125 50c. at 65 per cent. per annum, due 18 months hence? Ans. S present worth $3746 19c. 7m. 4. What ready money will discharge a debt of 1354l. 8s. due 8 years, 3 months, and 12 days hence, at 5gl. per cent. per annum? Ans. £1135 7s. 9d. DUTIES. DUTIES are assessed upon articles imported into the country, at a certain rate per pound, hundred, ton, gallon, &c. without respect to the value of the articles; or upon articles according to their actual cost. The latter are called ad valorem duties. The duties are computed in the former case, on the most obvious principles, as will be seen in the following EXAMPLES. 1. Required the duty on 987 of chocolate at 3 cents per pound. 3 $29.61 cents, Ans. 2. If the duty on molasses is 5 cents on a gallon when imported in an American vessel, and 10 per cent. more in a foreign vessel, what is the duty on 3950 gallons in both vessels? Ans. $197.50, and $217.25. 3. Required the duty on 10Cwt. 3qrs. 14 of cordage at $2.25 per Cwt. in an American vessel, and at 10 per cent. more in a foreign vessel ? Ans. $24-47 nearly, and $26.81. 4. What is the duty on 6hhds. of brown sugar, weighing 53Cwt. 2qrs. 20 tare 12 per 100, at 24cts. per pound in an American vessel, and at 10 per cent. more in a foreign vessel ? Ans $132, and $145.20. Duties ad valorem are estimated by adding 20 per cent, to the actual cost of the goods, &c. when imported from or beyond the Cape of Good Hope, and 10 per cent. when imported from any other places. Insurance, commission, &c. do not belong to the actual cost. The duties are computed in the following obvious manner. When the cost is reduced to Federal Money, add the per cent. to the cost, and then find the duty per cent. ad valorem. EXAMPLES. 1. What will be the duty on an invoice of goods, which cost £786 15s. sterling, at 15 per cent. ad valorem when imported in an American vessel, or at 10 per cent. more when imported in a foreign vessel from England? £ S. 786 15-3493.17 Add 10 per cent.= 349-317 3842-487 10 per cent. 384.2487 5 do. =192.12435 15 per cent.=$576-37-305 Ans. in Am. vessel. 2. What is the duty on goods, which cost in India, $2780-50, imported in an American ship, at 12 per cent. ad valorem? Ans. $417-075. BARTER Is the exchanging of one commodity for another, and teaches traders to proportion their quantities without loss.* CASE I. When the quantity of one commodity is given, with its value, or that of its integer, that is, of 1lb. 1cwt. lyd. &c. as also the value of the integer of some other commodity, to be given for it, to find the quantity of this; or, having the quantity thereof given, to find the rate of selling it. RULE. Find the value of the given quantity by the concisest method, then find what quantity of the other, at the rate proposed, you may have for the same money. Or, if the quantity be given, find from thence the rate of selling it. Or, As the quantity of one article is to its price, so, inversely, is the quantity of the other to its price. Or, as the price of one article is to its quantity; so inversely, is the price of the other to its quantity. EXAMPLES. 1. How much tea at 9s. 6d. per must be given in barter for 156 gallons of wine, at 12s. 33d. per gallon? Galls. *The Rules in Barter are only applications of the Rule of Three, and are easily understood. |